Embed Size (px)
Citation preview
DURHAM UNIVERSITYDepartment of Mathematical Sciences
Third- & Fourth-Year Mathematics2008-2009
Science LaboratoriesSouth RoadDurham Email: [email protected] 3LE Web: www.maths.dur.ac.uk
Contents
1 General Information 5
1.1 The Department . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.2 Useful Contacts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.3 Course Director . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.4 Natural Sciences Co-ordinator . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.5 Departmental Adviser . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.6 Registration for 3H . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.7 Consultation with Members of Staff . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.8 Staff-Student Consultation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.9 Students with Special Needs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.10 Illness and Absence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.11 Course Information . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.12 Computers, ICT and DUO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.13 Private Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.14 Books and Libraries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.15 Student Records . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.16 Smoking and Mobile Phones . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.17 Durham University Mathematical Society . . . . . . . . . . . . . . . . . . . . . . 11
1.18 Disclaimer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2 Examinations and Assessment 12
2.1 Regulations for B.Sc., M.Math. and M.Sci . . . . . . . . . . . . . . . . . . . . . . 12
2.2 University Assessment Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.3 Board of Examiners . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1
2.4 Plagiarism, Cheating and Collusion . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.5 Monitoring of Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.6 Submission of Formative Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.7 Submission of Summative-Assessed Work . . . . . . . . . . . . . . . . . . . . . . 14
2.8 Calculators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.9 Examinations and Progression . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.10 Illness and Examinations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
3 Learning and Teaching 16
3.1 The Mathematics Modules and Degrees . . . . . . . . . . . . . . . . . . . . . . . 16
3.2 Honours Natural Sciences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
3.3 Further Remarks on Degrees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
3.4 Booklists and Descriptions of Courses . . . . . . . . . . . . . . . . . . . . . . . . 18
3.4.1 Algebraic Geometry III and IV – MATH3321/MATH4011 . . . . . . . . . 20
3.4.2 Analysis III and IV – MATH3011/4201 . . . . . . . . . . . . . . . . . . . 22
3.4.3 Approximation Theory & Solutions of ODEs – MATH3081/MATH4221 . 24
3.4.4 Bayesian Methods III and IV – MATH3311/MATH4191 . . . . . . . . . . 26
3.4.5 Bayesian Statistics III and IV – MATH3341/MATH4031 . . . . . . . . . . 28
3.4.6 Communicating Mathematics III – MATH3131 . . . . . . . . . . . . . . . 30
3.4.7 Continuum Mechanics III and IV – MATH3101/MATH4081 . . . . . . . . 32
3.4.8 Decision Theory III – MATH3071 . . . . . . . . . . . . . . . . . . . . . . 34
3.4.9 Differential Geometry III – MATH3021 . . . . . . . . . . . . . . . . . . . 36
3.4.10 Dynamical Systems III – MATH3091 . . . . . . . . . . . . . . . . . . . . 38
3.4.11 Electromagnetism III – MATH3181 . . . . . . . . . . . . . . . . . . . . . 40
3.4.12 Elliptic Functions III and IV – MATH3221/MATH4151 . . . . . . . . . . 42
3.4.13 Galois Theory III – MATH3041 . . . . . . . . . . . . . . . . . . . . . . . 44
2
3.4.14 General Relativity III and IV – MATH3331/MATH4051 . . . . . . . . . . 46
3.4.15 Geometry III and IV – MATH3201/MATH4141 . . . . . . . . . . . . . . . 48
3.4.16 Independent Study III – MATH3161 . . . . . . . . . . . . . . . . . . . . . 50
3.4.17 Mathematical Biology III – MATH3171 . . . . . . . . . . . . . . . . . . . 52
3.4.18 Mathematical Finance III and IV – MATH3301/MATH4181 . . . . . . . . 54
3.4.19 Mathematics Teaching III – MATH3121 . . . . . . . . . . . . . . . . . . . 56
3.4.20 Number Theory III and IV – MATH3031/MATH4211 . . . . . . . . . . . 58
3.4.21 Operations research III – MATH3141 . . . . . . . . . . . . . . . . . . . . 60
3.4.22 Partial Differential Equations III and IV – MATH3291/MATH4041 . . . . 62
3.4.23 Probability III and IV – MATH3211/MATH4131 . . . . . . . . . . . . . . 64
3.4.24 Quantum Mechanics III – MATH3111 . . . . . . . . . . . . . . . . . . . . 66
3.4.25 Representation Theory & Modules III and IV – MATH3191/MATH4101 . 68
3.4.26 Solitons III and IV – MATH3231/MATH4121 . . . . . . . . . . . . . . . . 70
3.4.27 Statistical Mechanics III and IV – MATH3351/MATH4231 . . . . . . . . . 72
3.4.28 Statistical Methods III – MATH3051 . . . . . . . . . . . . . . . . . . . . 74
3.4.29 Stochastic Processes III and IV – MATH3251/MATH4091 . . . . . . . . . 76
3.4.30 Topology III – MATH3281 . . . . . . . . . . . . . . . . . . . . . . . . . . 78
3.4.31 Advanced Quantum Theory IV – MATH4061 . . . . . . . . . . . . . . . . 80
3.4.32 Algebraic Topology IV – MATH4161 . . . . . . . . . . . . . . . . . . . . 82
3.4.33 Project IV – MATH4072 . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
3.4.34 Riemannian Geometry IV – MATH4171 . . . . . . . . . . . . . . . . . . . 86
A Details of Modules and Programmes 88
3
4
1 General InformationWelcome to 3H or 4H Mathematics. This booklet has been written for third- and fourth-yearstudents registered for Single Honours degrees in Mathematics, but also for third-year NaturalSciences, Combined Arts and Social Sciences students, and third- and fourth-year Natural Sciencesstudents who are registered for the B. Sc. or M. Sci. Joint Honours Mathematics and Physicsdegrees, the M.Sci. Joint Honours Chemistry and Mathematics degree or the M.Sci. in NaturalSciences with Mathematics as a subject.
Whether you are entering the final year of your course, or the third year of a four year course, thereis a good choice of options and you can study those topics which you find most interesting. Youcan look forward to an enjoyable year.
This booklet contains information specific to your programme of study within the Department. Forinformation concerning general University regulations, examination procedures etc., you shouldconsult the Faculty Handbooks (www.dur.ac.uk/faculty.handbook) and the University Calen-dar, which provide the definitive versions of University policy. The web address of the Learningand Teaching Handbook, which contains information about assessment procedures amongst otherthings, is www.dur.ac.uk/learningandteaching.handbook/
You may find it useful to keep this booklet for future reference. For instance, prospective employersmight find it of interest.
An on-line version of this booklet may be found by following the link ‘Teaching > 3H Mathe-matics’ from the Department’s home page (www.maths.dur.ac.uk). If you intend to take mathsmodules at Level 2 this year, do consult the relevant booklet on-line.
1.1 The DepartmentBesides undergraduate teaching, the Department has a second main function — research. Just likeyou, lecturers and tutors are building on their existing expertise and trying to solve mathematicalproblems. Together with various administrative tasks, it is their main occupation outside the class-room. One difference, however, is that the problems you are asked to tackle should actually besolvable! The important thing is that Mathematics in Durham is living, developing and growing —and you are joining in.
1.2 Useful ContactsThe first point of contact for issues referring to a particular course or module should be the relevantlecturer. For more general questions or difficulties you are welcome to consult the Course Directoror the Head of Department. For issues involving University registration for mathematics modules,please see the Registration Co-ordinator.
Head of Department:Prof. M. Goldstein, (room CM207), Email: [email protected] Director:Prof M Menshikov, (room CM231c), Email: [email protected] Co-ordinator:Dr. S. Borgan, (room CM216), Email: [email protected] of Support Teaching:Dr. S. Borgan, (room CM216), Email: [email protected]
5
For each Joint Honours degree there is a designated member of staff from each participating de-partment whom you may contact if you wish to discuss any aspect of your joint degree course. Therelevant contacts in the Department are as follows:
Joint degrees with Physics:Prof. P. R. W. Mansfield, (room CM210), Email: [email protected] degree with Chemistry:Dr D. J. Smith, (room CM231a), Email: [email protected]
We may also wish to contact you! Please keep the Mathematics Office informed of your currentterm-time residential address and e-mail address.
1.3 Course Director
Prof M Menshikov, (room CM231c), Email: [email protected] Third- and Fourth-Year Course Director is responsible for overseeing the academic progressof third and fourth year students. If at any time you would like to discuss aspects of your course, orif there are questions about the Department which this booklet leaves unanswered, please contacthim.
1.4 Natural Sciences Co-ordinator
If you are a Natural Sciences student, and wish to discuss specific aspects of your programme,you may contact the liaison officer with Natural Sciences, Dr. J. Bolton, (room CM314), Email:[email protected]
1.5 Departmental Adviser
Each Single Honours Mathematics, Natural Sciences, Combined Arts and Social Sciences studentwho takes mathematics modules has a designated Departmental adviser who is a member of staffwho acts as an academic presence throughout the period of study of each advisee. The advisercontacts advisees at the beginning of each academic year to remind them of the opportunity theyhave to come and discuss any aspect of their academic life. In many cases, the adviser will be theperson to contact if you need a reference letter, you will meet your adviser at least once during theyear, when you discuss your exam results and register (if not final year) for the subsequent year.
If you need to contact your advisor and he/she is not available, please contact Dr. S. Borgan, (roomCM216), Email: [email protected]
1.6 Registration for 3H
You will register for the required number of modules in June. You may attend additional mod-ules during the first few weeks of the Michaelmas Term. If you then decide that you wantto change one or more of your modules you must see Dr. S. Borgan, (room CM216), Email:[email protected] Any such change must be completed during the first four weeks ofthe Michaelmas Term.
6
1.7 Consultation with Members of Staff
If you have any questions on the subject or are experiencing difficulties with a lecture course, youshould consult the member of staff giving the course as soon as possible. The consultation may be:
- immediately after the lecture,- by calling on the lecturer concerned in his or her office,- by email.
1.8 Staff-Student Consultation
The Board of Studies has two Staff-Student Consultative Committees ((A) for Honours Mathemat-ics and Natural Sciences and (B) for Auxilliary level 1 modules) to provide an effective means ofcommunication between staff and students.
The Committees first meet in the Michaelmas Term. They have student representatives from eachyear (each module for Committee (B) and if you have issues you wish to raise, you should contactyour year (or module) representatives.
The Staff-Student Consultative Committees also seek feedback from all students on all aspectsof Mathematics courses by way of a questionnaire during the penulimate week of Michaelmasand Epiphany terms. These are considered by the lecturers concerned, the Head of Departmentand the members of the Department’s Monitoring Committee. The Staff-Student ConsultativeCommittees report to the Board of Studies, the main decision-making body of the Department, andplay an active role in promoting high quality teaching in the Department. There are also studentrepresentatives to the Board of Studies, who act in an advisory capacity and also provide directfeedback to the student body.
Summary results of the questionnaires and minutes of the Staff-Student Committees are posted onthe relevant noticeboards (first floor corridor of the Department).
If you have concerns about teaching which are not covered by these meetings and questionnaires,contact can be made directly with the Staff-Student Consultative Committee Chairmen:Dr J. F. Blowey, (room CM325), Email: [email protected] for Committee (A).Dr. S. Borgan, (room CM216), Email: [email protected] for Committee (B).
1.9 Students with Special NeedsThe University is committed to full compliance with the aims of the Special Educational Needs andDisability Act 2001. Once a student has been accepted for a course of study, the University acceptsa responsibility to ensure appropriate provision for that student throughout his/her course. Studentswith disabilities can expect to be integrated into the normal University environment. They areencouraged and helped to be responsible for their own learning and so achieve their full academicpotential.
Durham University Service for Students with Disabilities (DUSSD) aims to provide appropriatecare and support for all Durham students with a disability, dyslexia, medical or mental healthcondition which significantly affects study. DUSSD can advise you and organise special academicfacilities if you have a disability and need some help. They will try to provide whatever support isnecessary to enable you to study effectively and to make full use of your opportunities at University.This help will be specific and appropriate to you and relevant to the courses you choose.
7
Special arrangements and facilities may well be required by disabled students when taking ex-aminations. These might include extra reading time or a separate quiet room and are intended tominimise the effects of disability, which are often exacerbated by examination conditions. DUSSDorganises all the requisite examination concessions for hearing-impaired, visually-impaired anddyslexic students. DUSSD also makes recommendations to departments for students with otherdisabilities who have regular support from the Service.
The Departmental Disability representative (DDR) is Dr. S. Borgan, (room CM216), Email:[email protected] and is the person to be contacted regarding any concerns, require-ments or problems in studying in the department.
For further advice, or to obtain a copy of the University’s Disability Statement, please contactDurham University Service for Students with Disabilities (DUSSD), Pelaw House, Leazes Road,Durham, DH1 1TA, Tel: 0191 334 8115 (Voice and Minicom),Email: [email protected].
1.10 Illness and Absence
If you miss tutorials or fail to hand in written work because of illness you must ask your College toinform the Department. If this work is summatively assessed (i.e., counts towards your final markfor a module), you must complete a self-certification of illness form.
If your academic performance is significantly affected by illness or other difficulties at any time,you should obtain documentary evidence as described in the Learning and Teaching Handbook ofthe University of Durham.The relevant section is: 6.3.16 Student Absence and Illness. This is accessible on-lineunder the address www.dur.ac.uk/learningandteaching.handbook/3.16.pdf and containslinks to downloadable self-certification forms and requests for a doctor’s certificate.
A member of the Department is liaising with the Colleges regarding illness and absences related toillness. Feel free to contact her if you feel it might be beneficial for you to discuss matters withinthe department.
Liaison officer (Colleges - Dept of Mathematical Sciences):Mrs F. Giblin ([email protected])
If you miss tutorials or fail to hand in written work for other reasons you should contact the CourseDirector as soon as possible.
1.11 Course Information
Term time in Durham is Michaelmas (10 weeks), Epiphany (9 weeks) and Easter (9 weeks). Thereare 22 teaching weeks, and the last seven weeks are dedicated to private revision, examinationsand registration for the subsequent academic year.
Timetables giving details of places and times of your commitments are available on Departmentalweb pages and noticeboards in the first floor corridor of the Department. It is assumed that youread them!You may access your own Maths timetable at www.maths.dur.ac.uk/teaching/ andthen clicking on the ‘My Maths timetable’ link.
8
Also, teaching staff often send you important information by e-mail to your local ‘@dur.ac.uk’address, and so you should scan your mailbox regularly (see below).
1.12 Computers, ICT and DUO
You are expected to use the internet — i.e., e-mail and the World-Wide Web (WWW) — andfacilities are provided by the Information Technology Service (ITS). You should take advantageof ITS instruction courses to make sure that you have a basic acquaintance with computers. Theweb-address is www.dur.ac.uk/ITS.
The Maths Department web-address is www.maths.dur.ac.uk and a valuable link is ‘Teaching’.Here besides lecture and tutorial timetables you will find material provided by lecturers. For thisthey may use ‘Durham University Online’ (DUO).
DUO is a virtual learning environment which is a collection of on-line resources including linksto web pages, lecture notes and exercise sheets/solutions, communication tools like email andassessment features such as formative quizzes. Your login area on DUO is where you can accessall on-line course materials offered by your lecturers.
Soon after your registration details have been entered onto the University’s student records system(Banner), you will automatically be enrolled by the Learning Technologies Team at the IT Serviceon the DUO courses related to the mathematics modules that you are taking. Details of how tologon to the DUO system are given at duo.dur.ac.uk and in the IT Service publication ‘Com-puting at Durham’. Individual lecturers will inform you about its use for their courses from timeto time during the year. The Department will also make use of the Announcements area in DUOto pass on important information to you so please get into the habit of logging in every day.
1.13 Private Study
‘An undergraduate module with effect from October 1998 is defined as a study unit comprising200 hours of SLAT (Student Learning Activity Time) per annum and lasts one academic year’(University of Durham Teaching and Learning Handbook). The total ‘contact time’ that a studentspends in lectures, tutorials, etc. amounts to around 30% of the total SLAT. You would be wise toplan how best to use the remaining 70% (140 hours for a 20-credit module, i.e., 6.4 hours per weekof a 22-week academic year per 20-credit module). Your tutor will be able to help you with this.This time is allocated within the module to be spent, not only in preparing submitted work (e.g.essays, assessed problems), but in private study of the lecture course material and in revision. Youare advised to organise your time in such a way that you are able to devote a number of hours eachweek to reviewing your lecture notes, reading around the subject and working through exercisesextra to those which have been set by the lecturer. By so doing you will be developing yourstudy and personal management skills and be giving yourself the best opportunity to gain a firmunderstanding of the topics as they unfold. By attending to any difficulties or misconceptions youhave as the course progresses you will be in an excellent position at the end of the course to makethe most of your revision time. Planning and preparation are the key to reducing examination stress.It was within this framework that the University developed PDPs (Personal Development Plans)which is a structured and supported process which will enable you to reflect on the mathematicalprocess and to discover, at a personal level, the best way for you to learn the mathematics you aretaught. To find out more visit duo.dur.ac.uk.
9
1.14 Books and Libraries
The main collection of mathematics books, including all the texts recommended in this booklet, ishoused in the University Library. Each College has its own library.
For mathematical sites on the Internet, visit www.yahoo.co.uk/Science/Mathematics/
1.15 Student Records
Your Department record file contains some or all of the following:
-Your UCAS form,-Annual Department registration forms,-Your plagiarism forms,-Annual examination results,-Copies of letters received from you and sent to you.
Lists of student names are used in the preparation of registers for tutorials, practicals and examplesclasses and in the examinations. All such Departmental computer files are registered under theData Protection Act. Each student’s marks for all examinations and assessed work are confidentialto the members of the Board of Examiners of the Department; aggregate marks are known to themembers of the Faculty Board of Examiners, College Principals and Senior Tutors, the Examina-tions Department, and the individual student.
1.16 Smoking and Mobile Phones
Please note that (i) smoking is not allowed in the Department, and (ii) mobiles must always beswitched off in teaching rooms.
10
1.17 Durham University Mathematical Society
∃MathSoc∀Durham University Mathematical Society, affectionately known as MathSoc, provides an oppor-tunity for maths students (or any one with an interest in maths) to meet away from lectures.
We arrange a variety of events throughout the year such as bowling, bar crawls, paintballing, aChristmas meal and the highlight of the year a trip to see Countdown being filmed! So you shouldfind something to interest you.
MathSoc also helps to arrange guest lectures in a wide range of aspects of maths such as What ismathematics? and A symmetry approach to viruses. These are at a level so that anyone with aninterest in maths can enjoy them and they will hopefully inspire an interest in a part of maths youmay not have seen before.
We have our own website www.durham.ac.uk/mathematical.society where you will find all themost up to date information about the society. Here you will also find our second hand book listwhich has many of the books needed for courses for much cheaper than you will find in the shops.
If you would like any more information about either the society itself or advice on any other aspectof the maths course that you dont want to ask the lecturers about feel free to get in touch with anyof the exec listed below or via the society email address [email protected].
TO JOIN:
Come see our stand at the freshers fair or email at any time it costs only 6 for life membership or3 for a year
This year’s Exec. is:President - Rachel Gillespie ([email protected])Treasurer - Matthew Jones ([email protected])Secretary - Caspar de Haes ([email protected])Social Secretary - Hannah Edwards ([email protected])Publicity Officer - Louisa Picciau (louisa.picciau.durham.ac.uk)
1.18 Disclaimer
The information in this booklet is correct at the time of going to press in May 2008 . The Univer-sity, however, reserves the right to make changes without notice to regulations, programmes andsyllabuses. The most up-to-date details of all undergraduate modules can be found in the FacultyHandbook on-line at www.dur.ac.uk/faculty.handbook/.
11
2 Examinations and Assessment
2.1 Regulations for B.Sc., M.Math. and M.Sci
The General Regulations for the B.Sc. and M.Math./M.Sci. degrees and the special regulations forthe courses described in this booklet are printed in Volume II of the current version of the DurhamUniversity Calendar, which is available for consultation in the main library or in the Departmentof Mathematical Sciences Office. An offprint of the B.Sc. and M.Sci./M.Math. regulations maybe obtained from the Science Faculty Office.
2.2 University Assessment Process
Full details of the University procedures for Examinations and Assessment may be found in theLearning and Teaching Handbook (www.dur.ac.uk/learningandteaching.handbook/)
2.3 Board of Examiners
The Board of Examiners is responsible for all assessment of Mathematics courses.Chair: Prof. A. Taormina (room CM302) Email: [email protected] Chair: Dr. S. Borgan, (room CM216), Email: [email protected]: Dr. P.S. Craig (room CM317) Email: [email protected]
2.4 Plagiarism, Cheating and Collusion
Working with your fellow students is perfectly acceptable, but joint work should be declared assuch. The University has a strict policy against plagiarism and other forms of cheating, a state-ment of which may be found in the Teaching and Learning Section in Volume I of your Faculty’sUndergraduate Handbook.
Plagiarism includes• The verbatim copying of another’s work without acknowledgement.• The close paraphrasing of another’s work by simply changing a few words, or altering the orderof the presentation, without acknowledgement.• Unacknowledged quotation of phrases from another’s work.• The deliberate and detailed presentation of another’s concept as one’s own.
Cheating includes• Communication with or copying from any other student during an examination.• Communication during an examination with any person other than a properly authorised invigi-lator or another authorised member of staff.• Introducing any written or printed material into the examination room unless expressly permittedby the Board of Examiners in Mathematical Sciences or course regulations.• Introducing any electronically stored information into the examination room, unless expresslypermitted by the Board of Examiners in Mathematical Sciences or course regulations.• Gaining access to unauthorised material during or before an examination.• The provision or assistance in the provision of false evidence or knowledge or understanding inexaminations.
Collusion includes• The collaboration, without official approval, between two or more students in the preparation andproduction of work which is ultimately submitted by each in an identical, or substantially similar,
12
form and/or represented by each to be the product of his or her individual efforts.• The unauthorised co-operation between a student and another person in the preparation andproduction of work which is presented as the student’s own.
2.5 Monitoring of Work
Under the general regulations of the University with regard to the keeping of terms, you are re-quired to complete written work to a standard satisfactory to the Chairman of the Board of Studies.In practice this means that you will be expected to hand in at least 75% of written work on time at astandard of grade C or better (see table below). To encourage this, your performance is monitoredby the course director.
Formative assessment1 of coursework occurs at all Levels, while summative continuous assessmentof coursework occurs for the auxiliary2 Level I Mathematics modules.
The purpose of formative and summative continuous assessment of coursework is to help the stu-dent at each stage of the learning process. It is designed to encourage effort all year long andprovides manageable milestones, in preparation for the summative assessment of end of year exam-inations. Course lecturers provide problems of an appropriate standard and length to the students,as well as assessment templates (model solutions) to the markers.
Each script is returned to the student with the grade written on it. The interpretation of grades is asin the table below.
The returned scripts should indicate clearly where errors and gaps in arguments occur, and the na-ture of errors. They should give brief indications as to the approach required, bearing in mind thatmodel solutions for all set problems will be provided to students by lecturers shortly after the mark-ing has occurred. The lecturer makes relevant model solutions available to students via the coursewebpage or/and Durham On-Line (DUO) shortly after they have submitted their assignments.
Remark: A grade C is deemed acceptable. D/E or a failure to hand in work is a demerit. If say 4questions of equal standard are set and 2 are answered very well and 2 are not tackled at all thenthere is close to 50% attainment, resulting in grade C.
In all cases, performance at marked written work can provide useful evidence for the Board ofExaminers if examination performance is adversely affected by illness or other circumstances.
1‘Summative’ assessment counts towards the overall mark for the module. ‘Formative’ assessment does not.2MATH1031, MATH1541, MATH1551, MATH1561, MATH1571, MATH1711
13
Grade Equivalent Mark Quality
A ≥ 80% Essentially complete and correct workB 60%—79% Shows understanding,
but contains a small number of errors or gapsC 40%—59% Clear evidence of a serious attempt at the work,
showing some understanding, but with important gapsD 20%—39% Scrappy work, bare evidence of understanding
or significant work omittedE <20% No understanding or little real attempt made
2.6 Submission of Formative Work
Students are required to submit work before the deadline for submission. No credit is given forlate work unless there is a prior arrangement with the lecturer.
2.7 Submission of Summative-Assessed Work
Students are required to submit work before the deadline for submission. They will be informedclearly of the deadline for a given piece of work — the submission dates will be displayed onDepartmental noticeboards (first floor corridor) and module webpages where appropriate.
If a student, for whatever reason, believes he/she is unable to submit a major piece of summativeassessed work by the due deadline, he/she should submit a written request for an extension to theChairman/Chairwoman of the Board of Examiners in Mathematical Sciences well in advance ofthe due deadline and explain his/her reasons, providing supporting evidence where appropriate.The Chairman/Chairwoman of the Board of Examiners concerned will have to consider whetherin his/her view the grounds offered by the student are sufficient to warrant an extension to theoriginal deadline. Normally the only grounds on which an extension will be granted are wherecircumstances beyond the control of the student have prevented submission.
If an extension is granted then the new deadline will be made clear to the student, in writing, andthe procedures with regard to meeting the new deadline should be those outlined in this policystatement. If a student is not granted an extension, and fails to submit a piece of summativeassessed work by the due deadline, the work will not be marked and a mark of zero will be recorded.
2.8 Calculators
Calculators are needed for some Maths modules and in the corresponding examinations. In theinterest of fairness, the Board of Studies in Mathematical Sciences has decided that only the sim-plest scientific types are allowed. In particular, you are not allowed to take to an examination anycalculator which is programmable, can display graphics, has facilities for text storage or commu-nications or can evaluate integrals or solve linear equations.
For details follow the links ‘Teaching > Exam info’ from the Department’s home-page.
14
2.9 Examinations and Progression
Details of examination papers structure may be found by following the links ‘Teaching > Examinfo’ from the Department’s home-page.
For all degrees, you must satisfy the University rules on progression. You should refer to Section 3of the Undergraduate Handbook Volume II-Core (Final Honours) for the precise rules for progres-sion from Year 3 to Year 4. Flowcharts have been designed to help you understand the implicationsof the progression regulations. They may be found at www.dur.ac.uk/faculty.handbook byfollowing the link ‘Student survival guide > Flowchart of Progression Regulations’
The pass mark for modules is 40%. As a Level 3 student, you cannot resit any module. However inthe case where you have failed 1 or 2 modules (possibly including 1 from Level 2, with marks of atleast 30% in these modules, and if your Level 3 average is at least 40%, you are considered to havepassed the ‘failed’ modules by compensation, provided the regulations of your degree programmedo not specifically exclude those modules from compensation. As a Level 4 student, you cannotresit any module, but can pass 1 module by compensation.
The following summarises the minimum requirements for an undergraduate student to leave theuniversity with some award. The numbers refer to the minimum number of modules passed in totaland where applicable at specific Levels. These numbers refer to passes after any allowed resits orcompensation.
Award Total Additional Level requirementsCertificate 5Diploma 10 Minimum 3 modules above Level 1Ordinary degree 15 Minimum 3 modules above Level 2Bachelors Honours degree 17 A Level 1 failed module onlyMasters Honours degree 23 A Level 1 failed module only
Note that the award of an Honours degree allows one module to be failed at Level 1 but all 6modules must be passed at Levels 2 and 3, and also at Level 4 for a Masters degree.
The final degree awarded depends on your performance in each year after your first one. The 2H,3H and 4H results are weighted in the ratios 2:3:4 to produce the final mark on which the degreeclassification is decided. The Department operates Scheme 1 (Mean/Weighted) for each of itsdegrees.
Final marks will be available on-line before a Departmental interview held in the last week of term.For non-final year students, the interview is an opportunity to discuss your progress and plans. Italso deals with formal registration for the fourth year and cannot be missed. For time and placesof interviews, see Departmental noticeboards after results are published.
Final year students may wish to chat with their adviser at the end of the Easter term too, and theyare welcome to organise a meeting after the exam period.
Official written statements are available from the Examinations Office, Old Shire Hall, after thesecond full week of July, on payment of the appropriate fee. Students must authorise, in writing,the Chairman of the Board of Examiners if they wish their marks to be disclosed in reply to writtenrequests from outside bodies such as EPSRC and other universities.
15
2.10 Illness and Examinations
If your academic performance is significantly affected by circumstances beyond your control – forinstance, illness or bereavement – at any time during your programme of study, and especially inthe period leading up to or during the examination period, you might wish to bring these mitigatingcircumstances to the attention of the Board of Examiners.
The Board of Examiners has discretion to take mitigating circumstances into account when makinga final decision on a student’s progression to the next year of study or on his/her class of degree.Students must inform the Board of Examiners before they meet, using the Mitigating Circum-stances form, which can be obtained from Colleges or downloaded fromhttp://www.dur.ac.uk/resources/learningandteaching.handbook/Section6/a6.04.pdf.Supporting evidence such as a doctor’s certificate, or other evidence from an independent profes-sional such as a counsellor or members of DUSSD, should be submitted with the form if availableand appropriate.
Students considering claiming Mitigating Circumstances are advised to read Section 6 of theLearning and Teaching Handbook of the University of Durham, accessible on-line under the ad-dresshttp://www.dur.ac.uk/learningandteaching.handbook/6/.The relevant section is:6.3.16.4. Evidence for Boards of Examiners.
3 Learning and Teaching
There are no formal tutorial arrangements in 3H or 4H. Most classes are smaller in size than in 2Hand you are expected to contact the lecturer concerned if you experience difficulties.
Written work is set regularly for each module and your solutions marked and graded A-E, reflectingyour attainment as a proportion of the total work set. Model solutions to work set are then providedfor reference on the appropriate website. It is important to devote time and effort to your solutions.Attempting problems adds to the enjoyment and excitement of mathematics as well as developingyour mathematical and problem-solving skills. It also enhances your ability to explain your ideasand methods to others in writing. This is one of the most valuable transferable skills you can gainfrom your studies.
3.1 The Mathematics Modules and Degrees
In Durham, students take the equivalent of six modules in each year, each single module beingworth 20 credits. The flowchart at the centre of the booklet shows the pre-requisites of all modulesoffered by the Department of Mathematical Sciences to students at level 3 and 4 in October 2008. It should be read in conjunction with the information in the Appendix where excluded combina-tions are spelt out. The Department may withdraw a module if it attracts very few students. YourDepartmental Adviser and the relevant lecturers will be happy to discuss your proposed choice ofmodules.
Each third- and fourth-year module, apart from Communicating Mathematics III, IndependentStudy Module and Project IV, has two contact hours per week during the Michaelmas and Epiphanyterms, and revision lectures in the Easter term. For the Independent Study Module there are twooffice hours per week during which you can come to discuss any problems relating to the material.
16
As a third/fourth-year Honours Mathematics student, you will have chosen one of the three degreeprogrammes in Table 1 of the Appendix. Each degree programme offers a great flexibility ofchoice in the modules you take at level 3 and 4. Of course, your choice of 2H options may limityour choice of modules in 3H and 4H, but if you are a Single Honours student, you will have atleast nine modules to choose from in 3H. The Single Honours degrees in Mathematics are firstdegree, eligible for local authority funding in the normal manner.
Each programme in Table 1 will qualify you for a wide range of employment and also for furtherstudy in programmes such as the PGCE, specialised M.Sc. courses which make use of mathematicsin other areas (Operational Research, Applied Statistics, Computation etc.) or research leading toa Ph.D.
The overall 3- or 4-year structure of the degree programmes is described in your Faculty’sThird/Fourth-Year Undergraduate Handbook under ’Degree Programme Frameworks’.
3.2 Honours Natural Sciences
Natural Sciences students who wish to include mathematics modules at level II and above as partof their programme must have taken Core Mathematics A (MATH1012) during the first year. Inaddition, Core Mathematics B1 (MATH1051) must be taken in year 1 or in year 2 (except as notedin the following paragraph). Subject to these constraints students may choose any mathematicsmodule for which they satisfy the prerequisites other than Independent Study III (MATH3161) andCommunicating Mathematics III (MATH3131). Please note the minimal requirements [2008 on-wards] listed on the first page of the Appendix regarding level 3 and level 4 mathematics modules.
A student who has taken Core Mathematics A in year 1 and who was unable to take Core Math-ematics B1 in year 2 (because he/she needed to take two Level I modules to begin a new sub-ject), should have taken Linera Algebra II (MATH2021), Statistical Concepts II (MATH2041)and one of Codes and Actuarial Mathematics II (MATH2131), Codes and Geometric Topology II(MATH2141), Probability and Actuarial Mathematics (MATH2161) and Probability and Geomet-ric Topology II (MATH2151) . He/she may now continue with Decision Theory III (MATH3071),Operations Research III (MATH3141) and Statistical Methods III (MATH3051) .
The BSc/MSci degrees in Mathematics and Physics and the MSci degree in Chemistry and Math-ematics have fixed programmes which are shown in Table 2 (Appendix).
The Natural Sciences mathematics adviser is:Dr. J. Bolton, (room CM314), Email: [email protected] .
3.3 Further Remarks on Degrees
The Master of Mathematics degree and the Master in Science Joint Honours in Mathematics andPhysics or Chemistry and Mathematics are particularly suitable if you wish to study your subjectin greater depth, for example with the intention of doing research. They aim to provide a math-ematical education comparable with that which may be gained elsewhere in Europe, in terms ofdepth and breadth. One third of the final year will be spent on project work, allowing an in-depthstudy of a particular topic.
17
3.4 Booklists and Descriptions of Courses
The following pages contain brief descriptions of the level 3 and 4 modules available to you.
Each module description is followed by a list of recommended books and a syllabus. For somemodules you are advised to buy a particular book, indicated by an asterisk; for others a choice oftitles is offered or no specific recommendation is given.
There are also suggestions for preliminary reading and some time spent on this during the summervacation may well pay dividends in the coming year.
Syllabuses, timetables, handbooks, exam information, and much more may be found atwww.maths.dur.ac.uk/teaching/, or by following the link ‘teaching’ from the Department’shome page (www.maths.dur.ac.uk).
The syllabuses are intended as guides to the modules. No guarantee is given that additional materialwill not be included and examined nor that all topics mentioned will be treated.
18
19
3.4.1 ALGEBRAIC GEOMETRY III AND IV – MATH3321/MATH4011 (38 lectures)
To be given in 2009 -2010
Algebraic geometry studies polynomials in several variables. It does so by means of a dictio-nary between algebraic notions (rings, ideals etc) and geometric notions (curves, surfaces, tangentspaces etc). The resulting theory is remarkably rich, bringing powerful algebraic tools to bear ongeometric problems, and bringing geometric insight to problems in algebra. The subject is centralin modern mathematics and borders on areas as diverse as number theory, differential geometryand mathematical physics.
This course is concerned with complex plane curves. Bezout’s theorem determines their intersec-tion behaviour, and has elegant applications to the geometry of their tangents and their singularpoints. It also gives rise to special behaviour possessed by curves of particular low degrees: forexample a plane cubic has a natural group structure and can be parametrised by elliptic functions.
Studying the field of rational functions on a variety ( = curve, surface, three manifold etc) leadsto the idea of ’birational geometry’, in which the variety can have quite different manifestationsin spaces of different dimensions. A famous example is the cubic surface in 3-space, whose ge-ometry is equivalent to that of 6 points in the plane. The birational geometry of a curve is muchsimpler than that of higher dimensional varieties and is governed by the Riemann-Roch theorem.This involves the ‘genus’ of the curve, which (over the complex numbers) can be interpreted as thenumber of handles if the curve is viewed as a real orientable surface.
Recommended Books
1. Reid: Undergraduate algebraic geometry, LMS Student Texts 12, Cambridge 1988, ISBN0-521-35662-82. Kirwan, Complex algebraic curves, LMS Student Texts 23, Cambridge 1992, ISBN 0-521-42353-83. Gibson, Elementary Geometry of Algebraic Curves, CUP 19984. Harris, Algebraic geometry - a first course, GTM 133, Springer 19925. Fulton, Algebraic curves, Benjamin 19696. Reid Undergraduate commutative algebra, LMS Student Texts 29, Cambridge 19957. Brieskorn, H. Knorrer, Plane algebraic curves, Birkhuser 19868. Clemens, A scrapbook of complex curve theory, Plenum 1980.
The course is based mainly on [2], and a bit on [1]. It is also largely parallel to [3], which has manyexamples and explanations. [4] is also excellent, and much more thorough. [5] is an old warhorse;[6] should be mentioned as a close relative of [1]. [7] and [8] are for those with stamina.
Preliminary Reading: Chapter 1 in either of [1] or [2].
4H reading material reference: G. Fisher, Plane algebraic curves, AMS, Student MathematicalLibrary, Vol. 15. (Chapter 7)
20
Outline of course Algebraic Geometry III/IV
Aim: To introduce the basic theory of algebraic varieties and birational geometry, with particularemphasis on plane curves.
Term 1 (20 lectures)
Plane Curves: Affine and projective plane curves over a field. Conics, Pappus’ Theorem. Uniquefactorisation in polynomial rings. Study’s lemma, irreducibility. Singular points, tangents, pointsof inflection. Dual plane, linear systems of curves.
Bezout’s Theorem: Resultants, weak form of Bezout. Applications to Pascal’s theorem, Cayley-Bacharach theorem, group law on a cubic. Intersection multiplicity, strong form of Bezout.
Term 2 (18 lectures)
Bezout’s Theorem: Applications. Flexes. Hessian, configuration of flexes of a cubic. Ellipticcurves, Weierstrass normal form.
Complex Curves as Real Surfaces: Basic topology and manifolds; degree-genus formula. Reso-lution of singularities and non-singular models.
4H reading material: a topic in the following area: parametrizing the branches of a curve byPuiseux Series
21
3.4.2 ANALYSIS III AND IV – MATH3011/4201 (38 lectures)
To be given in 2009 -2010
This course is an introduction into Analysis on differentiable manifolds, which are curved ana-logues of Euclidean spaces. Examples of manifolds are curves (in dimension one) and surfaces (indimension two). Manifolds of higher dimension play an important role in many areas of pure andapplied mathematics.
The course starts with elements of Functional Analysis: we will use techniques of Banach andHilbert spaces to prove the classical Picard existence theorem for ordinary differential equations.
We will then study differentiation and integration of functions on manifolds and discuss relatedgeometric objects: vector fields, flows, differential forms. We will discuss Stokes’ theorem fordifferential forms Z
Mdω =
Z∂M
ω
which generalizes on high dimensions the familiar fundamental theorem of calculusZ b
af ′(x)dx = f (b)− f (a).
The general Stokes theorem is a unique source of various integration formulae which appear inAnalysis, Geometry and Mathematical Physics.
Recommended Books
1. M. Spivak, Calculus on Manifolds, Benjamin.
2. D.M. Bressoud, Second year calculus, Springer-Verlag 1991
3. MIT Analysis II lecture notes, available from:
http://ocw.mit.edu/OcwWeb/Mathematics/18-101Fall-2005/CourseHome/index.htm
4. S. Lang, Analysis I, Addison-Wesley.
22
Outline of course Analysis III/IV
Term 1 (20 lectures)
Metric Spaces: Examples of metric spaces, completeness, Banach and Hilbert spaces, ContractionMapping Theorem, Picard Theorem for Ordinary Differential Equations.
Differentiation on Manifolds: Implicit Function Theorem, Inverse Function Theorem, Differen-tiable Manifolds (as submanifolds of Rn), Tangent spaces and Vector fields.
Term 2 (18 lectures)
Integration in Rn: Riemann Integrals in Rn, Sets of measure zero, Fubini’s Theorem, Partitionsof Unity.
Differential forms: Linear Algebra of alternating forms, Differential forms (mainly of low de-gree), examples, Lie derivative, exterior derivative.
Integration on Manifolds: Integration of Differential forms, Stokes’ Theorem, Green’s Theorem,Divergence Theorem.
4H Reading material:
Flows and vector fields on manifolds.
R. Darling, Differential forms and connections, Cambridge University Press, Cambridge, 1994.x+256 pp. ISBN 0-521-46800-0
23
3.4.3 APPROXIMATION THEORY & SOLUTIONS OF ODEs III AND IV –MATH3081/MATH4221 (38 lectures)
Dr M. P. J Jensen / Dr J. F. Blowey
In Approximation Theory one is concerned with approximating a function in an infinite-dimensionalvector space by a function in a finite dimensional subspace. For example one examines how a con-tinuous function can be approximated by a polynomial. Because Numerical Analysis is aboutthe transition from infinite-dimensional mathematical structures (such as differential equations) tofinite-dimensional ones, the question how this transition works on the level of functions is one ofthe most fundamental topics of Numerical Analysis.
Similarly, Numerical Analysis has been defined as ‘the study of algorithms for problems of con-tinuous mathematics’. An ordinary differential equation with appropriate initial or boundary con-ditions is an example of such a problem. We may know that the problem has a unique solutionbut it is rarely possible to express that solution as a finite combination of elementary functions.For example, how could we compute y(3) to some specified accuracy, given that y′+ xy = 1 andy(0) = 0? For this simple linear differential equation we can find an integrating factor and expressy(x) as an integral, but that only replaces one problem by another—try it. The vast number ofmethods available for approximating solutions of differential equations may be sub-divided intoa small number of families, and within each family specific formulae may be derived systemati-cally. Convergence and stability, in its various forms, are essential features of practical methods;techniques which allow us to investigate these properties will be established and used.
Specialist software will be used for computational work throughout the module.
Recommended BooksNo single book gives a satisfactory treatment of all the topics in the course, but most introductorybooks on numerical analysis have some relevant material. There is some approximation theory inChapter 15 of:R. Plato, Concise Numerical Mathematics, AMS, ISBN 082182953X,K.E. Atkinson, An Introduction to Numerical Analysis, 2nd edition, Wiley 1989, ISBN 0471500232,(alternative reference)G.M. Phillips and P.J. Taylor, Theory and Applications of Numerical Analysis, 2nd edition,Academic Press 1996, ISBN 0125535600, (alternative reference)Books devoted specifically to the numerical solution of differential equations include:U.M. Ascher and L.R. Petzold, Computer Methods for Ordinary Differential Equations andDifferential-Algebraic Equations, SIAM 1998, ISBN 0898714125J.R. Dormand, Numerical Methods for Differential Equations, CRC Press 1996, ISBN 0849394333A. Iserles, A First Course in the Numerical Analysis of Differential Equations, Cambridge1996, ISBN 0521556554,R.L. Burden and J.D. Faires, Numerical Analysis, 7th edition, Brooks/Cole 2001, ISBN 0534382169,(Chapter 5)
Preliminary Reading: Read as much as you like of Chapter 5 of Burden and Faires or Chapter 6of Atkinson.
24
Outline of course Approximation Theory & Solutions of ODEs III/IV
Aim: To build on the foundations laid in the level II Numerical Analysis module and to enable stu-dents to gain a deeper knowledge and understanding of two particular areas of numerical analysis.
Term 1 (20 lectures)
Approximation Theory: Piecewise polynomial approximation and spline functions. Best approxi-mation, with emphasis on minimax and near-minimax polynomial approximations. Approximationby rational functions. Trigonometric polynomials and fast Fourier transforms.
Term 2 (18 lectures)
Numerical Solution of Ordinary Differential Equations: Introduction to numerical methodsfor initial-value problems. Local and global truncation errors, convergence. One-step methods,with emphasis on explicit Runge-Kutta methods. Practical algorithms for systems of equations.Linear multistep methods, in particular the Adams methods. Predictor- corrector methods andestimation of local truncation error. Stability concepts. BDF and stiff problems. Shooting methodsfor boundary-value problems.
4H reading material: to be announced.
25
3.4.4 BAYESIAN METHODS III AND IV – MATH3311/MATH4191 (38 lectures)
To be given in 2009 -2010
This course provides a study of practical aspects of Bayesian Statistics. The underlying theme willbe to explain what the software package BUGS (Bayesian inference Using Gibbs Sampling) triesto do, how it works and how it might be used by a statistician. A small number of case studies willbe used in illustration.
The first term will begin with a review of the Bayesian statistics component of Statistical Conceptsand will then mostly be about the need for more complex models and the theory and practice ofhow to build and represent such models for realistic situations.
The second term will almost entirely be given over to the modern method of computation forBayesian statistics: Markov chain Monte Carlo. As part of the development, there is a substantialsection on methods for generating random numbers with particular distributions.
Students taking the level IV module will study, independently, numerical approximations to poste-rior distributions which do not require the use of Monte Carlo.
Recommended Books
The first book is a general introduction to the theory of Bayesian statistics. The second covers thesecond term (Markov chain Monte Carlo) well and also some of the first term material. The thirdmixes more advanced theory with computation and many examples. The fourth is a more appliedbook and contains many examples of the use of BUGS. None is completely suitable by itself as atext for this module.
P.M. Lee, Bayesian statistics: an introduction (2nd ed.), Arnold, 1997, ISBN 0340677856
D. Gamerman and H.F. Lopes, Markov chain Monte Carlo : stochastic simulation for Bayesianinference (2nd ed.), Chapman & Hall/CRC, 2006, ISBN 1-58488-587-4
A Gelman et al, Bayesian Data Analysis (2nd ed.), Chapman & Hall/CRC, 2004, ISBN 1-58488-388-X
P. Congdon, Bayesian Statistical Modelling, Wiley, 2001, ISBN 0471496006
Level IV reading material reference: Gamerman and Lopes (see above).
26
Outline of course Bayesian Methods III/IV - MATH3311/MATH4191
Aim: To provide an overview of practical Bayesian statistical methodology together with importantapplications.
Term 1 (20 lectures)
Review and introduction: Bayesian paradigm, conditional independence and conjugacy. Manip-ulation of multivariate probability distributions. Methods for computation: conjugacy, quadrature,Monte Carlo. Conjugacy for the normal linear model. Interpretation of Monte Carlo output. Con-cept of Markov chain Monte Carlo.
Hierarchical modelling: Motivation, latent variables, random effects, conjugacy and semi-conjugacy.
Bayesian graphical modelling: Directed acyclic graphs, Bayesian networks, conditional indepen-dence, moral graph, separation theorem. Doodles in WinBUGS. Specification of prior beliefs.
Term 2 (18 lectures)
Random number generation: generation of uniform pseudo-random numbers. Distribution func-tion method, rejection sampling, log-concavity, adaptive rejection sampling
Markov Chain Monte Carlo: Markov chains, equilibrium distribution. Gibbs sampling. Metropolis-Hastings: Metropolis random walk, independence sample, Gibbs sampling. Analysis and interpre-tation of MCMC output. WinBUGS.
Model comparison: Bayes factors, deviance information criterion (DIC).
Case studies.
4H reading material: Approximation of posterior distributions without use of Monte Carlo.
27
3.4.5 BAYESIAN STATISTICS III AND IV – MATH3341/MATH4031 (38 lectures)
Dr P. S. Craig / Mr D. A. Wooff
This course provides an overview of foundational and theoretical aspects of Bayesian statistics andan introduction to Bayes linear methods. It is concerned with deep intellectual questions abouthow humans should reason in the presence of uncertainty; in the absence of uncertainty there iswidespread agreement that the correct approach is logic. It should be seen as complementary tothe more practically and computationally oriented Bayesian Methods course.
The first term will discuss the foundations of subjective probability, the role of exchangeability, ex-ponential families and conjugate families of prior distributions, model selection and some aspectsof Bayesian non-parametric inference and/or robustness.
The second term forms an introduction to Bayes linear statistics. These are methods based onexpectation as a primitive, rather than probability. The material covers: expectation as a primitive,evaluation and interpretation of adjusted expectations, variances and covariances, summarisingadjustment using the resolution transform, belief diagnostics, partial adjustment and exploitingexchangeability.
Recommended Books
Parts of the first two books are recommended reading for first term material. Other reading will besuggested as appropriate.
The third book, written by members of the department, is the definitive reference for Bayes linearinference.
B. De Finetti, Probability, induction and statistics : the art of guessing, Wiley, 1972, ISBN0-471-20140-5J.M. Bernardo and A.F.M. Smith, Bayesian theory, Wiley, 1993, ISBN 0-471-92416-4M. Goldstein and D.A. Wooff, Bayes Linear Statistics, Wiley, 2007, ISBN 978-0-470-01562-9
Reading Material reference
This will be provided to Bayesian Statistics IV students by the lecturer at a suitable time inMichaelmas term.
28
Outline of course Bayesian Statistics III/IV - MATH3341/MATH4031
Aim: To provide an overview of the theoretical basis for Bayesian statistics and an introduction tothe theory and application of the Bayes linear approach
Term 1 (20 lectures)
Review: Bayesian paradigm, conditional independence and conjugacy. Manipulation of multivari-ate probability distributions.
Foundations: Rational basis for subjective probability and Bayesian statistics, exchangeabilityand de Finetti’s representation theorem, parametric modelling.
Exponential families: regular exponential families, canonical representation, sufficiency, conju-gacy, linearity of posterior expectation.
Non-parametrics/robustness: Introduction to Bayesian non-parametric inference or robustness.
Term 2 (18 lectures)
Bayes linear methods:
Expectation as a primitive; evaluation and interpretation of adjusted expectations, variances andcovariances; resolution transform and canonical structure; observed adjustment; belief diagnostics;partial adjustment; Bayes linear influence diagrams; exploiting exchangeability.
Case studies.
Level IV reading material: A topic in Bayesian non-parametrics or robustness to be specified bythe lecturer.
29
3.4.6 COMMUNICATING MATHEMATICS III – MATH3131
The aim is to take a topic in mathematics or statistics and to communicate the ideas, by a poster,a short presentation, and a written report, to an audience at the level of 3H. The emphasis is oncommunication of material at the 3H level.
The first term consists of independent guided study within a small group. Each group consistsnormally of 3 or 4 students. The second term consists of preparing an individual written report.During the second term, students give a group-linked presentation, accompanied by individualposters. The poster and presentation are assessed and count for 15% of the module mark.
There are no lectures. However, there is a short series of workshops at the beginning of the course.These address what constitutes effective communication in mathematics and statistics: how topresent mathematical material, how to use computer packages to write mathematical reports, howto lay out reports, how to convey statistical information effectively, and so forth.
The topic supervisor guides the technical content and organizes progress meetings about once perfortnight.
This module is a single 3H option so you should devote on average one-sixth of your efforts to it.The report deadline is the first Friday of the Easter term, May 1st 2009.
For details of topics and supervisors visit the 3H courses web pagewww.dur.ac.uk/teaching/3H/ and follow the prominent link.
To take this option, submit your preferred topics via the web link before the deadline given there.Then attend the allocation meeting in CM101 at 2:00pm on Friday 13th June 2008, where the aimis optimum matching of students and supervisors.
Please note that a supervisor may not accept 5 or more students, and a topic with fewer than 3takers may be withdrawn.
30
3.4.7 CONTINUUM MECHANICS III AND IV – MATH3101/MATH4081 (38 lectures)
To be given in 2009 -2010
The air we breathe, the water in our body, the earth on which we stand and many (most?) objectsaround us can, to a very good approximation, be modelled as either a fluid or an elastic mate-rial. In this course we will study the equations underlying such models and their solutions (e.g.,sound, water waves great and small, flows in pipes and around wings, etc.), with emphasis on fluidmechanics.
We start with a general kinematic description of fluid flow, followed by the equations governingthe motion of an ideal fluid (‘dry water’); specific examples of these are given and their limitationsdiscussed. We will then consider the more realistic Navier–Stokes equations and the equations fora compressible fluid.
Recommended Books
A.R. Paterson, A First Course in Fluid Dynamics, CUP, ISBN 0521274249A.J.M. Spencer, Continuum Mechanics, Longman, ISBN 0582442826G.A. Holzapfel, Nonlinear Solid Mechanics, Wiley, ISBN 0471823198G.K. Batchelor, An Introduction to Fluid Mechanics, CUP, ISBN 0521098173D.J. Acheson, Elementary Fluid Dynamics, OUP, ISBN 0198596790.
None of these covers everything but the book by Paterson gives reasonable coverage of the fluidspart of the course.
Preliminary Reading
This course is based on vector calculus (Analysis in Many Variables II). In particular, please re-view: gradient, divergence and curl (including Gauss’ and Stokes’ theorems); partial differentialequations and Fourier series.
M.J. Lighthill, An informal introduction to theoretical fluid mechanics, OUP, ISBN 0198536305.
4H reading material references
P. G. Drazin and W. H. Reid, Hydrodynamic Stability, Cambridge (1981), ISBN 0521525411U. Frisch, Turbulence, Cambridge (1996), ISBN 0521457130
32
Outline of course Continuum Mechanics III/IV
Aim: To introduce a mathematical description of fluid flow and other continuous media to fa-miliarise students with the successful applications of mathematics in this area of modelling. Toprepare students for future study of advanced topics.
Term 1 (20 lectures)
Description of fluid flows: continuum hypothesis, velocity field, particle paths and streamlines,vorticity and compressibility.
Euler equations: ideal (‘perfect’) fluid.
Bernoulli’s equation: irrotational flows, water waves.
Navier-Stokes equations3: viscous fluids and boundary conditions; boundary layers.
Term 2 (18 lectures)
Examples: flows in a channel and a pipe, vortices, aerodynamic lift [this will likely be spreadthroughout the course as appropriate].
Compressible flows: sound waves.
Elastic media: time permitting.
4H reading material: a topic in Hydrodynamic Stability and Turbulence.
3material partly covered in Term 2
33
3.4.8 DECISION THEORY III – MATH3071 (38 lectures)
Dr M. Troffaes
Decision theory concerns problems where we have a choice of decision, and the outcome of ourdecision is uncertain (which describes most problems!). Topics for the course include:
(1) Introduction to the ideas of decision analysis.
(2) Decision trees - how to draw and how to solve.
(3) Quantifying rewards, as utilities - informal ideas, formal construction, multi-attribute utility.
(4) Quantifying uncertainties as subjective probabilities - informal notions, formal construction,sensitivity analysis.
(5) Using data in decision making - statistical decision theory, Bayesian inference, sequential anal-ysis.
(6) Bargaining problems.
(7) Game theory.
(8) Group decisions.
(9) Representing decision problems using influence diagrams.
Recommended Books
J.Q. Smith, Decision Analysis - A Bayesian Approach, Chapman & Hall, 1988, (paperback).D.V. Lindley, Making Decisions, (2nd edition), Wiley (1985), paperback.S. French, Decision Theory: An Introduction to the Mathematics of Rationality, Ellis Hor-wood, Chichester (1986), (paperback).M.H. DeGroot, Optimal Statistical Decisions, McGraw-Hill.R.T. Clemen, Making Hard Decisions, (2nd edition), Duxbury Press (1995).
Preliminary Reading
While any of the recommended books would make a good preparation for the course (except DeGroot, which is rather too technical) you are advised to look at Making Decisions by Lindley,which gives an interesting and fairly painless introduction to some of the most important ideas.
34
Outline of course Decision Theory III
Aim: To describe the basic ingredients of decision theory, for individuals and for groups, and toapply the theory to a variety of interesting and important problems.
Term 1 (20 lectures)
Introduction to Decision Analysis: Decision trees, uncertainties and values, solution by backwardinduction. Perfect information.
Utility: Quantifying values - informal ideas and formal construction - Properties. Elicitation.Utility of money, global and local risk aversion.
Multiattribute Utility: Concepts, utility independence, elicitation.
Uncertainty: Quantification of uncertainty as probability, formal construction, properties. Use ofinformation in decision making. Bayes theorem. Value of information.
Term 2 (18 lectures)
Statistical Decision Theory: Bayes risk. Bayes decisions. General Solution. Conjugate analysis.General quadratic loss, Sequential decisions.
Group Decisions and Social Choice: Social welfare functions. Arrows theorem. Sufficient con-ditions for Utilitarianism.
Bargaining: Feasible regions. Rationality conditions. Nash point. Equitable distribution point.
Game Theory: Two person-zero sum games. Minimax theorem. Solutions of two player games.Prisoners dilemma.
Influence Diagrams: Graphical representation of decisions by influence diagrams. Conditionalindependence relations on diagrams. Operations on diagrams.
35
3.4.9 DIFFERENTIAL GEOMETRY III – MATH3021 (38 lectures)
Dr F. Tari / Dr W. Klingenberg
Differential geometry is the study of curvature. Historically it arose from the application of thedifferential calculus to the study of curves and surfaces in 3-dimensional Euclidean space. Todayit is an area of very active research mainly concerned with the higher-dimensional analogues ofcurves and surfaces which are known as n-dimensional differentiable manifolds, although therehas been a great revival of interest in surfaces in recent years. Differential geometry has beenstrongly influenced by a wide variety of ideas from mathematics and the physical sciences. Forinstance, the surface formed by a soap film spanning a wire loop is an example of a minimal surface(that is, a surface whose mean curvature is zero) but the ideas and techniques involved in analysingand characterising such surfaces arose from the calculus of variations and from Riemann’s attemptto understand complex analysis geometrically. The interplay of ideas from different branches ofmathematics and the way in which it can be used to describe the physical world (as in the case ofthe theory of relativity) are just two features which make differential geometry so interesting.
In order to keep the treatment as elementary and intuitive as possible this level III course will bealmost entirely devoted to the differential geometry of curves and surfaces, although most of thematerial readily extends to higher dimensions. The techniques used are a mixture of calculus,linear algebra, and topology, with perhaps a little material from complex analysis and differentialequations. The course will follow the notes of Woodward and Bolton, and the book by Do Carmois also be very suitable. You are recommended to buy either of these. Spivak’s books are interest-ing, well written and useful for reference. If you like the format of the Schaum series, then youmay find Lipschutz’s book helpful.
Recommended Books
M. do Carmo, Differential Geometry of Curves and Surfaces, Prentice-Hall.S. Lipschutz, Differential Geometry, Schaums Outline Series, McGraw-Hill.M. Spivak, Differential Geometry, Vols. II & III, Publish or Perish.* L.M. Woodward and J. Bolton, Differential Geometry (Preliminary Version). Duplicated copiesof this are available for students to buy (total cost approx. £8) from the Departmental Office.
Preliminary Reading
Try looking at the first two chapters of do Carmo or Woodward and Bolton, or Chapters 1-3 ofSpivak II. If you cannot get hold of these, then any book which deals with the differential geometryof curves and surfaces in Euclidean 3-space should be useful.
36
Outline of course Differential Geometry III
Aim: To provide a basic introduction to the theory of curves and surfaces, mostly in 3-dimensionalEuclidean space. The essence of the module is the understanding of differential geometric ideasusing a selection of carefully chosen interesting examples.
Term 1 (20 lectures)
Curves: Plane curves. Arc length, unit tangent and normal vectors, signed curvature, Fundamentaltheorem. Involutes and evolutes. Gauss map, global properties. Space curves. Serret-Frenetformulae. Fundamental theorem. [Global properties]
Surfaces in Rn: Brief review of functions of several variables including differential (with geomet-ric interpretation) and Inverse Function Theorem. Definition of regular surface in and Coordinaterecognition lemma. [Change of coordinates.] Curves on a surface, tangent planes to a surface inRn .
First Fundamental Form: Metric, length, angle, area. Orthogonal and isothermal coordinates.[Orthogonal families of curves on a surface.] [Some more abstract notions such as ‘metrics’ onopen subsets of Rn can be introduced here.]
Mappings of Surfaces: Definitions, differential, expressions in terms of local coordinates. [Con-formal mappings and local isometrics. Examples (conformal diffeomorphisms and isometries ofR2, S2 and helicoid; conformal diffeomorphism of S1(a)×S1(b)⊆ R4 onto torus of revolution inR3; RP2 as Veronese surface in R5 )].
Term 2 (18 lectures)
Geometry of the Gauss Map: Gauss map and Weingarten map for surfaces in R3 , second fun-damental form, normal (and geodesic) curvature, principal curvatures and directions. Gaussiancurvature K and mean curvature H. Second fundamental form as second order approximation tothe surface at a point. Explicit calculations of the above in local coordinates. Umbilics. Compactsurface in R3 has an elliptic point.
Intrinsic Metric Properties: Christoffel symbols, Theorema Egregium of Gauss. Expression ofK in terms of the first fundamental form. Examples. [Intrinsic descriptions of K using arc lengthor area]. Mention of Bonnet’s Theorem. [Surfaces of constant curvature].
Geodesics: Definition and different characterisations of geodesics. Geodesics on surfaces of revo-lution. Geodesic curvature with description in terms of local coordinates.
Minimal Surfaces: Definition and different characterisations of minimal surfaces. Conjugateminimal surfaces and the associated family. [Weierstrass representation.]
Gauss-Bonnet Theorem: Gauss-Bonnet Theorem for a triangle. Angular defect. Relationshipbetween curvature and geometry. Global Gauss-Bonnet Theorem. Corollaries of Gauss-BonnetTheorem. [The Euler-Poincare-Hopf Theorem.]
37
3.4.10 DYNAMICAL SYSTEMS III – MATH3091 (38 lectures)
Dr R. C. Johnson
How does a unicyclist balance upright? A simple model involves the ODE
ϕ = sinϕ−hcosϕ
for angle ϕ(t) of lean from vertical, where the rider’s skill enters through the function h(ϕ, ϕ).
For almost every h there is no formula to be found for ϕ(t). But do solutions exist? If so, how dothey behave - are there any where ϕ is always small and so the rider keeps going?
This course enables you to tackle many interesting questions like this by regarding an ODE as a’vector field on a phase space’ - i.e., as a dynamical system.
It uses straightforward analysis and linear algebra, and develops some topological ideas. Theresults are typically a mix of local approximations and overall properties. Numerical computationuses facilities provided.
Recommended Books
D.K. Arrowsmith and C.M. Place, Dynamical Systems, Chapman & Hall 1992, ISBN 0412390809
P.G. Drazin, Nonlinear Systems, CUP 1992, ISBN 0521406684
P. Glendinning, Stability, Instability and Chaos, CUP 1994, ISBN 0521425662
F. Verhulst, Nonlinear Differential Equations and Dynamical Systems, Springer (2nd Edition)1996, ISBN 3540609342
These are all paperback introductory undergraduate texts with plenty of examples. Their style,content and order of topics all differ, but each includes something on most of the course material.
Any one is likely to be helpful and there is no best buy. Instead arrange with your friends to getdifferent ones, and share.
Preliminary Reading
Read the introduction at maths.dur.ac.uk/∼dma0rcj/PDF/prelude.pdf
38
Outline of course Dynamical Systems III
Aim: To provide an introduction to modern analytical methods for nonlinear ordinary differentialequations in real variables.
Term 1 (20 lectures)
Introduction: Smooth direction fields in phase space. Existence, uniqueness and initial-valuedependence of trajectories.
Autonomous Systems: Orbits. Phase portraits. Equilibrium and periodic solutions. Orbital deriva-tive, first integrals.
Equilibrium Solutions: Linearisation and linear systems. Hartman-Grobman, stable-manifoldtheorems. Phase portraits for non-linear systems, computation. Stability of equilibrium, Lyapunovfunctions.
Term 2 (18 lectures)
Periodic Solutions: Flow, section, maps, fixed points. Brouwer’s Theorem, periodic forcing,planar cycles. Poincare-Bendixson and related theorems. Orbital stability.
Bifurcations: Hopf and other local bifurcations from equilibrium. Uses of bifurcations.
39
3.4.11 ELECTROMAGNETISM III – MATH3181 (38 lectures)
Dr M. Rangamani / Dr S. A. Abel
Electromagnetism constitutes the nineteenth century’s greatest contribution to our understandingof the fundamental laws of nature. It provides a unified mathematical description of electricity,magnetism and light and contains the seeds of special relativity.
The range of phenomena and applications of electromagnetism is immense. The course dealswith the most important of these, building from basic to more complex situations in a step-by-stepfashion.
The major topics covered are:
(i) electrostatics and dielectric media(ii) magnetostatics and properties of circuits with currents(iii) time-dependent fields and electromagnetic waves(iv) special relativistic formulation of electromagnetism
Recommended Books
The following books have helpful discussions:
R.P. Feynman, R.B. Leighton and M. Sands, Feynman Lectures on Physics Vol. 2, Addison-Wesley 1971; £30.69 (hardback)W.J. Duffin, Electricity and Magnetism, W.J. Duffin Publishing, 2001; £24.80 (paperback)
A classic, slightly more advanced book (nonetheless very useful in its simpler sections) is
J.D. Jackson, Classical Electrodynamics, Wiley 1998; £66.45 (hardback)
Many other books are available in college and university libraries.
Preliminary Reading
A prerequisite for the course is a good knowledge of vector analysis, as covered in the 2H Analysisin Many Variables course. To refresh your memory, you could read the early chapters of Spiegel’sVector Analysis (or equivalent).
Volume 1 of the Feynman Lectures on Physics is excellent background reading for more generalaspects of the course, especially the sections on relativity and electromagnetic radiation.
40
Outline of course Electromagnetism III
Aim: To appreciate classical electromagnetism, one of the fundamental physical theories. Todevelop and exercise mathematical methods.
Term 1 (20 lectures)
Electrostatics (12): From experiment: Coulomb’s law and vector superposition of forces dueto different charges. Units: discussion of the possibilities, choice. Electric fields due to pointcharges and to volume and surface distributions. Electric field E expressed in terms of electrostaticpotential φ . Differential equations for the electrostatic field. Force density for a charge distributionρ in an electrostatic field. Electric dipoles. Multipole expansion of the electrostatic potential.Dielectric media and electrostatic polarisation, the phenomenological D field. Perfect conductors;the force per unit area on a conductor in an electrostatic field. Electrostatic energy.
(It may be necessary to review some or all of the following topics from mathematical methods:vector algebra, vector analysis, solid angle, delta functions, Green’s functions, Fourier integraldecompositions.)
Magnetostatics (8): From experiment: the force between current-carrying loops. Current densityj and conservation of charge; integral form and continuity equation. Magnetic field B due to acurrent-carrying loop; extension to volume distribution of current. Differential equations for B.Vector potential A and its differential relations.
Term 2 (18 lectures)
Magnetostatics (3): Force density on a current density in a magnetic field. Magnetic dipoles andthe gyromagnetic ratio. Permeable media, magnetisation and the phenomenological H.
Time-dependent Fields and Maxwell’s Equations (10): From experiment: Faraday’s law. Maxwell’sequations with microscopic sources and in simple media. Potentials and gauge invariance. Waveequation. Energy and momentum conservation (including energy density, Poynting vector, stresstensor). Plane waves. Polarisation. Transmission and reflection at plane boundaries (betweendielectrics, between the vacuum and a conductor). Retarded potentials.
Special Relativistic Formulation of Electromagnetism (5): Invariances of Maxwell’s equations:the equations with microscopic sources expressed as tensor equations in Minkowski spacetime.Relativistic equation of motion for a charged particle in an external electromagnetic field.
41
3.4.12 ELLIPTIC FUNCTIONS III AND IV – MATH3221/MATH4151 (38 lectures)
Dr J. R. Parker
The name elliptic function arises from the problem of finding the arc length of an ellipse, whichleads to an integral that cannot be evaluated by elementary functions. The idea of ‘inverting’ suchintegrals, rather like finding sin−1(x) as an integral, led to the theory of doubly periodic functions.The development of that theory, by Jacobi and Weierstrass and others, was one of the high pointsof nineteenth century mathematical achievement. Moreover, it is an ingredient of much currentresearch, from number theory and geometry to representation theory and theoretical physics. Theproof of Fermat’s last theorem uses modular functions and forms, which are closely related toelliptic functions and which will be mentioned towards the end of the course.
In this course we will cover the classical theory and touch on a few applications. We will makeextensive use of techniques from complex analysis covered in MATH2011, which is a prerequisitefor this course. In addition we will use a little geometry and number theory.
Recommended Books
* G.A. Jones & D. Singerman, Complex Functions, Cambridge University Press 1987, ISBN052131366X; £33.
* J.V. Armitage & W.F. Eberlein, Elliptic Functions, Cambridge University Press, 2006,ISBN 0521785634; £27.
E.T. Copson, Introduction to the Theory of Functions of a Complex Variable, Oxford1935, ISBN 0188531451; out of print but available from Amazon at around £12.
E.T. Whittaker & G.N. Watson, A Course of Modern Analysis, Cambridge University Press[4th Edition 1996], ISBN 0521588073; £49
Derek F. Lawden, Elliptic Functions and Applications, Springer-Verlag, New York 1989,ISBN 0387969659; £66
All of these books contain more material than is needed for the course. We will use Jones andSingerman, Chapters 1, 3 and 6. The only problem is that it does not cover the theory of Jacobifunctions. The book by Armitage and Eberlein contains all the material for the course but takesit in a different order from the one we shall follow in the lectures. Both books are well-writtenand inexpensive and you are strongly recommended to buy one or both of them. There are goodtreatments of elliptic functions in many books on complex analysis, for example Chapters 20, 21and 22 of Whittaker and Watson or Chapters 13, 14 and 15 of Copson. Copson is out of printbut available used via Amazon and Whittaker and Watson is published as a relatively inexpensivepaperback. You may be able to find both of these in your college library. The book by Lawden hasa special emphasis on the applications in applied mathematics and engineering.
Preliminary Reading
Read carefully Chapter 1 of Jones & Singerman and try problems 1E to 1N at the end. Then browsethrough Chapter 3.
42
Outline of course Elliptic Functions III/IV
Aim: To introduce the theory of multiply-periodic functions of one complex variable and to developand apply it.
Term 1 (20 lectures)
Motivation: Elliptic integrals, sums of squares, simple pendulum.
General Properties of Elliptic Functions: Review of necessary complex analysis. Periodic func-tions. Necessary conditions on zeros, poles and residues. Topological properties of elliptic func-tions.
Weierstrass Functions: Convergence / divergence of ∑′ |ω|α. Weierstrass σ, ζ, ℘ functions. Dif-
ferential equation. Expression of general elliptic functions in terms of ℘(z) and ℘′(z). Construc-tion of elliptic functions with given zeros and poles, and with given principal parts. The additiontheorem.
Term 2 (18 lectures)
Jacobi Functions: The functions C(z), S(z) and D(z). The Jacobi functions cn(z), sn(z) anddn(z). The differential equation. The addition theorem. Jacobi functions of a given modulus.Elliptic integrals.
Modular Forms and Functions: Lattices tori and moduli. The modular group. Discriminant of acubic polynomial. The modular function J. Analytic properties of g2, g3, J and ∆. The λ function.Picard’s theorem.
4H reading material: to be announced.
43
3.4.13 GALOIS THEORY III – MATH3041 (38 lectures)
Prof. V. Abrashkin
The origin of Galois Theory lies in attempts to find a formula expressing the roots of a polynomialequation in terms of its coefficients. Evarist Galois was the first mathematician to investigatesuccessfully whether the roots of a given equation could in fact be so expressed by a formula usingonly addition, subtraction, multiplication, division, and extraction of n-th roots. He solved thisproblem by reducing it to an equivalent question in group theory which could be answered in anumber of interesting cases. In particular he proved that, whereas all equations of degree 2, 3 or4 could be solved in this way, the general equation of degree 5 or more could not. Galois Theoryinvolves the study of general extensions of fields and a certain amount of group theory. Its basicidea is to study the group of all automorphisms of a field extension. The theory has applicationsnot only to the solution of equations but also to geometrical constructions and to Number Theory.
Recommended Books
*I. Stewart, Galois Theory, Chapman and Hall, ISBN 0412345404; £23.99E. Artin, Galois Theory, Dover, ISBN0486623424; about £6J. Rotman, Galois Theory, SpringerM.H. Fenrick, Introduction to the Galois Correspondence, BirkhaserD.J.H. Garling, A Course in Galois Theory, Cambridge University PressP.J. McCarthy, Algebraic Extensions of Fields, ChelseaH.M. Edwards, Galois Theory, SpringerB.L. Van der Waerden, Modern Algebra, Vol. 1, Ungar.
44
Outline of course Galois Theory III
Aim: To introduce the way in which the Galois group acts on the field extension generated by theroots of a polynomial, and to apply this to some classical ruler-and-compass problems as well aselucidating the structure of the field extension.
Term 1 (20 lectures)
Introduction: Solving of algebraic equation of degree 3 and 4. Methods of Galois theory.
Background: Rings, ideals, a ring of polynomials, fields, prime subfields, factorisation of polyno-mials, tests for irreducibility.
Field Extensions: Algebraic and transcendental extensions, splitting field for a polynomial, nor-mality, separability.
Fundamental Theorem of Galois Theory: Statement of principal results, simplest properties andexamples.
Term 2 (18 lectures)
Galois Extensions with Simplest Galois Groups: Cyclotomic extensions, cyclic extensions andKummer Theory, radical extensions and solvability of polynomial equations in radicals.
General Polynomial Equations: Symmetric functions, general polynomial equation and its Ga-lois group, solution of general cubic and quartic, Galois groups of polynomials of degrees 3 and 4,non-solvability in radicals of equations of degree ≥ 5.
Finite Fields: Classification and Galois properties, Artin-Schreier Theory, construction of irre-ducible polynomials with coefficients in finite fields.
Ruler and Compass Constructions: Definition, criterion for constructibility, impossibility oftrisecting angle, etc., condition on n for constructibility of n-gon, constructions (e.g. 17-gon).
45
3.4.14 GENERAL RELATIVITY III AND IV – MATH3331/MATH4051 (38 lectures)
To be given in 2009 -2010
Towards the end of the last century, it became clear that Newtonian mechanics breaks down inthree different areas: the world of the very small, the world of the very fast, and the world ofthe very massive. To cope with these, physicists invented three different extensions of Newtonianmechanics: quantum mechanics, special relativity and general relativity.
Before beginning General Relativity, we briefly review special relativity. General Relativity in-volves curved spacetime, and gravity is embodied in it. We sketch the notions of differential ge-ometry which allow us to discuss curvature in four dimensions. We study Einstein’s field equationswith applications to the classical tests of general relativity, black holes and cosmological models.
Recommended Books
Start by browsing some of the links on the course web page, and by looking at one or more non-technical books such as:R. Geroch, General Relativity from A to B, University of Chicago Press 1981K.S. Thorne, Black Holes and Time Warps: Einstein’s Outrageous Legacy, W.W. Norton, NewYork 1995R.M. Wald, Space, Time and Gravity: the Theory of the Big Bang and Black Holes, Universityof Chicago Press 1992.
The books below are all worth looking at.Relatively easy:J.Hartle, GRAVITY: An introduction to Einstein’s General Relativity, Addison Wesley 2003L.P. Hughston and K.P. Tod, An Introduction to General Relativity, Cambridge 1994B.F. Schutz, A First Course in General Relativity, Cambridge 1985 W. Rindler, Introduction toSpecial Relativity, Oxford 1991R. d’Inverno, Introducing Einstein’s Relativity, Oxford 1992Relatively hard:S. Carroll, Lecture Notes on General Relativity, Addison Wesley 2004
http://pancake.uchicago.edu/∼carroll/notes/C.W. Misner, K.S. Thorne and J.A. Wheeler, Gravitation, Freeman 1973R.M. Wald, General Relativity, University of Chicago 1984Preliminary Reading:As preliminary reading, Hartle is to be recommended for a more intuitive approach, and the first3 chapters of Houghston & Tod, the first 4 chapters of Schutz, or the final 3 chapters of Rindler’sIntroduction to Special Relativity, would be useful.4H reading material references:R. Geroch and G.T. Horowitz, Global structure of spacetimes, in the book General Relativity:An Einstein centenary survey, ed. by S.W. Hawking and W. Israel, Cambridge U. Press 1979S. W. Hawking and G.F.R. Ellis, The Large Scale Structure of Space-Time, New York: Cam-bridge U. Press, 1975
46
Outline of course General Relativity III/IV
Aim: To appreciate General Relativity, one of the fundamental physical theories. To develop andexercise mathematical methods.
Term 1 (20 lectures)
Introduction to General Relativity: differences between GR & SR, gravity as geometry, equiva-lence principle.
Special Relativity: spacetime diagrams, line element, vectors and tensors, electromagnetism,stress-energy tensor.
Differential Manifolds: spacetime is a manifold, coordinates and coordinate transformations, tan-gent vectors, tensors revisited.
Metric: distance relationships, light cones, Riemann normal coordinates.
Covariant Derivative: inadequacy of partial derivatives, parallel transport, connection coeffi-cients, differentiating tensors, metric connection, geodesics.
Curvature: Riemann tensor, characterisation of flat space, parallel transport around closed curves,commutation formulae, Bianchi identity, Einstein tensor, geodesic deviation.
Term 2 (18 lectures)
General Relativity: equivalence principle, physics in curved spacetime, Einstein’s equations, lin-earized theory and Newtonian limit, Einstein-Hilbert action.
Black Holes: spherical symmetry, Schwarzschild solution, geodesics, solar-system applications,event horizon and Kruskal coordinates, black hole formation.
Cosmology: isotropy and homogeneity, FRW metric, examples of cosmologies, Hubble law, par-ticle horizons.
4H reading material: a topic related to global structures of spacetime (causal properties and topol-ogy) .
47
3.4.15 GEOMETRY III AND IV – MATH3201/MATH4141 (38 lectures)
Dr N. Peyerimhoff / Dr M.V. Belolipetsky
The course deals with the relationship between group theory and geometry as outlined by FelixKlein in the late 19C in his famous ‘Erlanger Program’. This will be discussed through differenttypes of geometries in 2-dimensions, namely Euclidean, affine, projective, spherical and hyperbolicgeometry.
Each of these geometries may be characterized as the study of properties of figures which arepreserved by a certain group of transformations. For example, Euclidean geometry of the planeis concerned with properties which depend on the notion of distance. The corresponding groupof distance preserving transformations is the Euclidean group consisting of translations, rotations,reflections and glide reflections. The affine group, which contains the Euclidean group, is the groupof transformations of the plane which send lines to lines and the corresponding geometry, whichis more general than Euclidean geometry, is called affine geometry. Projective geometry, which isa further generalization, provides a powerful tool for describing many aspects of the geometry ofthe plane.
Another classical geometry which is called spherical has its origin in the cartography as it is relatedto the geometry of the Earth surface. Passing through its “absolute limit”, the Mobius geometrydealing with the group of Mobius transformations of the extended complex plane, we will en-counter hyperbolic geomtry. Its discovery in the first half of the 19C has had a profound effect onthe foundations of mathematics and on various aspects of mathematical physics, especially rela-tivity theory. The study of geometry of the hyperbolic plane is the final and essential part of thecourse.
Recommended BooksH.S.M. Coxeter, Introduction to Geometry, Wiley, published 1963. A superb introduction togeometry and full of good things. Many ideas from the course are presented in an elementary way.E. Rees, Notes on Geometry, Springer. A nice and compact set of notes containing a good dealbut not all of the material for the course.D. Brannan, M. Esplen, J. Gray, Geometry, Cambridge University Press. An excellent and pre-cisely written textbook with a lot of examples and exercises. Will be used as a basic reference formost of the course except hyperbolic geometry.J. Anderson, Hyperbolic Geometry, Springer-Verlag. The book provides a selfcontained intro-duction to the subject.S. Katok, Fuchsian groups, Chicago Lectures in Mathematics. This is a one more excellent set oflecture notes, the first 3 chapters provide a solid introduction to hyperbolic geometry.E.B. Vinberg (Ed.), Geometry II, Encyclopaedia of Mathematical Sciences, Vol. 29, Springer-Verlag. Presents another view on the subject which has many overlaps with the course, can be usedas a source for independent further study.
Preliminary ReadingThe following is general background reading: Encyclopaedia Britannica (articles on Geometryand Perspective).M. Greenberg, Euclidean and non-Euclidean Geometry, Freeman.D. Hilbert and S. Cohn-Vossen, Geometry and the Imagination, Chelsea.M. Kline, Mathematics in Western Culture, (Chapter 26), Pelican.
48
Outline of course Geometry III/IV
Aim: To give students a basic grounding in various aspects of plane geometry. In particular, toelucidate different types of plane geometries and to show how these may be handled from a grouptheoretic viewpoint.
Term 1 (20 lectures)
Introduction to Klein’s Erlanger Program: The relationship between group theory and geome-try.
Isometries of R2: The Euclidean group as group of isometries, generators, fixed points. Conjugacyclasses. Discrete subgroups of the Euclidean group. Fundamental domains. Torus and Klein bottleas quotients of R2 by discrete subgroups which act freely.
Affine Transformations of R2: The affine group, triple transitivity. Conjugacy classes (statementonly) and examples. Proof that every collineation is affine. Affine geometry. Affine ratio. Triangleof reference, affine coordinates and collinearity. Ceva and Menelaus Theorems.
Projective Geometry: Projective plane RP2 as an extension of R2 , RP2 as S2/{±1}, RP2 \{p} as Mobius band. Homogeneous coordinates. Fundamental Theorem of Projective Geometry.Projective group, affine group as a subgroup. Collineations are projective. Lines. Projectionfrom a point. Duality. Desargues and Pappus Theorems and their duals. Cross-ratio, harmonicquadrangles.
Term 2 (18 lectures)
Spherical Geometry: Isometries of R3 and S2. Brief review of finite subgroups of O(3) and reg-ular solids, fundamental domains. Spherical trigonometry. Spherical version of Ceva’s Theorem.The Riemann sphere and stereographic projection. Stereographic projection is circle preservingand conformal. Metric in terms of complex numbers.
Mobius Transformations: Mobius transformations. Triple transitivity. Inversion, the inversivegroup. Similarities of R2 and isometries of S2 as elements of the inversive group. Circle-preservingtransformations of S2 are inversive. Orthogonal circles. Coaxal circles. Cross-ratio. Characteriza-tion of a circle through three points in terms of cross-ratio.
Hyperbolic Geometry: Upper half-plane model of the hyperbolic plane. The hyperbolic metric.Hyperbolic lines. PSL(2,R) as a transitive group of isometries. Hyperbolic area and the Gauss-Bonnet formula. Hyperbolic trigonometry.Poincare and Klein disc models of hyperbolic plane. Euclid’s axioms, parallel postulate. Parallellines in the hyperbolic plane. Angle of parallelism. The importance of hyperbolic geometry in thehistory of mathematics.Conjugacy classes of isometries. Discrete subgroups of PSL(2,R). Fundamental domains. Ex-amples of tessellations of the hyperbolic plane. Brief treatment (by example) of surfaces of genusgreater than 1 as quotients of the hyperbolic plane by discrete subgroups which act freely.
4H reading material: to be announced.
49
3.4.16 INDEPENDENT STUDY III – MATH3161
To be given in 2009 -2010
The aim of this module is to develop student self-sufficiency by enabling you to learn for yourselfsome selected topics.
There are no lectures. A book which is suitable for independent study at this level is specified. Thesyllabus is defined by that book or identified parts of it. Problems will be set and marked as forother Level 3 modules.
*S. M. Ross, An Elementary Introduction to Mathematical Finance: Options and other top-ics, second edition, CUP 2003, ISBN 0521814294
In the world of finance a call option gives the holder the right, but not the obligation, to buy somecommodity at an agreed price at or before a specified time. For options to be useful in the worldeconomy it is necessary that their value be, at least approximately, computable and for call optionsthis is done via the Black-Scholes formula.
This short textbook (270 pages, price £28) has as its main objective the explanation and derivationof the Black-Scholes formula. This requires considerable use of probability. The book containssome introductory chapters treating what is needed but a good understanding of Core MathematicsA Probability will be of definite benefit. The text then moves on to consider basic notions likeinterest rates and present values and the key concept of arbitrage. Combining these ideas witha simple approximate model of geometric Brownian motion, a simple derivation of the Black-Scholes formula is presented.
The reminder of the book looks at: generalising the ideas treated to cover the use of expected utility(in place of present value); nonstandard options that require simulation methods for a computationneeded for Black-Scholes; models of commodity price changes which generalise the geometricBrownian motion model used earlier.
Students taking this module may well be asked to use a computer package (e.g. MAPLE, MAT-LAB, R ...) to do some simple simulations and fit some basic linear models to data.
In this course an introduction to option pricing is given based on a stochastic modelling approach.The pde methods for solving the resulting equations will not be explored in detail.
Preliminary Reading
Chapters 1 and 2 of the book by Ross.
50
Outline of course Independent Study III
Aim: To develop student self-sufficiency by enabling the students to learn for themselves someselected topic in mathematical sciences.
Geometric Brownian Motion: Geometric Brownian Motion, derivation, Brownian Motion.
Interest Rates and Present Value Analysis: Interest rates, present value analysis, rate of return,continuously varying interest rates.
Pricing Contracts via Arbitrage
The Arbitrage Theorem: Arbitrage theorem and its proof, multiperiod Binomial model.
The Black-Scholes Formula: Black-Scholes formula and its properties, Delta hedging arbitragestrategy, derivation of the Black-Scholes formula.
Additional Results on Options: Call options on dividend-paying securities, American put op-tions, jumps to Geometric Brownian motion, estimating the volatility parameter.
Valuing by Expected Utility: Limitations of the arbitrage pricing, expected utility, portfolio se-lection problem, (conditional) value at risk, Capital Asset Pricing model, mean variance analysisof risk-neutral-priced call options, rates of return.
Exotic Options: Barrier options, Asian and lookback options, Monte Carlo simulation, pricingexotic options by simulation, more efficient simulation estimators.
Beyond Geometric Brownian Motion Models
Autoregressive Models and Mean Reversion: Autoregressive model, valuing options by theirexpected returns, mean reversion.
51
3.4.17 MATHEMATICAL BIOLOGY III – MATH3171 (38 lectures)
Prof. B. Straughan
Mathematical Biology is one of the most rapidly growing and exciting areas of applied mathe-matics. Mathematicians can help biologists to understand very complex problems by developingmodels for biological situations and then providing a suitable solution. Of course, the mathemati-cian must also listen to the biologist to see what important questions need answering!
Mathematics is now being applied in a wide array of biological and medical contexts and profes-sionals in this field are recognising the benefits of having mathematicians working alongside. Forexample, ECG readings of the heart are an active area where mathematicians are developing waysof recognising the signal without the need for an experienced cardiologist to read every recording.MRI brain scans present an image recovery problem where mathematicians are leading the way insolution. These are two of the many exciting areas mathematicians are helping the biological andmedical world. In this course we shall examine some fundamental biological problems and seehow one can go about developing a model to describe the biology and predict some of the featuresnecessary.
Recommended Books
J.D. Murray, Mathematical Biology I. An Introduction, Springer, ISBN 0387952233; (about£20)J.D. Murray, Mathematical Biology II. Spatial models and biomedical applications, Springer,ISBN 0387952284; about £57Segel, Lee, A. Modelling dynamic phenomena in molecular and cellular biology, CambridgeUniversity Press, ISBN 052127477X; about £20Keener, J. and Sneyd, J. Mathematical Physiology, Springer, ISBN 0387983613; about £50
None of these books covers the course entirely. However, the course is covered by all four. Segel’sbook is easiest to read. Murray’s are excellent. Keener and Sneyd is a very good account of moremedical applications of mathematics.
52
Outline of course Mathematical Biology III
Aim: Study of non-linear differential equations in biological models, building on level 1 and 2Mathematics.
Term 1 (20 lectures)
Introduction to the Ideas of Applying Mathematics to Biological Problems
Reaction Diffusion Equations and their Applications in Biology: Reaction diffusion and chemo-taxis mechanisms. Application of the classical diffusion equation to dispersal of insects. Realisticmodelling of insect dispersal with a nonlinear diffusion equation. Application of a nonlinear dif-fusion equation to patterns formed by herrings and gulls. Linear and nonlinear stability for thediffusion equation. Chemotaxis and slime mould aggregation.
ODE Models in Biology: Enzyme kinetics. A little ODE stability theory. The chemostat andbacteria production.
Term 2 (18 lectures)
The Formation of Patterns in Nature: Pattern formation mechanisms, morphogenesis. Questionssuch as how does a tiger get its stripes. Diffusion driven instability and pattern formation. Thefamous work of Turing dealing with how patterns occur in nature. Conditions under which Turinginstabilities will not form.
Epidemic Models and the Spread of Infectious Diseases: Epidemic models. Spread of infectiousdiseases. Simple ODE model. Spatial spread of diseases. Spatial spread of rabies among foxes.Stability in epidemics and predator - prey models.
53
3.4.18 MATHEMATICAL FINANCE III AND IV –MATH3301/MATH4181 (38 lectures)
Dr I. M. MacPhee / Dr M. Jensen
Finance is one of the fastest developing areas in the modern banking and corporate world. This,together with the sophistication of modern financial products, results in a need for new mathe-matical models and modern mathematical methods. The demand from financial institutions forwell-qualified mathematicians is great.
This course describes the modeling of financial derivative products from an applied mathemati-cian’s viewpoint. We shall examine options and markets, asset price random walks, the Black-Scholes model, and the mathematics associated with this. A significant part of the course is de-voted to numerical methods for solving the Black-Scholes equations, including finite differenceand Monte-Carlo schemes.
Recommended Books
*D. Higham, Introduction to Financial Option Valuation: Mathematics, Stochastics and Com-putation, CUP, ISBN 0521547571, (the main reference for the course though it doesn’t cover ev-erything).R. Seydel, Tools for Computational Finance, Springer, ISBN 3540279237, (complementary toHigham’s book).S. M. Ross, An Elementary Introduction to Mathematical Finance, 2nd Edition, CUP, ISBN0521814294, (emphasises the probability approach).P. Wilmott, S. Howison and J. Dewynne, The Mathematics of Financial Derivatives: a studentintroduction, CUP, ISBN 0521497892, (rather more advanced than the texts listed above).
Preliminary Reading
Chapter 1 of Wilmott et al., ‘An introduction to options and markets’.
4H reading material references
Material will be selected from the books by D. Higham and by Wilmott, Howison and Dewynnewhich are listed above.
54
Outline of course Mathematical Finance III/IV
Aim: To provide an introduction to the mathematical modelling of financial derivative products.
Term 1 (20 lectures)
Introduction to options and markets. Interest rates and present value analysis, asset price randomwalks, pricing contracts via arbitrage.
The Black-Scholes formula. The Black-Scholes formula for the geometric Brownian price modeland its derivation.
Term 2 (18 lectures)
Partial Differential Equations.
The Black-Scholes model revisited.
Numerical methods.
Pricing exotic options by simulation.
More general models (if time). Limitations of arbitrage pricing, volatility, valuing by expectedutility, portfolio selection.
4H reading material: One of the following topics: American options (obstacle problems, freeboundary problems), exotic options, historical volatility.
55
3.4.19 MATHEMATICS TEACHING III – MATH3121 (38 lectures)
Dr D. J. Smith / Dr S. K. Darwin
The course will focus on school mathematics from an advanced standpoint and provides studentswith an opportunity to consider teaching mathematics as a career with no obligation. Students areencouraged to reflect on current issues and pupils’ learning in secondary schools, and to examinethese matters in the light of their own mathematical experience.
Term 1 will start with 3–4 weeks considering current issues in mathematics teaching and learning.Students will then spend about 5 sessions in small groups at local secondary schools on plannedobservation. Visits will be followed by themed group seminars.
Term 2 will include sessions concerned with the history and ideas of various teaching approachesand their background from an advanced standpoint. Students will give a presentation on theirMathematics essay topic which is agreed with the lecturers.
A meeting will take place at 2.15p.m. in CM101 on Monday, June 9th 2008 where more informa-tion will be provided.
Due to the programme of school visits, the number of students on the course is limited to 40. Toregister an interest all eligible students are asked to complete an on-line formwww.dur.ac.uk/douglas.smith/password/mt.php. As the number interested is likely to ex-ceed the cap, selection will be based on a short question set on Wednesday, June 4th 2008 whichstudents will have one week to prepare, for instance an essay on “Discuss whether or not all sixth-form students should study Mathematics at some level”. Students will be informed of the outcomeof the selection process on Friday, June 13th 2008 prior to registration.
Assessment of the module is based on: (a) a 2 12 hour examination; (b) an essay on the school visits;
(c) the Mathematics essay and its related presentation.
Recommended Books
The National Curriculum (Mathematics), HMSO, ISBN 0112708838.The Cockcroft Report, Mathematics Counts, HMSO, ISBN 0112705227.John Mason et al., Thinking Mathematically, Addison-Wesley, ISBN 0201102382G. Polya, Induction and Analogy in Mathematics, Princeton.H.S.M. Coxeter, Introduction to Geometry, Wiley, ISBN 0471182834.
Preliminary Reading
A knowledge of the structure, though not the detail, of national curriculum mathematics would beuseful. As a contrast, have a look at the approach used in “Thinking Mathematically”.
56
Outline of course Mathematics Teaching III
Aim: To encourage the students to reflect on the content, method and learning of mathematics fromthe points of view of school and university and in so doing to gain a fuller understanding of thenature and foundations of the subject and also to acquire skills in presentation of particular topicsthrough talks and seminars.
School Mathematics (4): General background. Pupils’ learning problems. Implications of recentinitiatives including: Cockcroft report, GCSE, National Curriculum.
Topics in School Mathematics (2): Number, algebra, geometry and applied mathematics fromboth pupil and mathematical viewpoints.
Pupils Learning Mathematics (12): Collection of examples from school visits and analysis anddiscussion of material collected from pupil mathematical viewpoints.
School Mathematics from an Advanced Standpoint (20): After some introductory lectures, thestudents are invited to give seminars on a selection of topics whose scope is indicated by thefollowing syllabus. Clearly there is not time to cover everything and other topics in the same spiritmay be proposed: the syllabus is therefore only an indication. Whenever possible students areinvited to pick topics that can be related to other options in the department.
The idea of proof, both from an advanced standpoint and at school level. The development of thereal number system; the real numbers in analysis and the theory of measure; the school stand-point. The solution of equations and the development of algebra, with an emphasis on constructionproblems. Quadratic forms and geometry; the foundations of Euclidean geometry. Backgroundto school geometry - as it was, as it is and as it might be. Arithmetic and number investigationsin school. Other topics from Arithmetic, Algebra, Analysis, Geometry and Applied and Applica-ble Mathematics as proposed by the student and agreed by the lecturer, all in the spirit of Klein’s’Elementary Mathematics from an Advanced standpoint’.
57
3.4.20 NUMBER THEORY III and IV– MATH3031/MATH4211 (38 lectures)
Dr H. Gangl / Dr M. D. KerrWith the recent (1994) resolution of Fermat’s Last Theorem by Andrew Wiles, the study of Dio-phantine Equations (equations where the unknowns are to be integers) has come again to the fore-front of mathematics. In this course we study some of the techniques, mainly based on ideas offactorization and congruence, which have been applied to the Fermat and other Diophantine equa-tions. We find, for example, the number of solutions in positive integers to
x4 + y4 = z4 (the case n = 4 of the Fermat equation)x2 = y3−2 (a case of Mordell’s equation) and
x2−5y2 = 1 (a case of Pell’s equation, closely related to the Fibonacci numbers).(The answers are 0, 1 and ∞, respectively.)Our methods lead us to study number systems (rings) other than that of the ordinary integers (e.g.the Gaussian integers). And we consider what we can do when things do not go according to plan.E.g. in the set of numbers of the form a+bi
√5 with a and b∈Z, we find that 6 has two completely
different irreducible factorizations, 2×3 and (1+ i√
5)(1− i√
5).When uniqueness of factorization breaks down in this way, we are led to consider “ideal num-bers” (which lead to the notion of an ideal). We use this idea to regain a form of uniqueness offactorization.A further goal of the course is the classification of the ideals in a ring. We show, using a prettyapplication of the ‘Geometry of Numbers’, that there are only finitely many ideal classes. Thisleads to the solution of more sorts of diophantine equation.The course provides insight into this fascinating subject and through many examples makes clearthe motivation of algebraic number theory.Recommended Books*I.N. Stewart and D.O. Tall, Algebraic Number Theory and Fermat’s Last Theorem, A.K.Peters, 2001, ISBN 1568811195P. Samuel, The Algebraic Theory of Numbers, Hermann/Kershaw, 1973, ISBN 0901665061Harvey Cohn, Advanced Number Theory, Dover publications, 1980, ISBN 048664023XG.H. Hardy and E.M. Wright, An Introduction to the Theory of Numbers, OUP.A. Baker, A Concise Introduction to the Theory of NumbersK.F. Ireland and M.I. Rosen, A Classical Introduction to Modern Number Theory, SpringerJ.P. Serre, A Course in Arithmetic, SpringerBorevich and Shafarevich, Number Theory, Academic Press*Students are recommended to buy this book or one of the previous editions (entitled just Alge-braic Number Theory). This is a readable account with some historical details and motivationand some further material. Samuel is more abstract and concise but is otherwise also good. Cohn,also, covers much of the course nicely. The other books (in rough order of sophistication) arerecommended for background and further reading.Preliminary ReadingDip into any of the above books except perhaps the last two. In particular, Baker and Hardy andWright give the general flavour of Number Theory well. Revise and extend 2H work by readingAllenby “Rings, Fields and Groups” Chap. 3 esp. 3.5-3.9, or Baker Chap. 7 (you’ll need to referback to Chap. 6 for one or two definitions).
58
Outline of course Number Theory III/IVAim: To provide an introduction to Algebraic Number Theory (Diophantine Equations and IdealTheory).Term 1 (20 lectures)Diophantine Equations Using Elementary Methods: xy = n, x2− y2 = n, Pythagorian triples,x4 + y4 = z2. [Other examples using factorization in Z, e.g. cases of y2 = x(x−a)(x−b)].Unique Factorization or the Lack of it: Recap properties of integers and polynomial rings. Ex-amples such as Z [
√−6 ] . Factorization in monoids (prime vs. irreducible). Applications e.g.
x2 +d = yn with d = 2 and n = 3.Ideals: Principal and multi-generator ideals. Sum, product, associative and distributive laws forideals. Manipulation of generators. Prime and maximal ideals and their characterizations by theirquotient rings. Preliminary discussion of ideal factorization. Examples of recovery of uniquenessof factorization using ideals. Principal ideal domains (⇒ UFD).Euclidean Rings: Norms, in particular Euclidean norms (give PIDs). Examples.Fields: Algebraic number fields. Vector spaces over such. Field extensions: index; tower theorem;simple extensions and minimum polynomials; monogeneity (no proof). Conjugates over Q. Ho-momorphisms of fields. Norms and traces over Q. Matrix formulae for these.Algebraic Integers: The several characterizations of algebraic integers; int(K) is a ring. Integral-ity of norm and trace.Quadratic Fields and Integers: int(Q [
√d ]). Easy Euclidean quadratic fields [up to int(Q [
√29 ])
if time]. Primes. Representation of integers by binary quadratic forms where there is unique fac-torization in the relevant quadratic field.Term 2 (18 lectures)The Discriminant and Integral Bases: Definition, formal properties and calculation of the dis-criminant of a basis of a field over Q. Revision of torsion-free abelian groups. Determinant/indexformula. Integral bases. Discriminants of lattices. Discriminant/index formula. Determination ofan integral basis.Factorization of Ideals: Fractional ideals. Unique factorization and inverses of ideals. The idealgroup. Prime ideals. Factorization of (p) (Dedekind’s theorem) and of other ideals. [Mentionramification and the discriminant.]The Ideal Class Group: The class group. Minkowski’s theorem. Finiteness and determination ofthe class group (full proof of bound formula only for quadratic fields if time is short). Representa-tion of integers by definite binary quadratic forms (problems rather than general theory). Furthercases of x2 +d = yn .Dirichlet’s Unit Theorem: Proof for quadratic fields. Representation of integers by non-definitebinary quadratic forms.4H reading material: to be announced.
59
3.4.21 OPERATIONS RESEARCH III – MATH3141 (38 lectures)
Dr O. Hryniv / Prof. M. MenshikovAs its name implies, operations research involves ”research on operations”, and it is applied toproblems that concern how to conduct and coordinate the operations (activities) within an organi-zation. The nature of the organization is essentially immaterial, and, in fact, OR has been appliedextensively in such diverse areas as manufacturing, transportation, construction, telecommunica-tions, financial planning, health care, the military, and public services to name just a few.This course is an introduction to mathematical methods in operations research. Usually, a mathe-matical model of a practical situation of interest is developed, and analysis of the model is aimedat gaining more insight into the real world. Many problems that occur ask for optimisation of afunction f under some constraints. If the function f and the constraints are all linear, the simplexmethod is a powerful tool for optimisation. This method is introduced and applied to several prob-lems, e.g. within transportation.Many situations of interest in OR involve processes with random aspects and we will introducestochastic processes to model such situations. An interesting area of application, addressed in thiscourse, is inventory theory, where both deterministic and stochastic models will be studied andapplied. Further topics will be chosen from: Markov decision processes; integer programming;nonlinear programming; dynamic programming.
Recommended Books* F.S. Hillier and G.J. Lieberman, Introduction to Operations Research, 8th ed., McGraw-Hill2004, ISBN 007123828xThis excellent book (softcover, £43.99) covers most of the course material (and much more), andthe course is largely developed around this textbook. It is well written and has many useful exam-ples and exercises. It also contains some PC software that can be used to gain additional insight inthe course material. This is the best textbook we know of for this course (the 6th and 7th editionare equally useful).
There are plenty of books with ’operations research’, ’mathematical programming’, ’optimisation’or ’stochastic processes’ in the title, and most of these contain at least some useful material. Beaware though that this course is intended to be an introduction, and most books tend to go ratherquickly into much more detail.
Preliminary ReadingThe first two chapters of the book by Hillier and Lieberman provide an interesting introductioninto the wide area of operations research, and many of the other chapters start with interestingprototype examples that will give a good indication of the course material (restrict yourself to thechapters on topics mentioned above). Appendices 2 and 4 of this book could be read to refreshyour understanding of matrices and the notion of convexity. Most other books within this area(see above) have introductory chapters that provide insight into the topic area, and the possibleapplications of OR.
60
Outline of course Operations Research III
Aim: To introduce some of the central mathematical models and methods of operations research.
Term 1 (20 lectures)Introduction to Operations Research: Role of mathematical models, deterministic and stochasticOR.Linear Programming: LP model; convexity and optimality of extreme points; simplex method;duality and sensitivity; special types of LP problems, e.g. transportation problem.Networks: Analysis of networks, e.g. shortest-path problem, minimum spanning tree problem,maximum flow problem; applications to project planning and control.Introduction to Stochastic Processes: Stochastic process; discrete-time Markov chains.
Term 2 (18 lectures)Inventory Theory: Components of inventory models; deterministic models; stochastic models.Markov Decision Processes: Markovian decision models; the optimality equation; linear pro-gramming and optimal policies; policy-improvement algorithms; criterion of discounted costs;applications, e.g. inventory model.Further Topics Chosen From: Integer programming: Model; alternative formulations; branch-and-bound technique; applications, e.g. knapsack problem, travelling salesman problem.Nonlinear programming: Unconstrained optimisation; constrained optimisation (Karush-Kuhn-Tucker conditions); study of algorithms; approximations of nonlinear problems by linear problems;applications.Dynamic programming: Characteristics; deterministic and probabilistic dynamic programming.
61
3.4.22 PARTIAL DIFFERENTIAL EQUATIONS III AND IV –MATH3291/MATH4041 (38 lectures)
Dr J. P. ColemanThe topic of partial differential equations (PDEs) is central to mathematics. It is of fundamentalimportance not only in classical areas of applied mathematics, such as fluid dynamics and elasticity,but also in financial forecasting and in modelling biological systems, traffic flow, blood flow in theheart and many other processes of practical interest.In this module we are concerned with both theoretical and numerical analysis of PDEs. For somevery simple equations with appropriate boundary conditions we can find explicit solutions. Thatwill be our starting point and it will suggest what our concerns should be in more complicatedsituations. Modern computer power and the availability of numerical algorithms make it easy togenerate numerical approximations for solutions of PDEs. That makes a sound theoretical under-standing all the more important. In particular, it is essential to be clear that the computer is askedto solve a sensible problem.The first term is devoted to a thorough grounding in aspects of the theory of PDEs, together withthe study of some methods for finding explicit solutions when that is possible. The second termis concerned mainly with finite difference methods — their derivation and the study of importantconcepts such as convergence, consistency and stability. Finite difference methods lead us toapproximate a PDE by a (usually large) set of algebraic equations, so we need efficient methodsfor solving algebraic systems; in particular, some iterative methods for linear systems will bediscussed.Recommended BooksNo particular book is recommended for purchase. The first two books listed are concerned withanalytical aspects and are relevant to the first part of the module. The others are useful for thesecond term.J. Ockendon, S. Howison, A. Lacey and A. Movchan, Applied Partial Differential Equations,revised edition, Oxford University Press 2003, ISBN 0198527713;E.C. Zachmanoglou and D.W. Thoe, Introduction to Partial Differential Equations with Appli-cations, Dover 1986, ISBN 0486652513;K.W. Morton and D.F. Mayers, Numerical Solution of Partial Differential Equations, secondedition, Cambridge University Press 2005, ISBN 0521607930;G.D. Smith, Numerical Solution of Partial Differential Equations: Finite Difference Methods,third edition, Oxford University Press 1985, ISBN 0198596502;A. Iserles, A First Course in the Numerical Analysis of Differential Equations, CambridgeUniversity Press 1996, ISBN 0521556554.Preliminary Reading For the flavour of the first part of the course look at Chapter 1 of Ockendonet al. or Chapter III of Zachmanoglou and Thoe.Reference for 4H Additional Reading Specified parts of the book by Ockendon et al. listedabove.Other references will be mentioned during the year.
62
Outline of course Partial Differential Equations III/IV
Aim: To develop a basic understanding of the theory and methods of solution for partial differen-tial equations, and of the ideas of approximate (numerical) solution to certain partial differentialequations.Term 1 (20 lectures)First-order equations and characteristics. Conservation laws.Systems of first-order equations, conservation laws and Riemann invariants.Hyperbolic systems and discontinuous derivatives. Acceleration waves.Classification of general second order quasi-linear equations and reduction to standard form foreach type (elliptic, parabolic and hyperbolic).
Term 2 (18 lectures)Energy methods for parabolic equations. Well-posed problems.Maximum principles for parabolic equations.Finite difference solution to parabolic and elliptic equations.Stability and convergence for solution to finite difference equations.Iterative methods of solving Ax = b.
4H reading material will be on one of the following topics:strict form of the maximum principle for parabolic equations;error estimates for numerical solutions of partial differential equations;further aspects of non-linear partial differential equations.
63
3.4.23 PROBABILITY III AND IV – MATH3211/MATH4131 (38 lectures)
Dr O. Hryniv / Prof. M. Menshikov
Modern probability theory began around 1930 with A.N. Kolmogorov, and continues to flourish.The theory has many applications, probabilistic reasoning being essential to statistics, or whenevera model for an unpredictable future is needed. At the same time, the subject appeals to puremathematicians with a taste for real analysis. Indeed, measure theory, the branch of analysisconcerned with areas and volumes of sets, and the theory of integration, are closely related tothe foundations of probability.In this course we shall cover topics such as independence, the law of large numbers, the centrallimit theorem, conditional probability and expectation, in greater depth than was possible in thefirst and second years probability courses.
Recommended BooksG. R. Grimmett and D.R. Stirzaker, Probability and Random Processes, Clarendon Press, Ox-ford, also Oxford University Press, ISBN 0198572220.(It will also be possible to use this book for the course Stochastic Processes III/IV if you take thatcourse in 2009–2010).The book Probability: theory and examples by R. Durrett published by Duxbury Press, (2ndedition: 1996, ISBN 0-534-24318-5) extends and complements the course material.
Preliminary ReadingYour notes from the 2nd year probability course. The Grimmett-Stirzaker book has a review ofbasic probability in Chapters 1-4. Alternatively you could look at:Y. A. Rozanov, Probability theory, a concise course, Dover, ISBN 0486635449 (paperback).(This is inexpensive and covers the basics.)D. Stirzaker, Elementary Probability, Cambridge University Press. ISBN 0521833442 ( paper-back). (This has many interesting examples and exercises.)
64
Outline of course Probability III/IV
Aim: This module continues on from the treatment of probability in 2H modules MATH2151and MATH2161. It is designed to build a logical structure on probabilistic intuition; to studyclassical results such as the Strong Law of Large Numbers, the Central Limit Theorem, and someapplications; to discuss some modern developments in the subject. Students completing this courseshould be equipped to read for themselves much of the vast literature on applications to biology,physics, chemistry, mathematical finance, engineering and many other fields.
Term 1 (20 lectures)Introductory examples: from finite to infinite spaces.Probability as Measure: σ-fields. Measure and probability spaces. Main properties.Random Variables as measurable functions. Distributions. Generating and characteristic func-tions.Convergence almost surely, in probability, in L2. Borel-Cantelli lemmas. Kolmogorov 0-1 law.
Term 2 (18 lectures)General theory: probability spaces and random variables.Integration: Expectation as integral. Inequalities. Monotone and dominated convergence.Limit results: Laws of large numbers. Weak convergence. The central limit theorem. Largedeviations.Topics Chosen From: Random graphs. Percolation. Ergodic theorems. Stochastic integral.
4H reading material: To be announced.
65
3.4.24 QUANTUM MECHANICS III – MATH3111 (38 lectures)
Dr B. Doyon
Quantum theory has been at the heart of the enormous advances that have been made in our un-derstanding of the physical world over the last century, and it also underlies much of modern tech-nology, e.g., lasers, transistors and superconductors. It gives a description of nature that is verydifferent from that of classical mechanics, in particular being non-deterministic and non-local, andseeming to give a crucial role to the presence of ‘observers’.The course begins with a brief historical introduction, in which the reasons for the failure of clas-sical ideas will be explained, and then introduces the defining postulates and reviews the necessarymathematical tools – basically linear vector spaces and second order differential equations. Appli-cations to simple systems, which illustrate the power and characteristic predictions of the theory,are then discussed in detail. Some problems are solved using algebraic methods, others involvesolutions of partial differential equations.
Recommended BooksF. Mandl, Quantum Mechanics, Wiley 1992, ISBN 0471931551; £25.L.I. Schiff, Quantum Mechanics, McGraw Hill 1969, ISBN 0070856435; £39.A. Messiah, Quantum Mechanics, 2 Volumes, North Holland 2000, ISBN 0486409244; £20P. Dirac, The Principles of Quantum Mechanics, OUP 4th Ed. 1981, ISBN 0198520115; £25A Sudbery, Quantum Mechanics and the Particles of Nature, CUP 1986, ISBN 0521277655.Out of print but in libraryLandau and Lifshitz, Quantum Mechanics. (Non-relativistic Theory) Butterworth-Heinemann1982, ISBN 0080291406; £35Any of the above covers most of the course. There are many other books which are quite similar.
Preliminary ReadingBecause quantum theory is conceptually so different from classical mechanics it will be usefulto have read descriptive, non-mathematical, accounts of the basic ideas, for example from theintroductory chapters of the books above, or from general ‘popular’ accounts e.g., in Polkinghorne,‘The Quantum World’; Pagels, ‘The Cosmic Code’; Squires, ‘Mystery of the Quantum World’, etc.
66
Outline of course Quantum Mechanics III
Aim: To give an understanding of the reasons why quantum theory is required, to explain its basicformalism and how this can be applied to simple situations, to show the power in quantum theoryover a range of physical phenomena and to introduce students to some of the deep conceptualissues it raises.
Term 1 (20 lectures)Problems with Classical Physics: Photo-electric effect, atomic spectra, wave-particle duality,uncertainty principle.Formal Quantum Theory: Vectors, linear operators, hermitian operators, eigenvalues, completesets, expectation values, commutation relations. Representation Theory, Dynamics: Schrodingerand Heisenberg pictures.Spectra of Operators: Position operator, Harmonic Oscillator, Angular momentum.
Term 2 (18 lectures)Waves and the Schrodinger Equation: Time-dependent and time-independentSchrodinger equation. Probability meaning of |Ψ|2 . Currents. Plane-waves, spreading of a wavepacket.Applications in One Dimension: Square well, harmonic oscillator, square barrier, tunnelling phe-nomena.Three dimensional problems Three-dimensional harmonic oscillator, Hydrogen atom, other sys-tems.
67
3.4.25 REPRESENTATION THEORY AND MODULES III AND IV –MATH3191/MATH4101 (38 lectures)
To be given in 2009 -2010A module M over a ring R is an (additive) abelian group supplied with a method of multiplyingelements of M by elements of R. For example, a module over a field K is simply a vector spaceover K and a module over Z is just an abelian group. Like the concepts of group, ring and vectorspace before it, the concept of a module over a ring brings under one roof a number of apparentlydifferent mathematical ideas and has applications in many areas of mathematics.In this course we generalise the Fundamental Theorem of Abelian Groups to get a similar structuretheorem for modules over, e.g., polynomial rings and from this we deduce canonical forms for ma-trices over any field. On another tack, we find that module theory over matrix rings is surprisinglystraightforward.A substantial topic in the course is that of group representations, which are, quite literally, rep-resentations of groups as groups of matrices. These representations correspond to modules overa ring, called the group ring. However, special to this topic is the neat and satisfying calculusof group characters which provides a very striking solution to the problem of determining all therepresentations of a finite group.
Recommended BooksM.E. Keating, A first course in module theory, Imperial College Press, ISBN 186094096XG.James & M.Liebeck, Representations and characters of groups, CUP, ISBN 0521445906B. Hartley & T.O. Hawkes, Rings, modules and linear algebra, C&H, ISBN 0412098105W.Adkins & S.Weintraub, Algebra: an approach via module theory, Springer, ISBN 0387978399S. Lang, Algebra (3rd Ed.), Addison Wesley, ISBN 0201555409J.-P. Serre, Linear Representations of finite groups, Springer-Verlag, ISBN 0387901906V.E. Hill, Groups and Characters, Chapman & Hall 2000, ISBN 1584880384The books by Keating and James & Liebeck together cover most of the course, as does the bookby Lang (good value for the budding algebraist but a bit tough). The book by Hartley and Hawkescovers much of the ground of Keating and is in some ways preferable (but it is out of print). BothAdkins & Weintraub and Serre provide alternative viewpoints.
Preliminary ReadingRevise ring theory and polynomials and read a bit about Euclidean Rings and Modules: e.g. inKeating. (Compare the work of ch. 10.5 with work in ANT II on abelian groups).
4H reading referencesIn addition to the above recommended books the following will be usefulJ. R. Silvester, Introduction to algebraic K-theory, Chapman & Hall, ISBN 0412237407J. W. Milnor, Introduction to algebraic K-theory, Princeton UP, ISBN 0691081018Bruce A. Magurn, An algebraic introduction to K-theory, CUP, ISBN 0521800781Hyman Bass, Algebraic K-theory.The prescribed reading will probably be chosen from the first two chapters of the book by Milnor(with some supporting notes). The book by Silvester is good and in the library but it is otherwisedifficult to obtain. The book by Bass, an awkward read, both mathematically and typographically,but good in parts, can be downloaded as a massive (37.2Mb) pdf file fromwww.math.uni-bielefeld.de/˜rehmann/DML/BOOKS/index.html.
68
Outline of course Representation and Modules III/IV
Aim: To develop and illustrate the theory of modules and that of complex characters of finitegroups.Term 1 (20 lectures)Rings and Ideals: Revision of the definitions of rings, subrings and of (two-sided) ideals and ringhomomorphisms. Examples, especially of non-commutative rings (which have not been muchused previously).Modules and Submodules: Definition of (left and right)-modules, submodules and module ho-momorphisms. Examples. Relation between matrix representations of a group and modules overthe group ring. Quotient modules, kernels and the homomorphism theorems. One sided ideals andcyclic modules. Direct sums, their relationship with internal sum. Endomorphism rings, projec-tions and idempotents. The Chinese Remainder Theorem. Simple and indecomposable modules(are not necessarily the same thing). Free modules and presentations.Modules over Principal Ideal Domains: Examples of principal ideal domains. gcd’s. (Finite)presentation of finitely generated modules. Diagonalization of matrices by integral operations.Fundamental theorem of f.g. modules over PID’s. Relevance to abelian groups. The torsion, orderand exponent of a module.Similarity of Matrices over a Field: Equivalence of square matrices under similarity and modulesover K[X ] under isomorphism. Companion matrices. Jordan canonical form and rational canonicalform. Application to functions of matrices.(Finite dimensional) Semisimple Algebras: Matrix rings over skew fields and their modules.Semisimple algebras, Schur’s lemma. Structure theorems for simple and semi-simple algebras(especially over C). Modules over such algebras.Term 2 (18 lectures)Modules over a Group algebra: The group algebra KG, where K is a field and G is a (chieflyfinite) group. Examples of modules over KG. Maschke’s Theorem. Failure (for e.g. Z2C2).Matrix representations of G and their relationship with modules over KG. The character of amatrix representation or of a KG-module.Representations of Groups over C: Characters of representations over C, revise conjugacyclasses. Irreducible characters. Orthogonality of irreducible characters and of the correspondingidempotents. Number and degree of irreducible characters.Construction of the character table of a finite group, orthogonality properties. Products of charac-ters. Induced representations [Frobenius reciprocity, the Artin induction theorem].Mention Burnside’s prqs theorem and other applications. [Representations over R and Q, YoungTableaux].4H reading material: a topic related to: Modules of finite length; Modules over Dedekind rings;and/or Projective modules and Grothendieck groups.
69
3.4.26 SOLITONS III AND IV – MATH3231/MATH4121 (38 lectures)
Prof. P. E. Dorey / Prof. R. S. Ward
Solitons occur as solutions of certain special nonlinear partial differential equations. In general,nonlinear equations cannot be handled analytically. However, soliton equations (which arise nat-urally in mathematical physics) have beautiful properties; in particular, they admit solutions thatone can write down explicitly in terms of simple functions. Such solutions include solitons: local-ized lumps which can move around and interact with other lumps without changing their shape.The course will describe a number of these soliton equations, explore their solutions, and studysolution-generating techniques such as Backlund transformations and inverse scattering theory.
Recommended Books*P.G. Drazin and R.S. Johnson, Solitons: An Introduction, CUP 1989, ISBN 0521336554T. Dauxois and M. Peyrard, Physics of Solitons, CUP 2006, ISBN 0521854210G. Eilenberger, Solitons: Mathematical Methods for Physicists, Springer 1981, ISBN 354010223XG.L. Lamb, Elements of Soliton Theory, Wiley 1980S. Nettel, Wave Physics, Springer 1995, ISBN 3540585044R. Remoissenet, Waves Called Solitons, Springer 1999, ISBN 3540659196M. Toda, Theory of Nonlinear Lattices, Springer [2nd Ed] 1988See alsoR.K. Bullough and P.J. Caudrey (eds), Solitons, Springer 1980G.B. Whitham, Linear and Nonlinear Waves, Wiley 1974
The book by Drazin and Johnson covers most of the course, and is available in paperback.
Preliminary ReadingRead chapter 1 of Drazin and Johnson, and study the exercises for that chapter (note that theanswers are given at the end of the book!).Explore the website www.ma.hw.ac.uk/solitons/ (in particular the movies and the pages onScott-Russell’s soliton).
70
Outline of course Solitons III/IVAim: To provide an introduction to solvable problems in nonlinear partial differential equationswhich have a physical application. This is an area of comparatively recent development which stillpossesses potential for growth.Term 1 (20 lectures)Nonlinear Wave Equations: Historical introduction; John Scott Russell and the first experimentalobservation of a solitary wave (1834); Korteweg and de Vries’ wave equation (1905). The sine-Gordon equation and its connection with surfaces of constant negative Gaussian curvature. Solitonscattering (experimental, John Scott Russell; theoretical, Skyrme and Perring). Properties of non-linear wave equations; dispersion, dissipation: dispersion law for linear equations; group velocityand phase velocity. Examples.Progressive Wave Solutions: Derivation of the sine-Gordon equation as the limit of the motionunder gravity of a set of rigid pendula suspended from a torsion wire. Progressive wave solutions:D’Alembert’s solution recalled, and progressive wave solutions of KdV and sine-Gordon equa-tions. Topological lumps and the Bogomolnyi bound.Backlund Transformations for Sine-Gordon Equation: Sine-Gordon as the only Lorentz-invariantequation with auto-Backlund transformations; the Liouville equation and its solution by Backlundtransformations of solutions of the free field equation. Generation of multisoliton solutions ofsine-Gordon by Backlund transformations. Theorem of permutability. Two-soliton and breathersolutions of sine-Gordon. Lorentz-invariant properties. Asymptotic limits of two-soliton solutions.Discussion of scattering.Backlund Transformations for KdV Equation: The same analysis repeated for the KdV equa-tion.Conservation Laws in Integrable Systems: Conservation laws for the wave equation, sine-Gordon and KdV equations. Relation to conservation of soliton number and momentum in scatter-ing.Hirota’s Method: Hirota’s method for multisoliton solutions of the wave and KdV equations.Term 2 (18 lectures)The Nonlinear Schrodinger Equation: The nonlinear Schrodinger equation and its connectionwith the Heisenberg ferromagnet.The Inverse Scattering Method: Discussion of the initial value problem; motivation for the in-verse scattering method. The inverse scattering method for the KdV equation. Lax pairs. Dis-cussion of spectrum of Schrdinger operator for potential; Bargmann potentials. Scattering theory;asymptotic states, reflection and transmission coefficients, bound states. Marchenko equations.Multisoliton solutions of KdV.The Inverse Scattering Method: Two-Component Equations: MKdV equation and Miura trans-formation. Two-component Lax pair formulation of inverse scattering theory. Same for sine-Gordon equation.Toda equations: Conservation laws in integrable finite-dimensional models. The Toda moleculeand the Toda chain. Conservation laws for the Toda molecule.Integrability: Hamiltonian structures for the KdV equation. Integrability. Hierarchies of equa-tions determined by conservation laws.4H reading material: to be announced.
71
3.4.27 STATISTICAL MECHANICS III/IV – MATH3351/MATH4231 (38 lectures)
Dr V. E. Hubeny / Prof. F. E. CorriganStatistical Mechanics seeks to use statistical techniques to develop mathematical methods neces-sary to deal with systems with large number of constituent entities, like a box of gas with whichcomprises of a large number of individual gas molecules. It provides a framework for relating themicroscopic properties of individual atoms and molecules to the macroscopic or bulk properties ofmaterials that can be observed in everyday life, therefore explaining thermodynamics as a naturalresult of statistics and mechanics (classical and quantum) at the microscopic level.We will start by discussing basic concepts in thermodynamics and then proceed to see how theseideas can be derived from a statistical viewpoint. As we go along, we will learn to apply theconcepts to various interesting physical phenomena, such as neutron stars, Bose-Einstein conden-sation, phase transitions, etc..Recommended Books*R. K. Pathria, Statistical Mechanics, Second Edition, Elsevier 1996, ISBN 07050624698K. Huang, Statistical Mechanics, Wiley; 2 edition 1987, ISBN 0471815187F. Reif, Fundamentals of Statistical and Thermal Physics, McGraw-Hill, 1975, ISBN 070518009R. Blowey and M. Sanchez, Introductory Statistical Mechanics, Oxford science publications,1999, ISBN 0198505760L. D. Landau and E. M. Liftshitz, Statistical Physics, 3rd Edition, Part 1, Buttterworth Heine-mann, 1980, ISBN 0750633727
* This will be the main textbook for the course. Huang covers most of the material as well. Reifis a good source for basics of thermodynamics. Landau and Liftshitz is a classic book and anexcellent resource for advanced topics.Preliminary Reading Dip into any of the above books except perhaps the last one. In particular,the first few chapters of Reif should provide a good starting point.
72
Outline of course Statistical Mechanics III/IV - MATH3351/MATH4231
Aim: To develop a basic understanding of dynamics and behaviour ofsystems with a large numberof constituents.
Term 1 (20 lectures)Thermodynamics: Thermal equlibrium, the laws of thermodynamics. Equations of state, idealgas law.Probability distributions: Review of probability distributions and random walks.Classical statistical mechanics: Statistical basis of thermodynamics: microstates, macrostatesand the thermodynamics limit. Ideal gas. Gibbs paradox and entropy. Microcanonical, canonicaland grand-canonical ensembles.
Term 2 (18 lectures)Distributions and identical particles: Maxwell-Boltzmann distribution. Bose and Fermi distri-butions, parastatistics.Physical phenomena: Black-body radiation, magnetisation, neutron stars (Chandrashekar limit).Phase transitions: Mean field theory, Ising model, Landau-Ginzburg theory of phase transitions.
Level IV reading material: To be announced.
73
3.4.28 STATISTICAL METHODS III – MATH3051 (38 lectures)
Dr J. Einbeck
The course introduces widely used statistical methods. Theoretical issues are down-played. Thecourse should be of particular interest to those who intend to follow a career in statistics or whomight choose to do a fourth year project in statistics.Topics include: statistical computing using R; multiparameter problems (multivariate normal dis-tribution, likelihood theory); principal component analysis; regression (general linear model, diag-nostics, influence, outliers, transformations, variable selection); analysis of variance; generalisedlinear models (simple examples, exponential family, link functions, deviance).There is no one recommended book but the books below more than cover the course material. Inparticular, the books by Anderson and Weisberg give a quite good coverage of term 1 and term2, respectively. The books by Fahrmeir/Tutz and Hastie et al. go partly well beyond the coursematerial, but are nevertheless nicely readable.
Recommended BooksT. W. Anderson, An Introduction to Multivariate Statistical Analysis, Wiley 1984, ISBN 0471360910M. J. Crawley, Statistical Computing – An Introduction to Data Analysis using S-Plus, Wiley2002, ISBN 0471560405,L. Fahrmeir and G. Tutz, Multivariate Statistical Modelling based on Generalized Linear Mod-els (Second edition), Springer 2001, ISBN 9780387951874,T. Hastie, and R. Tibsihirani and J. Friedmann, The Elements of Statistical Learning, Springer2001, ISBN 0387952845W.J. Krzanowski, An Introduction to Statistical Modelling, Arnold 1998, ISBN 0340691859,J. Neter, M. Kutner, C. Nachtsheim, W. Wasserman, Applied Linear Statistical Methods (severaleditions with different combinations of authors 1974–2004), ISBN 0256117365Pawitan, J , In All Likelihood, Oxford 2001, ISBN 0198507658J.A. Rice, Mathematical Statistics and Data Analysis, Duxbury 1995, ISBN 0534209343S. Weisberg, Applied Linear Regression, Wiley 1985, ISBN 0471879576.
Preliminary ReadingChapters 3.1-3.3, 8, 9.1-9.4 (Michaelmas) and 12, 14 (Epiphany) in Rice’s book.
74
Outline of course Statistical Methods III
Aim: To provide a working knowledge of the theory, computation and practice of statistical meth-ods.
Term 1 (20 lectures)Statistical Computing: Introduction to statistical software for data analysis, and for reinforcingstatistical concepts.Multivariate Analysis: Multivariate normal distribution, covariance matrix, Mahalanobis dis-tance, principal component analysis.Likelihood Estimation: Likelihood- and scorefunction for multiparameter models, Fisher Infor-mation, confidence regions, method of support, LR tests, profile likelihood.
Term 2 (18 lectures)Regression: General linear model: regression, analysis of variance, designed experiments, diag-nostics, influence, outliers, transformations, variable selection, correlated errors, lack-of-fit tests.Generalised linear models: Framework, exponential families, likelihood and deviance, standarderrors and confidence intervals, prediction, analysis of deviance, residuals, over-dispersion.
75
3.4.29 STOCHASTIC PROCESSES III AND IV –MATH3251/MATH4091 (38 lectures)
To be given in 2009 -2010
A stochastic process is a mathematical model for a system evolving randomly in time. For example,the size of a biological population or the price of a share may vary with time in an unpredictablemanner. These and many other systems in the physical sciences, biology, economics, engineeringand computer sciences may best be modelled in a non-deterministic manner.More technically, a stochastic process is a collection of random quantities indexed by a time param-eter. Typically these quantities are not independent, but have their dependency structure specifiedvia the time parameter. Specific models to be covered include Markov chains in discrete and con-tinuous time, Poisson processes and Gaussian processes.
Recommended BooksThe book by Grimmett and Stirzaker is available in paperback and recommended for purchase.The book “Probability and random processes” by G. R. Grimmett and D. R. Strizaker (3rd ed.,OUP 2001) covers most topics discussed in the lectures.S. M. Ross, Introduction to probability models, Academic Press, New York 1997.S. Karlin and H. M. Taylor, A first course in stochastic processes, Academic Press, New York.W. Feller, An introduction to probability and its applications, Volume I., Wiley (hard but aclassic).J. R. Norris, Markov chains, Cambridge University Press, 1998, ISBN 0521633966 (Paperback).
Preliminary ReadingYour 1H probability notes. The books above also begin with reviews of probability theory, e.g.chapters 1-3 of Grimmett and Stirzaker. Also, D.R. Stirzaker, Elementary probability theory,CUP 1994. This has many worked examples and exercises, and covers some of the course as wellas the preliminaries.
4H reading material referencesThe book by Grimmett and Stirzaker recommended above, and W. Feller, An introduction toprobability and its applications, Volume I, Wiley (1968) and Volume II, Wiley (1971).
76
Outline of course Stochastic Processes III/IV
Aim: This module continues on from the treatment of probability in 2H modules MATH2151 andMATH2161. It is designed to introduce mathematics students to the wide variety of models ofsystems in which sequences of events are governed by probabilistic laws. Students completingthis course should be equipped to read for themselves much of the vast literature on applications toproblems in mathematical finance, physics, engineering, chemistry, biology, medicine, psychologyand many other fields.
Term 1 (20 lectures)Probability Revision Conditional expectation, sigma fields.Examples of Markov Chains Branching processes, Gibbs sampler.Discrete Parameter Martingales The upcrossing lemma, almost sure convergence, the backwardmartingale, the optional sampling theorem, examples and applications.Discrete Renewal Theory The renewal equation and limit theorems in the lattice case, examples.
Term 2 (18 lectures)General Renewal Theory The renewal equation and limit theorems in the continuous case, excesslife, applications. The renewal reward model.Poisson Processes Poisson process on the line, relation to exponential distribution, marked/compoundPoisson processes, Cramer’s ruin problem, spatial Poisson processes.Continuous Time Markov Chains Kolomogorov equations, birth and death processes, simplequeueing models.Topics Chosen From: stationary Gaussian processes, Brownian motion, percolation theory, con-tact process.
4H reading material: a topic related to one of Markov chains, random walks, applications toqueueing.
77
3.4.30 TOPOLOGY III – MATH3281 – (38 lectures)
Dr V. Kurlin
Topology is a mathematical theory which explains many interesting natural phenomena: it helpsto study equilibrium prices in economics, instabilities in behaviour of dynamical systems (forexample, robots), chemical properties of molecules in modern molecular biology and many otherimportant problems of science. Topological methods are used in most branches of pure and appliedmathematics.The course is an introduction to topology. We shall start by introducing the idea of an abstracttopological space and studying elementary properties such as continuity, compactness and con-nectedness. A good deal of the course will be geometrical in flavour. We shall study closedsurfaces and polyhedra, spend some time discussing winding numbers of planar curves and theirapplications. Orbit spaces of group actions will provide many interesting examples of topologicalspaces.Elements of algebraic topology will also appear in the course. We shall study the fundamentalgroup and the Euler characteristic and apply these invariants in order to distinguish between variousspaces. We shall also discuss the concept of homotopy type; this material will be further developedin the course ‘Algebraic Topology’.
Recommended BooksM.A. Armstrong, Basic Topology, Springer-Verlag 1983, ISBN 3540908390R. H. Crowell and R. H. Fox, Introduction to Knot Theory, 1963.
W. Fulton, Algebraic topology, a first course, 1995Preliminary readingTo get an idea of what topology is about one may consult the first two chapters of Armstrong. Anyof the following two books may also serve as an enjoyable introduction to topology:W. G. Chinn and N.E. Steenrod, First Concepts of Topology, Random House, New York, 1966C.T.C. Wall, A Geometric Introduction to Topology, Addison Wesley, Reading, Mass., 1972.
78
Outline of course Topology III
Aim: To provide a balanced introduction to Point Set, Geometric and Algebraic Topology, withparticular emphasis on surfaces.
Term 1 (20 lectures)Topological Spaces and Continuous Functions: Topological spaces, limit points, continuousmaps, homeomorphisms, compactness, product topology, connectedness, path-connected spaces,quotient topology, topological spaces as configuration spaces in robotics.Topological Groups and Group Actions: Orthogonal groups, connected components of O(n),the concept of orientation, topological groups, quaternions, group actions, orbit spaces, projectivespaces, lens spaces.The Fundamental Group4: Paths, the fundamental group, simply connected spaces, the funda-mental group of the punctured plane, winding number of a planar curve with respect to a point,degree of a map S1 → S1, Borsuk’s theorem about odd maps f : S1 → C−{0}, Borsuk - Ulamtheorem, invariance of dimension, ham sandwich theorem, Lusternik - Schnirelmann - Borsuktheorem.
Term 2 (18 lectures)Basic concepts of homotopy theory: Homotopy and homotopy equivalence, retracts and defor-mation retracts, homotopy type, induced homomorphisms of fundamental groups, homotopy in-variance of the fundamental group, Brouwer fixed point theorem, Van Kampen theorem (withoutproof), applications, fundamental groups of graphs.Elements of geometric topology: Polyhedra, triangulations of topological spaces, Euler charac-teristic, homotopy invariance of χ(X) (without proof), additive property of the Euler characteristic,closed surfaces, models of the Klein bottle and of the projective plane, connected sums, handles,orientable and non-orientable surfaces, surgery, classification theorem for surfaces.
4material partly covered in Term 2
79
3.4.31 ADVANCED QUANTUM THEORY IV – MATH4061 (38 lectures)
Dr M. Zamaklar
The course will introduce quantum field theory (QFT) by bringing together concepts from classicalLagrangian and Hamiltonian mechanics, quantum mechanics and special relativity.We will first present the quantisation of fields using the canonical procedure. By studying theexample of φ4 theory, we will explain the basic ingredients required to tackle the quantisation ofinteracting field theories in general.In the second part of the course we will examine the idea of path integrals in quantum mechanicsand field theory. We will then explain the basic concepts of renormalisation and regularisation onthe example of scalar φ4 theory.This course should prove interesting and useful especially to students who might want to pursuean interest in what is currently understood about the fundamental nature of matter.
Recommended BooksThere are many QFT books so you will find no shortage of material in the library. For example,the following introductory books cover much of the material in this course (and more). You arenot required to purchase any of these books, but you might find them useful.B Hatfield, Quantum Field Theory of Point Particles and Strings (Frontiers in Physics S.),1999, Perseus Publishing.M.E.Peskin and D.V.Schroeder, An Introduction to Quantum Field Theory, 1995, Addison-Wesley Advanced Book Program.A.Zee, Quantum Field Theory in a Nutshell, 2003 Princeton University Press.
80
Outline of course Advanced Quantum Theory IV
Aim: To introduce quantum field theory using strings as the primary example, and to developstring theory sufficiently to show that its spectrum includes all elementary particles thus unifyingthe fundamental forces.
Term 1 (20 lectures)Action principles and classical theory: Review of Lagrangian formulation of classical field the-ory. World-line action for free relativistic particle, symmetries, equations of motion, formulationwith intrinsic metric. Nambu-Goto action, symmetries, formulation with intrinsic metric, equa-tions of motion, boundary conditions, simple solutions.
Quantisation of free scalar fields: Canonical quantisation of free scalar fields on a circle and ona finite interval, Fock space. Application to strings, problem of time-like oscillators.
Virasoro algebra: String constraints as generators of conformal transformations. Form of centralterm. Representations of the algebra. Calculation of central charge for free bosonic fields.
Term 2 (18 lectures)Spectra: Physical state condition. Statement of no-ghost theorem, critical dimension. Spectrumof open string. Connection to gauge theory. Discussion of non-Abelian gauge symmetry and itsimportance for the Standard Model. Spectrum of closed string. Connection to General Relativity.Discussion of compactification.Spinning string: Gauge fixed action. Ramond and Neveu-Schwarz boundary conditions. Super-Virasoro algebra. Spectrum.D-branes: Coupling to extended objects via boundary conditions.
81
3.4.32 ALGEBRAIC TOPOLOGY IV – MATH4161 (38 lectures)
Prof. M. Farber
The basic method of Algebraic Topology is to assign an algebraic system (say, a group or a ring)to each topological space in such a way that homeomorphic spaces have isomorphic systems.Geometrical problems about spaces can then be solved by ‘pushing’ them into algebra and doingcomputations there. This idea can be illustrated by the theory of fundamental group which isfamiliar from the Topology III course.In the course ‘Algebraic Topology’ we shall meet several other theories where ‘topology gets rep-resented into algebra’. We shall discuss the theory of covering spaces with its amazing interplaybetween geometric properties of topological spaces and algebraic properties of their fundamen-tal groups. We shall study homology theories for topological spaces and the ways they can becomputed in terms of simplicial or cell structures.Having homology theories at our disposal we shall progress in two different directions: (1) intohomological algebra, i.e. studying homological functors using algebraic tools and (2) into ho-motopy theory where we shall get more advanced information about sets of homotopy classes ofcontinuous maps and will be able to compute these sets explicitly in many cases.
Recommended BooksAllen Hatcher, Algebraic topology, Cambridge university press, 2002; This book is available freeof charge from www.math.cornell.edu/˜hatcherM.A. Armstrong, Basic topology, Springer-Verlag, 1983A. Dold, Lectures on Algebraic topology, Springer-Verlag, 1980W. Fulton, Algebraic topology (a first course), Springer - Verlag, 1995.E. Spanier, Algebraic topology, 1966
82
Outline of course Algebraic Topology IV
Aim: The module will provide a deeper knowledge in the field of topology ( a balanced introductionhaving been provided in Topology III (MATH3281))Term 1 (20 lectures)Homotopy properties of cell complexes: Cell complexes, homotopy extension property, mainconstructions (mapping cones and cylinders, products), examples.
Review of the fundamental group: Definition of the fundamental group, main properties, cal-culation for the circle, Van Kampen’s theorem, algorithm for computing fundamental group of apolyhedron.
Theory of covering spaces: Lifting properties, classification of covering spaces, groups of cover-ing translations, examples.Homotopy theory: Homotopy groups, Whitehead’s theorem, cellular approximation theorem,fiber bundles, exact homotopy sequences.
Term 2 (18 lectures)Elements of homological algebra: Chain complexes, homology, chain homotopy, exact sequences,Euler characteristic.Homology theory of topological spaces: Singular homology of topological spaces, homotopyinvariance, axioms of homology theory, Mayer - Vietoris sequences, relation between homologyand the fundamental group, the Hurewicz theorem, algorithms for computing homology groups ofcell complexes.
83
3.4.33 PROJECT IV – MATH4072
Each project is two terms’ work for a group of typically 2 to 4 students who collaborate on anopen-ended mathematical or statistical problem. Each student writes an individual report on anaspect of the work, and gives an oral presentation with accompanying poster during the Epiphanyterm. The presentation and poster are assessed and count towards 10% of the module mark.There are no lectures; the emphasis is on developing understanding in depth and encouraging cre-ativity through reading, discussion and calculation. The Project supervisor gives general guidanceand organizes progress meetings at least once a fortnight.A project is equivalent to two 4H options so you should devote on average one-third of your effortto it. The report deadline is the first Friday of the Easter term, 1st May 2009.The Project topics and supervisors, with further details, are available via the 4H courses web page(www.maths.dur.ac.uk/teaching/4H/) - follow the prominent link.Choose your preferred topics, via the above web link, before the deadline given there. Then attendthe project allocation meeting scheduled for CM101 3:30pm on Friday 13th June 2008, where theaim is optimum matching of students and supervisors.
84
3.4.34 RIEMANNIAN GEOMETRY IV – MATH4171 (38 lectures)
Dr N. Peyerimhoff / Dr F. TariThis course is based on the course Differential Geometry III as a prerequisite.In the first term we introduce Riemann’s concept of a manifold as a space with locally Euclideancoordinates and an intrinsic method of measuring distances and angles and discuss it in severalexamples. We also introduce geodesics via an variational approach and discuss parallel transportof vector fields along curves.The second term is concerned with the notion of curvature and its influence on the geometry of theunderlying space. We study spaces of constant curvature, present a global curvature comparisontheorem (Bonnet-Myers) and discuss applications.The aim of this course is to develop acquaintance with more general geometric spaces than just theEuclidean space and to the geometric meaning of curvature.
Recommended BooksM. Do Carmo, Riemannian Geometry, Birkhauser 1992S. Gallot, D. Hulin, J. Lafontaine: Riemannian Geometry, Springer 2004J. Lee, Riemannian Manifolds : An Introduction to Curvature, Springer GTM 1997F. Morgan: Riemannian Geometry: A Beginner’s Guide, Jones and Bartlett Publishers, 1998P. Peterson, Riemannian Geometry, Springer GTM 1997
86
Outline of course Riemannian Geometry IV
Aim: This module will provide a knowledge of the intrinsic geometry of Riemannian manifolds.This is a significant generalisation of the metric geometry of surfaces in 3-space.
Term 1 (20 lectures)The metric geometry of Riemannian manifolds. Geodesics. Various notions of curvature, and theireffect on the geometry of a Riemannian manifold.
Term 2 (18 lectures)Second variation formula, global comparison theorems with applications.
87
A Details of Modules and Programmes
All mathematics modules are open, except forMATH3121 (Mathematics Teaching III) — tied to G100, G103, G104 and NatSci and CombinedHonours programmes.MATH3131 (Communicating Mathematics III) — tied to G100, G104, CFG0, QRV0 and QRVAMATH3161 (Independent Study III) — tied to G100, G103 and G104
G100: B.Sc. in MathematicsG103: Master of MathematicsG104: B.Sc. in Mathematics (European Studies)
• If you do MATH3131 (Communicating Mathematics III), you cannot take another project modulein another department.• If you do MATH3121 (Mathematics Teaching III), you cannot take another into schools modulein another department.• The double module Project IV is open to students doing an M.Sci. in Natural Sciences, providedthe minimal requirements for taking a Level 4 mathematics module are met, provided the corequi-site of one other Level 4 mathematics module and provided no Level 4 Project is taken in anotherdepartment.
88
Level 1MATH1012 Core Mathematics A [P: A level Math at grade A; C:-;
EC: 1551, 1561 & 1571]MATH1051 Core Mathematics B1 [P: A level Math at grade A; P/C:1012;
EC: 1551, 1561 & 1571]MATH1041 Core Mathematics B2 [P: A level Math at grade A; C:1012 & 1051;
EC: 1551, 1561 & 1571 ]MATH1711 Data Analysis, Modelling & Simulation [P: A level Math at grade C or above;
C:-; EC: 1541]MATH1031 Discrete Mathematics [P: A level Math at grade C or above; C:-; EC:-]MATH1551 Maths for Engineers & Scientists [P: A level Math at grade C or above;
C:-; EC: 1012, 1561 & 1571]MATH1561 Single Mathematics A [P: A level Math at grade C or above; C:-;
EC: 1012 & 1551]MATH1571 Single Mathematics B [P: A level Math at grade C or above; C:1561;
EC: 1551]MATH1541 Statistics [P: A level Math at grade C or above; C:-; EC: 1711]
Level 2MATH2061 Algebra & Number Theory II [P: 1012 ; C: 2021; EC: 1551, 1561, 1571]
MATH2031 Analysis in Many Variables II [P: 1012; P/C: 1051; EC: 1551, 1561, 1571
MATH2141 Codes & Geometric Topology II [P: 1012 ; C:-; EC: 2131,2511 & 2571]
MATH2571 Codes & Probability II [P: 1012 ; C:-; EC: 2131, 2141,2151 & 2561]
MATH2011 Complex Analysis II [P: 1012; P/C: 1051; EC: 1551, 1561,1571 & Contours‡]
MATH2561 Contours & Probability II [P: 1012 ; P/C: 1051;EC: 1551, 1561, 1571, 2011, Contours* and Prob‡]
MATH2021 Linear Algebra II [P: 1012; C: -; EC: 1551, 1561 & 1571]
MATH2071 Mathematical Physics II [P: 1012; P/C1041 or PHYS1111 or PHYS1122);EC: 1551, 1561 & 1571]
MATH2051 Numerical Analysis II [P: 1012; P/C: 1051; EC: 1551, 1561 & 1571]
MATH2041 Statistical Concepts II [P: 1012 ; C:-; EC: 1551, 1561 & 1571]
Contours‡ 2111 or 2121 or 2171 or 2561.Prob‡ 2151 or 2161 or 2561 or 2571
P: Pre-requisite; C: Co-requisite; P/C: Pre- or Co-requisite; EC: Excluded combination.
CHEM1012: Core 1A Chemistry;PHYS1111: Fundamental Physics A;PHYS1122: Foundations of Physics I;PHYS2511: Foundations of Physics II;
89
Level 3MATH3131 Communicating Mathematics III (A3) [P: At least 3 Maths modules taken in 2ndyear, at least two of which are at Level 2; C: At least two other Level 3 maths module;EC: 3121 or Level 3 project modules in any other Dept.]MATH3121 Mathematics Teaching III (A3) [P: At least 3 Maths modules taken in 2nd year,
at least two of which are at Level 2;C: At least two other Level 3 maths modules; EC: 3131, GEOL3251, PSYC3191 & PHYS3611]
List AMATH3321 Algebraic Geometry III (A1) [P: 2021, 2061 & (2011 or Contours‡);
C:-; EC: 4011]MATH3011 Analysis III (A1,) [P: 2011 & 2031 ; C:-; EC: 4201]
MATH3081 Approx. Th. & Solns to ODEs III (A2)[P: 2051 and (1051∗ or 1∗) ; C:† ;EC: 4221]
MATH3311 Bayesian Methods III (A1) [P:2041 and (1051∗ or 1∗); C: †; EC: 4191]
MATH3341 Bayesian Statistics III (A2) [P: 2021 & 2041; C:-; EC: 4031]
MATH3101 Continuum Mechanics III (A1) [P: 2031 & (1∗ & (1041 or PHYS1111 orPHYS1122 ) or (1041 & 1051)∗); C: †; EC: 4081]
MATH3071 Decision Theory III (A3) [P: 2041 or P/C: 2021; EC: -]
MATH3021 Differential Geometry III (A3) [P: 2031; C:-; EC: -]
MATH3091 Dynamical Systems III (A3) [P:2031 & (2011 or Contours‡);C:-; EC: -]
MATH3181 Electromagnetism III (A3) [P: 2031 & (2071 or PHYS2511); C:-; EC:-;]
MATH3221 Elliptic Functions III (A2) [P: 2011 & (1051∗ or 1∗); C:†; EC: 4151]
MATH3041 Galois Theory III (A3) [P: 2021 & 2061 ; C:-; EC: -]
MATH3331 General Relativity III (A1) [P: 2021, 2031 & (2071 or PHYS2511); C:-; EC: 4051]
MATH3201 Geometry III (A2) [P: 2021, 2031, 2061 & (2011 or Contours‡);C:-; EC: 4141]
MATH3161 Independent Study III (A3) [P: 2H Honours Mathematics; C:-; EC: -]
MATH3171 Mathematical Biology III (A3) [P: 2031; C:-; EC: -]
MATH3301 Mathematical Finance III (A1) [P:(1051∗ and 1∗) or 2∗; C:-; EC: 4181]MATH3031 Number Theory III (A2) [P: 2021 & 2061 ; C:-; EC: 4211]
MATH3141 Operations Research III (A3) [P: 2021 ; C:-; EC: -]
MATH3291 Partial Differential Equations III (A3) [P: 2031 & (1051∗ or 1∗) ; C: †; EC: 4041]
MATH3211 Probability III (A2) [P: 2031, (2011 or Contours‡) & Prob‡); ; C:-; EC: 4131]
MATH3111 Quantum Mechanics III (A3) [P: 2021, 2031 & 2071; C:-; EC: -]
MATH3191 Rep. Theory & Modules III (A1) [P: 2021 & 2061; C:-;EC: 4101]
MATH3231 Solitons III (A2) [P: (2021, 2031 & (2011 or Contours‡) ; C:-; EC: 4121]MATH3351 Statistical Mechanics III (A2) [P: 2031 & (1051∗ or 1∗); C:-; EC: 4231]MATH3051 Statistical Methods III (A3) [P: 2021 & 2041 ; C:-; EC: -]MATH3251 Stochastic Processes III (A1) [P: 2021, 2031 & Prob‡; C:-; EC: 4091]MATH3281 Topology III (A3) [P: 2021, 2031 & (2011 or Contours‡); C:-; EC: 4021]∗ If taken in Year 2. 1∗ One Level 2 mathematics module.2∗ Two Level 2 mathematics modules. † One Level 3 mathematics module.Contours‡ 2111 or 2121 or 2171 or 2561. Prob‡ 2151 or 2161 or 2561 or 2571
90
Level 4MATH4072 Project IV (B3) [P: 5‡ ; C: one level 4 maths module; EC:†]List BMATH4061 Advanced Qntm. Th. IV (B3) [P: 5†, and including 3111 or PHYS3522; C:-; EC: -]
MATH4011 Algebraic Geometry IV (B1) [P: 3*; C:-; EC: 3321]
MATH4161 Algebraic Topology IV (B3) [P: 5† , and including 3281 ; C:-; EC: -]
MATH4201 Analysis IV (B1) [P: 3* ; C:-; EC: 3011]
MATH4221 Approx. Th. & Solns to ODEs IV (B2) [P:5† , and including 2051; C:-; EC: 3081]
MATH4191 Bayesian Methods IV (B1) [P:2041; C:-; EC: 3311]
MATH4031 Bayesian Statistics IV (B2) [P: 5†, and including 2021 and 2041 ; C:-; EC: 3341]
MATH4081 Continuum Mechanics IV (B1) [P: 5†, and including 2031 &(1041 or PHYS1111 or PHYS1122); C:-; EC: 3101]
MATH4151 Elliptic Functions IV (B2) [P: 5†, and including 2011; C:-; EC: 3221]
MATH4051 General Relativity IV (B1) [P: 3* And 2†; C:-; EC: 3331]
MATH4141 Geometry IV (B2) [P: 3* And 2†; C:-; EC: 3201]
MATH4181 Mathematical Finance IV (B1) [P: 5† or (4† & COMP3361) ; C:-; EC: 3301]MATH4211 Number Theory IV (B2) [P: 5†,and including 2021 & 2061 ; C:-; EC: 3031]
MATH4041 Partial Diff. Eqns IV (B3) [P: 2031 & (4‡ or (3†, & COMP3361)); C:-; EC: 3291]
MATH4131 Probability IV (B2) [P: 3*.And, 2†.;C:-; EC: 3211]
MATH4101 Rep. Th. & Moduless IV (B1) [P: 5†, and including 2021 & 2061; C:-; EC: 3191]
MATH4171 Riemannian Geometry IV (B3) [P: 5†, and including 3021; C:-; EC: -]
MATH4121 Solitons IV (B2) [P: 3* And, 2†; C:-;EC: 3231]MATH4231 Statistical Mechanics IV (B2) [P: 5†,and including 2031; C:-; EC: 3351]MATH4091 Stochastic Processes IV (B1) [P: 3*; C:-; EC: 3251]
3* : See Level 3 module.† See appendix A.2† in addition, a minimum of two maths modules at Level 3.3† Three maths modules in Years 2 and 3, with at least one module at Level 34† Four maths modules in Years 2 and 3, with at least one module at Level 34‡ Four maths modules in Years 2 and 3, with at least two modules at Level 35† Five maths modules in Years 2 and 3, with at least two modules at Level 3.5‡ Five or more maths modules in Years 2 and 3, with at least two modules at Level 3
P: Pre-requisite; C: Co-requisite; P/C: Pre- or Co-requisite; EC: Excluded combination.A2,B2: Modules taught in alternate years, starting in 2008 - 2009; A1,B1: Modules taught inalternate years, starting in 2009 - 2010; A3,B3: Modules taught every year.CHEM1012: Core 1A Chemistry; PHYS1111: Fundamental Physics A; PHYS1122: Foundationsof Physics I; PHYS2511: Foundations of Physics II; PHYS3522: Foundations of Physics III.COMP3361: Integrative Module - e-Science and Physics.
91
Table 1: Honours degrees in Mathematics*
B.Sc. Mathematics (G100)Year 1: Core Mathematics A, Core Mathematics B1, Core Mathematics B2 and Level 1 open modulesto the value of 40 credits chosen from those offered by any Board of Studies.
Year 2: Analysis in Many Variables II, Complex Analysis II (or Contours and Probability II)Linear Algebra II and mathematics modules to the value of 60 credits.
Year 3: Communicating Mathematics III or Mathematics Teaching III, and either modules to the valueof 100 credits chosen from List A or modules to the value of 80 credits chosen from List A AND oneopen 20 credit module chosen from those offered by any other Board of Studies.
Master of Mathematics (G103)Year 1: Core Mathematics A, Core Mathematics B1, Core Mathematics B2 and Level 1 open modulesto the value of 40 credits chosen from those offered by any Board of Studies.
Year 2: Analysis in Many Variables II, Complex Analysis II (or Contours and Probability II),Linear Algebra II and mathematics modules to the value of 60 credits.
Year 3: Modules to the value of 120 credits chosen from Mathematics Teaching III and List A.
Year 4: Project IV and either modules to the value of 80 credits chosen from List Bor modules to the value of 60 credits chosen List B ANDan open 20 credit module (at level 4) chosen from those offered by any other Board of Studies.
BSc Mathematics (European Studies) (G104)Year 1: Core Mathematics A, Core Mathematics B1, Core Mathematics B2 and Level 1 open modulesto the value of 40 credits chosen from those offered by any Board of Studies, of which at least 20credits must be an appropriate language module. The language requirement does not apply to studentsspending the year abroad at Trinity College, Dublin.
Year 2: Analysis in Many Variables II, Complex Analysis II (or Contours and Probability II),Linear Algebra II and mathematics modules to the value of 60 credits.
Year 3: Students must study and be assessed in a mathematics programme (together, possibly, withother topics) in a European university under the ERASMUS/SOCRATES Programme.
Year 4: Communicating Mathematics III or Mathematics Teaching III, and either modules to the valueof 100 credits chosen from List A or modules to the value of 80 credits chosen from List A ANDan open 20 credit module chosen from those offered by any other Board of Studies.
*The Ordinary degree regulations are the same as the Honours degree regulations.
92
Table 2: Mathematics module content of the BSc/MSci Mathematics and Physics, and MSci Chem-istry and Mathematics Programmes. Joint Honours students also take modules prescribed by thepartner department.
B.Sc. JH Natural Sciences (Mathematics and Physics) (CFG0)
Year 2: Analysis in Many Variables II, Complex Analysis II (or Contours and Probability II)and Linear Algebra II.
Year 3: Level 3 mathematics modules to the value of 40 credits, and possibly an extra 20credit Level 3 mathematics module.
MSci JH Natural Sciences (Mathematics and Physics) (NatSci3)
Year 2: Analysis in Many Variables II, Complex Analysis II and Linear Algebra II.
Year 3: Level 3 mathematics modules to the value of 60.
Year 4: Level 4 mathematics modules to the value of 40 credits from list B and Project IV(optional).
MSci JH Natural Sciences (Chemistry and Mathematics) (NatSci1)
Year 2: Analysis in Many Variables II, Linear Algebra II and Mathematical Physics II.
Year 3: Electromagnetism III, Quantum Mechanics III and a Level 2 or 3 mathematicsmodule to the value of 20 credits.
Year 4: Level 4 mathematics modules to the value of 40 credits from List B, and Project IV(optional).
93
