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Tufts University Department of Mathematics Course Descriptions Fall 2012 Contents: Math 61 (old number: 22): Discrete Mathematics p. 2 Math 63 (old number: 41): Number Theory p. 3 Math 70 (old number: 46): Linear Algebra p. 4 Math 72 (old number: 54): Abstract Linear Algebra p. 5 Math 87: Mathematical Modeling and Computation p. 6 Math 112: History of Mathematics p. 7 Math/CS 126: Numerical Analysis p. 8 Math 135: Real Analysis I p. 9 Math 145: Abstract Algebra p. 10 Math 151/ME 150: Applications of Advanced Calculus p. 11 Math 161: Probability p. 12 Math 211: Analysis p. 13 Math 215: Algebra p. 14 Math 218: Algebraic Toplogy p. 15 Math 250-01: Coxeter Groups and Hyperbolic Geometry p. 16 Math 250-02: Measure Theoretic Probability p. 17 Math 250-03: Linear Partial Differential Equations p. 18

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Tufts University Department of Mathematics

Course Descriptions Fall 2012

Contents:

Math 61 (old number: 22): Discrete Mathematics p. 2

Math 63 (old number: 41): Number Theory p. 3

Math 70 (old number: 46): Linear Algebra p. 4

Math 72 (old number: 54): Abstract Linear Algebra p. 5

Math 87: Mathematical Modeling and Computation p. 6

Math 112: History of Mathematics p. 7

Math/CS 126: Numerical Analysis p. 8

Math 135: Real Analysis I p. 9

Math 145: Abstract Algebra p. 10

Math 151/ME 150: Applications of Advanced Calculus p. 11

Math 161: Probability p. 12

Math 211: Analysis p. 13

Math 215: Algebra p. 14

Math 218: Algebraic Toplogy p. 15

Math 250-01: Coxeter Groups and Hyperbolic Geometry p. 16

Math 250-02: Measure Theoretic Probability p. 17

Math 250-03: Linear Partial Differential Equations p. 18

Math 61 Discrete MathematicsCourse Information

Fall 2012

Block: H (Tues Thus 1:30 - 2:30; Fri 2:30-3:20)Instructor: George McNinchEmail: [email protected]: Bromfield-Pearson 112Office hours: (Spring 2012) Mon 11:00 - 12:00, Wed and Fri 1:00-2:0Phone: (617) 627-6210

Prerequisites: Math 11 (new number: 32), Comp 11 or consent.

Text: Discrete Mathematics and its Applications (7th edition), Kenneth H. Rosen. McGrawHill 2012.

Course description:This course (Math 61, formerly called Math 22) is an introduction to discrete mathe-

matics. In mathematics, discrete is the opposite of continuous. In calculus, for example, westudy things that vary continuously, whereas in discrete mathematics, we focus on the com-binatorics of things that can be counted and vary only in discrete steps. We will examinesets, permutations, equivalence relations and properties of the integers, we will introducesome fundamental notions in graph theory and we will study and apply the basic principlesof propositional logic (p implies q and all that). In short, we will open the door to somefascinating and fundamental topics in mathematics.

This course is intended as a bridge from the more computationally oriented courses likecalculus to upper level courses in mathematics and computer science. We will spend a lotof time learning to read and write correct proofs of mathematical statements and we willpractice these skills on a variety of problems.

This course, which is co-listed as Comp 61 (formerly Comp 22), is required for computerscience majors. It counts toward the math major and is, in fact, highly recommended formath majors (or possible math majors) who want a first glimpse of higher mathematics.

A second section of this course (Math 61-02) is being taught by Professor Cowen fromthe Computer Science Department in the E+ block.

Math 63 Number TheoryCourse Information

Fall 2012

Block: K+ (Mon Wed 4:30 – 5:15)Instructor: Alberto LopezEmail: [email protected]: Bromfield-Pearson 106Office hours: (Spring 2012) Monday 3:00 – 4:00 and Thursday 9:30 – 10:20Phone: (617) 627-2357

Prerequisites: Math 11 (new number: 32) or consent.

Text: (Tentative) Elementary Number Theory, 7th edition, by David M. Burton.McGrawHill, 2011.

Course description: Have you ever thought about the similarities among the manyshapes that we find in nature? Think of a seashell or the tail of a seahorse and then look at thedrawing below. Do you see their relation with the sequence of numbers 1, 1, 2, 3, 5, 8, 13, . . . ?

Number theory is the study of positive integer numbers 1, 2, 3, . . . Informally, we candescribe it as the search for hidden relationships among numbers. It is a field that, forcenturies, has captivated some of the greatest mathematical minds in history. It is not onlyone of the most beautiful fields of mathematics, but also the most accessible – the onlybackground knowledge you need to start thinking about number theory is to understandthe operations of addition, multiplication, subtraction and division on the set of positiveintegers.

In this course we will start by studying integer numbers and some of their properties. Wewill next move to the core of the course: the study of prime numbers, that is, the atoms ofmathematics, the numbers you build other numbers from! We will also learn about modulararithmetic – “the mathematics of remainders” – a new way to count that imitates the wayclocks work. Throughout the course, you will see interesting questions to which an answer isknown, you will see problems to which the answer is unknown, and you will see applicationsof number theory. In particular, we will discuss some applications to cryptography, thedesign and implementation of secret codes.

Your grade will be based on a midterm exam and a final exam, as well as weekly assign-ments. These will constitute a significant part of your grade.

Math 70 Linear AlgebraCourse Information

Fall 2012

Block: D+ (TR 10:30-11:45)Instructor: Mary [email protected]

Office: Bromfield-Pearson 4Office hours: (Spring 2012)M 10:30-11:30, T 2:30-3:30, orby appointmentPhone: (617) 627-5045

Block: G+ (MW 1:30-2:45)Instructor: Alberto [email protected]

Office: Bromfield-Pearson 106Office hours: (Spring 2012)M 3:00-5:30, R 9:30-10:30, or byappointmentPhone: (617) 627-2357

Block: G+ (MW 1:30-2:45)Instructor: Joe [email protected]

Office: Bromfield-Pearson 216Office hours: (Spring 2012)By appointmentPhone: (617) 627-2843

Prerequisites: Math 12 (new number: 34) or 17 (new number: 39) or consent.

Text: Linear Algebra and its applications, 4th edition, by David Lay. Pearson-Addison-Wesley,2011.

Course description: Linear algebra is the study of matrices, vector spaces, and linear transfor-mations. In Math 70 we start by studying systems of linear equations. For example, 2x+3y = 5 is alinear equation in the unknowns x and y, whereas ex = y is nonlinear. The study of linear equationsquickly leads to useful concepts such as vector spaces, dimension (you will learn about four-, five-,and infinite-dimensional spaces), linear transformations, and eigenvalues. These concepts will helpyou solve linear equations efficiently, and more importantly, they fit together in a beautiful wholethat will give you a deeper understanding of mathematics.

Linear algebra arises everywhere in mathematics (you will use it in almost every one of our upperlevel courses) as well as in physics, chemistry, economics, biology, and a range of other fields. Evenwhen a problem involves nonlinear equations, as is often the case in applications, linear systems stillplay a central role, since the most common methods for studying nonlinear systems approximatethem by linear systems.

Among the applications of linear algebra are solutions of linear systems of equations, determiningconic curves and quadric surfaces not in standard form, the second-derivative test in vector calculus,computer graphics, linear economic models, and differential equations. Linear algebra also entersin further study of calculus, where the derivative is viewed as a linear transformation.

Mathematics majors and minors are required to take linear algebra (Math 46 [new number 70]or Math 54 [new number 72]) and are urged to take it as early as possible, as it is a prerequisitefor most upper-level mathematics courses. The course is also useful to majors in computer scienceand engineering, as well as those in the natural and social sciences. In addition to computationand problem-solving, the course will introduce the students to axiomatic mathematics and simpleproofs.

Math 72 Abstract Linear AlgebraCourse Information

Fall 2012

Block: G+ (Monday, Wednesday 1:30 - 2:45)Instructor: Genevieve WalshEmail: [email protected]

Office: Bromfield-Pearson 213Office hours: (Spring 2012) Tuesday Thursday Friday 1-2Phone: (617) 627-4032

Prerequisites: Math 12 (new number 34) or 17 (new number 39) or consent.

Text: Linear Algebra Done Right, 2nd Edition, by Sheldon Axler. Springer, 1997.

Course description: Linear algebra is the study of vector spaces, matrices and lineartransformations. Linear algebra provides a fundamental link between our geometric intuitionof the world around us and the powerful tools of algebra.

The results and concepts of linear algebra are essential in virtually every branch of highermathematics, from the most pure to the most applied. In fact, one could almost define highermathematics as the study of ideas that are based on linear algebra, except that this definitionwould include most of physics as well!

This course is the honors version of Math 70. From the beginning, we will adopt a moresophisticated point of view. In particular, proofs and theory will play a greater role than inMath 70. We will present concepts in full generality, while emphasizing geometric examples.

This course is recommended for all those wanting to learn linear algebra who also enjoya little extra challenge. Math 72 of course counts as a replacement for Math 70 whereverMath 70 is required as a prerequisite.

There will be three exams in this course, two during the semester and one at the end,and there will be weekly assignments. Students are asked, but not required, to participatein class by occasionally presenting their ideas.

Math 87 Mathematical Modeling and ComputationCourse Information

Fall 2012

Block: H+ (Tuesday, Thursday 1:30-2:45)Instructor: Scott MacLachlanEmail: [email protected]

Office: Bromfield-Pearson 212Office hours: (Fall 2011) Thursday 10:00 - 12:00 and Friday 1:30-2:30Phone: (617) 627-2356

Prerequisites: Math 12 (new number: 34) or 17 (new number: 39) or consent.

Text: none

Course description: This course is about using elementary mathematics and computingto solve practical problems. Single-variable calculus is a prerequisite; other mathematicaland computational tools, such as elementary probability, matrix algebra, elementary combi-natorics, and computing in Matlab, will be introduced as they come up.

Mathematical modeling is an important area of study, where we consider how mathe-matics can be used to model and solve problems in the real world. This class will be drivenby studying real-world questions where mathematical models can be used as part of thedecision-making process. Along the way, we’ll discover that many of these questions are bestanswered by combining mathematical intuition with some computational experiments.

Some of the problems that we will study in this class include:

1. The managed use of natural resources. Consider a population of fish that has a nat-ural growth rate, which is decreased by a certain amount of harvesting. How muchharvesting should be allowed each year in order to maintain a sustainable population?

2. The optimal use of labor. Suppose you run a construction company that has fixednumbers of tradespeople, such as carpenters and plumbers. How should you decidewhat to build to maximize your annual profits? What should you be willing to pay toincrease your labor force?

3. Project scheduling. Think about scheduling a complex project, consisting of a largenumber of tasks, some of which cannot be started until others have finished. Whatis the shortest total amount of time needed to finish the project? How do delays incompletion of some activities affect the total completion time?

Math 112 History of MathematicsCourse Information

Fall 2012

Block: I+ (Monday, Wednesday 3:00-4:15)Instructor: Moon DuchinEmail: [email protected]: Bromfield-Pearson 113Office hours: TBDPrerequisites: Math 12 or 17 (new numbers 34 or 39), or consent of the instructor.

Text: John Stillwell, Mathematics and its History, 3rd edition.

Course description:This course will articulate themes for thinking about the development and refinement of math-

ematics over time and across cultures. Rather than attempt to proceed chronologically, or haveparticular focus on one mathematical period, we will instead use this thematic organization to cutacross the timeline and visit episodes from antiquity to the present.

I. Introductiontopics: definitions, theorems, counterexamples, “the method of proofs and refutations”math concepts: proof, polyhedra, V − E + F , concepts of topology

II. Artifacts and Algorithmstopics: tablets and papyri, Babylonian and Egyptian mathematics, al-Khwarizmi and the quadraticformula, geography and the transmission of ideasmath concepts: Egyptian multiplication algorithms, the Chinese Remainder Theorem and otherearly number theory, the Euclidean algorithm, place value

III. Institutions and Abstractiontopics: educational societies and institutions, “Greek” mathematics, Zeno and Aristotle, Islamicmathematics and the House of Wisdom, early European mathematical duels, Cantor and theCatholic Church, Noether, Bourbaki, and the New Mathmath concepts: number and anthyphairesis, continued fractions, the cubic formula, notions ofinfinity and cardinality

IV. Symbols and Meaningtopics: π from the Bible to the Nazis, zero, trigonometry, early calculus, the development of algebra,Euler, Ramanujan, the story of Cmath concepts: trig formulas, infinite series, the complex plane and its motions

V. People and Mathematical Practicetopics: Galois and the cult of genius, Poincare vs. Hilbert, conjecture and aesthetics, “pure math”math concepts: groups and geometry, the Prime Number Theorem, the Riemann Hypothesis,Geometrization

Math/CS 126 Numerical AnalysisCourse Information

Fall 2012

BLOCK: E+: Wednesday, Friday 10:30-11:45 a.m.INSTRUCTOR: James AdlerEMAIL: [email protected]: Bromfield-Pearson 109OFFICE HOURS: Spring 2012 Tues. 4:00-5:30pm and Fri. 11:30am-1:00pmPHONE: 7-2354

PREREQUISITES: For Math 126/CS 126: Math 38 (new number: 51) and experience pro-gramming in a language such as C, C++, Fortran, Matlab, etc.

TEXT: TBD

COURSE DESCRIPTION:“Numerical analysis is the study of algorithms for the problems of continuous mathe-

matics.” (L. N. Trefethen, “The definition of numerical analysis”, SIAM News, November1992.) Continuous mathematics means mathematics involving the continuum of real orcomplex numbers.

There are many striking examples illustrating the importance of the neccessity of ac-curate numerical solution methods. Here are just three of them. (1) In 1991, a Norwegianoffshore oil platform called the Sleipner A sank. The resulting economic loss was esti-mated at $700 million. The post-accident investigation traced the problem to a faulty nu-merical method for predicting shear stresses. (2) Ordinary differential equations are usedto model the spread of an infection through a population. Numerical solution techniquesare usually the only practical option for solving these problems, but without an under-lying understanding of the properties of both the model and the algorithm, numericalresults are not to be trusted. (3) CT scanners are used throughout the world for medicaldiagnostics. This technology is based on numerical algorithms for the solution of certainintegral equations. Then, of course, there are the more simple questions, “How does mycalculator compute the square root of 2??”

This course is an introduction to the field, treating linear algebra fairly lightly, insteademphasizing numerical solution of nonlinear equations and differential equations, nu-merical integration techniques, and unconstrained optimization, often in the context ofapplications. (For a thorough treatment of numerical linear algebra, take Math 128/CS128 or its 200-level branch.) Computer programming will be a substantial component ofthe homework, with Matlab as the suggested tool.

Math 135 Real Analysis I

Course Information

Fall 2012

Block: G+ (Mon Wed 1:30–2:45)Instructor: Fulton GonzalezEmail: [email protected]

Office: Bromfield-Pearson 203Office hours: (Spring 2012) Tue 1–2:30p.m., Thu 10:30 a.m.–12:00 p.m.Phone: (617) 627-2368

Block: F (Tue Thu Fri 12:00–12:50)Instructor: Loring TuEmail: [email protected]

Office: Bromfield-Pearson 206Office hours: (Spring 2012) Mon 1:30–2:30 p.m., Wed 3–4 p.m., or by appointmentPhone: (617) 627-3262

Prerequisites: Math 13 or 18, and 46, or consent. (In the new numbering scheme, Math42 or 44, and 70, or consent.)

Text: Elementary Classical Analysis, Second Edition by Jerrold E. Marsden and MichaelJ. Hoffman, W. H. Freeman and Company, 1993, New York.

Course description:

Real analysis is the rigorous study of real functions, their derivatives and integrals. Itprovides the theoretical underpinning of calculus and lays the foundation for higher math-ematics, both pure and applied. Unlike Math 11, 12, and 13, where the emphasis is onintuition and computation, the emphasis in real analysis is on justification and proofs.

Is this rigor really necessary? This course will convince you that the answer is an un-equivocal yes, because intuition not grounded in rigor often fails us or leads us astray. Thisis especially true when one deals with the infinitely large or the infinitesimally small. Forexample, it is not intuitively obvious that, although the set of rational numbers contains theset of integers as a proper subset, there is a one-to-one correspondence between them. Thesetwo sets, in this sense, are the same size! On the other hand, there is no such correspondencebetween the real numbers and the rational numbers, and therefore the set of real numbersis uncountably infinite.

In this course, we will study the topology of the real line and Euclidean space, compact-ness, connectedness, continuous mappings, and uniform convergence. The topics constituteessentially the first five chapters of the textbook. Along the way, we will encounter theoremsof calculus, such as the intermediate-value theorem and the maximum-minimum theorem,but in a more general setting that enlarges their range of applicability.

In addition to introducing a core of basic concepts in analysis, a companion goal of thecourse is to hone your skills in distinguishing the true from the false and in reading andwriting proofs.

Math 135 is required of all math majors. A math minor must take Math 135 or 145 (orboth).

Math 145 Abstract AlgebraCourse Information

Fall 2012

Block: D+ (Tuesday, Thursday 10:30-11:45)Instructor: montserat teixidorEmail: [email protected]: Bromfield-Pearson 115Office hours: (Fall 2011) Tuesday and Friday 10:30 -11.30 and by appointmentPhone: (617) 627-2358

Prerequisites: Linear Algebra to be called Math 70 and 72, (n/’ees Math 46 or 54).

Text: A first course in Abstract Algebra seventh edition by John B. Fraleigh , AddisonWesley 2003.

Course description: Algebra, along with Analysis and Geometry is one of the mainpillars of mathematics. It has ancient roots, especially in Europe, India and China. Histori-cally, algebra was concerned with the manipulation of equations and, in particular, with theproblem of finding the roots of polynomials. This is the algebra that you know from highschool. There are clay tablets from 1700B.C. that show that the Babylonians knew how tosolve quadratic equations. The solutions to cubic and fourth degree polynomial equationswere solved in Italy during the Renaissance. About the time of Beethoven, a young Frenchmathematician Evariste Galois made the dramatic discovery that for polynomials of degreegreater than four, no similar solution exists. To do this, Galois introduced the branch ofmathematics known as group theory. This was not, of course, the end of the story. Algebrahas continued its development to the present day most notably with the classification offinite simple groups and with Andrew Wiles proof of Fermat’s Last Theorem.

The concept of a group is now one of the most important in mathematics. Roughlyspeaking, group theory is the study of symmetry. There are deep connections betweengroup theory, geometry and number theory. Groups pop up in every area of mathematics:symmetry groups can be used to find solutions to differential equations, associating groups(and rings) to topological spaces allows to distinguish among them. Groups also appear inthe attempts of physicists to describe the basic laws of nature.

In Math 145, we introduce the concepts of group and ring. Their properties mimic thearithmetic properties of numbers and polynomials. In Math 146, we describe the connectionGalois discovered between group theory and the roots of polynomials.

Math 151ME 150

Applications of Advanced CalculusCourse Information

Fall 2012

Block: J+ (Tu, Th 3:00-4:15)Instructor: Christoph BorgersEmail: [email protected]: Bromfield-Pearson 215Office hours: (Spring 2012) Mo 2:30–4:30, Tu/Th 4:00–4:30, and by appointmentPhone: (617) 627-2366

Prerequisite: Math 38 (new number: 51)

Please note: Math 152 will not be offered in the Spring of 2013. Both Math 151and Math 152 will be offered again during the academic year 2013/2014.

Text: Applied Partial Differential Equations with Fourier Series and Boundary Value Prob-lems, 4th Edition by Richard Haberman, Pearson Prentice-Hall (2004)

Course description:Most deterministic models in the sciences are partial differential equations. The weather,

the global climate, air flow around airplanes, blood flow in the heart, the propagation ofelectro-magnetic waves, the forces in bridges and buildings, the growth of tumors, the prop-agation of signals in nerve cells in the brain, the spread of epidemics, and countless otherphenomena are modeled using partial differential equations. Quite arguably there is nobranch of Mathematics with greater impact on the world.

The emphasis in this course is on linear partial differential equations. Three fundamental,prototypical examples are (1) the diffusion equation (describing the diffusion of a pollutant instill air, for instance), (2) the Poisson equation (the fundamental equation of electrostatics),and (3) the wave equation (describing the propagation of sound for instance). You will learnabout mathematical properties of the solutions of these and similar equations. In very specialcases, it is possible and useful to write down explicit, analytic expressions for the solutions,and you will learn about examples of that. Fourier analysis is a central tool in this subject,and therefore an introduction to Fourier analysis is a part of the course.

By far the most important solution techniques for partial differential equations are nu-merical methods. The study of such methods is a large and rapidly growing branch of Math-ematics. Although this is not a course on numerical analysis, we will study the simplest,most fundamental numerical methods in those places where it is conceptually clarifying.

Math 161 ProbabilityCourse Information

Fall 2012

Block: F+ (Tuesday, Thursday 12:00 - 1:15)Instructor: Kei KobayashiEmail: [email protected]: TBAOffice hours: (Spring 2012) not at Tufts, but reachable by emailPhone: TBA

Prerequisites: Math 13 (new number: 42) or consent.

Text: To be determined.

Course description: While it has been motivated by simple applications such as coinflipping and weather prediction, probability theory is a well-established branch of mathe-matics which analyzes random events observed in a wide variety of fields, including naturalscience, social science, computer science, medicine, engineering, and finance.

Math 161 is an introductory probability course. Students taking the course should have aworking knowledge of single variable calculus plus multiple integrals and partial derivatives.Math 162, which builds on Math 161, is a course on statistics. The emphasis in this course ison both concepts and applications. The goal is to learn the basic ideas of probability theoryin an intuitive, yet mathematical way, and get a feeling for some of the many ways in whichthe ideas are used. The concepts to be studied are essential for further study of stochasticprocesses or statistics.

Topics to be covered include sample spaces and events, conditioning and independence,discrete and continuous random variables and their distributions, expectations and variances,jointly distributed random variables, conditional probability and conditional expectations,Chebyshev’s inequality, the weak and strong laws of large numbers, moment generatingfunctions, and the central limit theorem. A highlight of the course is the central limittheorem. It is among the most famous and important theorems in all of mathematics. Itexplains why the distribution of so many random quantities in nature can be described usingthe function (2π)−1/2e−x2/2, whose graph is the famous “bell curve.”

Assessment will be based on exams and weekly problem sets.

Math 211 AnalysisCourse Information

Fall 2012

Block: M+ (Mon Wed 6:00 – 7:15)Instructor: Fulton GonzalezEmail: [email protected]: Bromfield-Pearson 203Office hours: (Spring 2012) Tuesdays 1:00 – 2:30; Thursdays 10:30 – 12:00Phone: 617 627 2368

Prerequisites: Math 135 or equivalent, Math 136 is recommended.

Text: Real and Complex Analysis, 3th Edition, by by Walter Rudin, 3rd. ed. (1986),McGraw-Hill.

Course description: Define the function f(x) on the interval [0, 1] by

f(x) =

{1 if x is rational,

0 if x is irrational.

It is easy to see that the Riemann integral1 of f(x), defined as the limit of Riemann sums,does not exist. However, since almost all; i.e., all except a countable number of, real numbersare irrational, one would, in a sense, want the integral of f(x) to exist and be equal to zero.Lebesgue’s theory of integration, which first appeared in his famous 1904 book [Lecons surintegration et la recherche des fonctions primitives, Gauthier-Villars, Paris, 1904; secondedition, 1928] treats discontinuous functions as “natural” objects to integrate and pavesthe way for integration on spaces besides Euclidean space (e.g. topological groups). It isimmediate from Lebesgue’s construction that the integral of f(x) exists and equals zero.

In this course, we will introduce measure theory and the tools needed to define the Lebesgueintegral. We will explore important concepts such as the great convergence theorems (i.e.,under what conditions is lim

∫fn =

∫lim fn?), L

p spaces, Banach and Hilbert spaces, andthe differentiation of measures. The course will roughly follow the first seven or eight chaptersof Rudin’s book.

There will be two exams - a midterm and a final - as well as weekly problem sets.

1Which was actually formally defined by Cauchy.

Math 215 AlgebraCourse Information

Fall 2012

Block: D+ (Tues Thur 10:30-11:45)Instructor: George McNinchEmail: [email protected]: Bromfield-Pearson 112Office hours: (Spring 2012) Mon 11:00 - 12:00, Wed and Fri 1:00-2:0Phone: (617) 627-6210

Prerequisites: Math 145 or consent.

Text: D.Dummit, S. Foote, Abstract Algebra, Prentice Hall.

Course description:

Algebra is an important part of modern mathematics, both as a subject in itself, and asa ubiquitous tool. Historically, algebra was concerned with the manipulation of equationsand, in particular, with the problem of finding the roots of polynomials; this is the algebrayou know from high school. Modern algebra systematically studies groups, rings, fields andmodules. Roughly speaking, group theory is the study of symmetry; of course, symmetriesare important in mathematics but also in physics, chemistry and other physical sciences.The study of rings and fields is part of arithmetic – i.e. of number theory – and of geometry– since the geometry of a space may be investigated by study of (suitable) rings of functionson the space. The study of modules over rings is in some sense a generalization of linearalgebra.

The course is suitable for students who have taken at least one semester of undergraduateabstract algebra and wish to acquire a solid foundation in algebra.

Math 218 Algebraic TopologyCourse Information

Fall 2012

Block: H+, T TH 1:30 - 2:45 p.m.Instructor: Kim RuaneEmail: [email protected]

Office: Bromfield-Pearson 211Office hours: (Spring 2012) Tuesday 3-4,Wednesday 2-3, and Friday 11:30 - 12:30Phone: 7-2006

Prerequisites: Math 135 and 145, or equivalent. Graduate standing or consent of instruc-tor.

Text: Algebraic Topology, by Allen Hatcher. Cambridge University Press, 2002. Alsoavailable at: http://www.math.cornell.edu/ hatcher/AT/ATpage.html.

Course description: Algebraic Topology is the study of algebraic invariants associatedto topological spaces. This course will approach this study from a decidedly geometricviewpoint. We will begin by reviewing some underlying geometric notions, such as homotopy.We will then define the fundamental group, and compute it for lots of examples, using VanKampen’s theorem as our main tool. We will also use the fundamental group to understandcovering spaces, and define a correspondence between subgroups of the fundamental groupof a space and covers of that space. Given a group G, we will construct a space whosehomotopy type depends only on its fundamental group G.

Next we will turn to an abelian theory, homology. Although somewhat more complicatedto define, this is an extremely useful tool. This theory assigns a sequence of abelian groupsto a space, called the homology groups. The first of these groups is the abelianization ofthe fundamental group. Homology groups can be computed naturally using a cell complex.Finally, we will study cohomology and Poincare duality for manifolds.

Throughout, examples and geometric constructions will be emphasized, with a particularemphasis on 2- and 3-dimensional manifolds, graphs, and 2-dimensional complexes.

Math 250-01 Coxeter Groups and Hyperbolic GeometryCourse Information

Fall 2012

Block: E+ (Monday, Wednesday 10:30 - 11:45)Instructor: Genevieve WalshEmail: [email protected]: Bromfield-Pearson 213Office hours: (Spring 2012) Tuesday Thursday Friday 1-2Phone: (617) 627-4032

Prerequisites: Graduate standing or permission of the instructor. Graduate level alge-braic topology preferred. (May be taken concurrently.)

Text:None required, but we will use material from several sources, including parts of:M. Kapovich, Hyperbolic Manifolds and Discrete Groups (Birkhauser 2000)M. Davis, The Geometry and Topology of Coxeter Groups (Princeton University

Press 2008)All sources will be on reserve at the library.

Course description:There is a deep connection between hyperbolic geometry and geometric group theory, and

this course will explore various aspects of that connection, focusing on the case of Coxetergroups. To get our intuition going, we will first study various aspects of hyperbolic 2- and 3-manifolds, including criteria for a manifold to be hyperbolic. We will also study the structureof hyperbolic orbifolds. These are quotients of H3 by a group of isometries, which may havetorsion. Many, many examples will be included such as hyperbolic knot complements withquotients that are hyperbolic orbifolds. Then we will investigate Coxeter groups, in par-ticular right-angled Coxeter groups, and criteria for these groups to be Gromov-hyperbolic.Sometimes hyperbolic Coxeter groups can be realized as the orbifold-fundamental groups ofhyperbolic reflection groups, and these examples will be ruthlessly exploited.

As time permits, there may be short presentations by students.

Math 250-02 Measure Theoretic ProbabilityCourse Information

Fall 2012

Block: F+ (Tuesday, Thursday 12:00-1:15)Instructor: Marjorie HahnEmail: [email protected]: Bromfield-Pearson 202Office hours: (Spring 2012) by appointmentPhone: (617) 627-2363

Prerequisites: Math 211 or consent.

Text: TBA

Course description: Probability theory at the measure theoretic level is a subject thatis mathematically beautiful and practical. We will learn the probabilistic techniques andconcepts required for research in probability and important uses/applications of probabilityto other areas of mathematics (analysis, dynamical systems, geometry, many areas of appliedmathematics) as well as the sciences and economics. This course provides the probabilisticfoundations necessary for further study of stochastic processes and theoretical statistics.Measure theory is essential but no prior probability is required.

The material will be developed in a manner that leads to an understanding of two im-portant kinds of stochastic processes (random processes evolving in time): random walksand Brownian motion. Random walks are processes in discrete time with random jumps.Brownian motion is the most famous continuous time stochastic process, modeling manycomplicated phenomena with fractal-like paths. An understanding of random walks requiresthe laws of large numbers, which in turn requires random variables, independence, 3 modesof convergence for random variables, the Borel-Cantelli Lemmas, and the Zero-one Laws.Another mode of convergence, called weak convergence, is required to obtain the famouscentral limit theorem which accounts for the importance of the normal distribution. Fouriertransforms will be developed as a tool for characterizing random variables and proving weakconvergence results. Random walks extended to stair-shaped paths in continuous time canbe suitably scaled in space and time to converge weakly to Brownian motion, yielding a pow-erful approximation technique. A look at Markov chains and Martingales introduces broaderclasses of dynamic probability models and the important notion of conditional expectations.

Homework problems will be sufficient to aid your understanding of the material. In lieuof exams, the course will culminate with short lectures by each student.

Math 250-03 Linear Partial Differential EquationsCourse Information

Fall 2012

Block: J+ (Tuesday, Thursday 3:00-4:15)Instructor: Scott MacLachlanEmail: [email protected]: Bromfield-Pearson 212Office hours: (Fall 2011) Thursday 10:00 - 12:00 and Friday 1:30-2:30Phone: (617) 627-2356

Prerequisites: Math 135 or permission of the instructor.

Text: to be determined

Course description:Partial differential equations are the principal language of mathematical science, and this

course will provide the student with a working knowledge of that language. We shall deriveand analyze the important prototypical linear partial differential equations for wave motion,diffusion, and potential theory. We shall learn how to classify partial differential equations,how to solve them using separation of variables and integral transforms, and how to provethat solutions exist and are unique.

Along the way, we will learn why a piano sounds different from a harpsichord, how mem-branes and beams vibrate, and why wine should be stored in deep cellars. The mathematicaltools that we will master include elements of vector calculus, linear algebra, ordinary differ-ential equations, Sturm-Liouville problems, special functions, Fourier series, eigenfunctionexpansions, Fourier transforms, and Green’s functions.

Math 250 differs from Math 151 in that it covers this material at a deeper level. Notably,we will study the theory of existence and uniqueness of solutions to PDEs, and we will studythe necessary functional analysis and calculus of variations needed for this.