Course Information Book

Department of Mathematics

Tufts University

Spring 2019

This booklet contains the complete schedule of courses offered by the Math Department in the Spring 2019 semester, as well as descriptions of our upper-level courses and a few lower-level courses.

For descriptions of other lower-level courses, see the University catalog.

If you have any questions about the courses, please feel free to contact one of the instructors.

Descriptions may not be available for all courses*

Course schedule pp. 2-3 Math 61 p. 4

Math 87 p. 5

Math 123 p. 6

Math 128 p. 7

Math 135 p. 8

Math 136 p. 9

Math 146 p. 10

Math 150-01 p. 11

Math 158 p. 12

Math 162 p. 13

Math 213 p. 14

Math 250-02 p. 15

Math 252 p. 16

Mathematics Major Concentration Checklist pp. 17-18

Applied Mathematics Major Concentration Checklist pp. 19-21

Mathematics Minor Checklist p. 22

Jobs and Careers, Math Society, and SIAM p. 23

Block Schedule p. 24

Spring 2019 Mathematics Course Schedule

Subject Number Section Title Instructor First Instructor Last Block Time EnrolLocation

Math 0010 01 Conic Sections Andrew Izsak H+ TR - 1:30-2:45PM 40Eaton Hall, 208 - ITS Lab

Math 0016 01 Symmetry Gail Kaufmann DM - 09:30AM-10:20AM, TR - 10:30AM-11:20AM 45 BP 002

Math 0019 01 Social Choice Rylee Lyman GWF WF - 01:30PM-02:45PM 40 BP 101Math 0019 02 Social Choice Mary Glaser H+ TR - 1:30-2:45PM 40 BP 002Math 0021 01 Intro Probability and Statistics Linda Garant D+ TR - 10:30AM-11:45AM 30 BP 003Math 0021 02 Intro Probability and Statistics Linda Garant F+TR TR - 12:00PM-01:15PM 35 BP 002

Math 0032 01 Calculus 1 Gail Kaufmann F TRF - 12:00-12:50PM 78 Eaton 201Math 0032R 01RA Calculus 1R Gail Kaufmann BF F - 8:30AM-09:20AM 32 BP 006Math 0032R 01RB Calculus 1R Gail Kaufmann CT T - 9:30-10:20AM 32 BP 006Math 0032R 01RC Calculus 1R Gail Kaufmann EF F - 10:30-11:20AM 32 BP 006Math 0032 02 Calculus 1 Zbigniew Nitecki E MWF 10:30-11:20 90 Pearson 104Math 0032R 02RA Calculus 1R TA TA BR R - 8:30-9:20AM 32 BP 006Math 0032R 02RB Calculus 1R TA TA FR R - 12:00-12:50PM 32 BP 006Math 0032R 02RC Calculus 1R TA TA LR R - 4:30-5:20PM 32 BP 006

Math 0034 01 Calculus 2 Michael Chou DM - 09:30AM-10:20AM, TR - 10:30AM-11:20AM 90 Braker 001

Math 0034R 01RA Calculus 2R TA TA BR R - 8:30-9:20AM 32 BP 003Math 0034R 01RB Calculus 2R TA TA FR R - 12:00-12:50PM 32 BP 003Math 0034R 01RC Calculus 2R TA TA LR R - 4:30-5:20PM 32 BP 003Math 0034 02 Calculus 2 Boris Hasselblatt B TRF - 08:30AM-09:20AM 90 Robinson 253Math 0034R 02RA Calculus 2R TA TA CT T - 9:30-10:20 32 BP 005Math 0034R 02RB Calculus 2R TA TA FT T - 12:00-12:50 32 BP 003Math 0034R 02RC Calculus 2R TA TA JT T - 3:00-3:50 32 BP 003

Math 0034 03 Calculus 2 Robert Kropholler HTR - 1:30AM-2:20PM, F - 2:30PM - 3:20PM 90 Braker 001

Math 0034R 03RA Calculus 2R TA TA AW W - 8:30-9:20AM 32 BP 006Math 0034R 03RB Calculus 2R TA TA CW W - 9:30-10:20AM 32 BP 006Math 0034R 03RC Calculus 2R TA TA KW W - 4:30-5:20PM 32 BP 006

updated: 1/3/2019

Spring 2019 Mathematics Course Schedule

Math 0042 01 Calculus 3 Zachary Faubion C TWF - 09:30AM-10:20AM 90 Braker 001Math 0042 01RA Calculus 3R TA TA BR R - 8:30-9:20AM 32 BP 005Math 0042 01RB Calculus 3R TA TA FR R - 12:00-12:50PM 32 BP 007Math 0042 01RC Calculus 3R TA TA JR R - 3:00-3:50PM 32 BP 005Math 0042 02 Calculus 3 Caleb Magruder E MWF 10:30-11:20 90 Braker 001Math 0042 02RA Calculus 3R TA TA AW W - 8:30-9:20AM 32 BP 003Math 0042 02RB Calculus 3R TA TA KW W - 4:30-5:20PM 32 BP 003Math 0042 02RC Calculus 3R TA TA MW W - 6:00-6:50PM 32 BP 003

Math 0042 03 Calculus 3 Zachary Faubion HTR - 1:30AM-2:20PM, F - 2:30PM - 3:20PM 80 Anderson 206

Math 0042 03RA Calculus 3R TA TA CW W - 9:30-10:20AM 32 BP 002Math 0042 03RB Calculus 3R TA TA GW W - 1:30-2:20PM 32 Pearson 112Math 0042 03RC Calculus 3R TA TA MW W - 6:00PM-6:50PM 32 BP 006

Math 0051 01 Diff Eq Boris Hasselblatt F TRF 12:00-12:50PM 80 Anderson 206Math 0051 01RA Diff Eq R TA TA BR R - 8:30-9:20AM 32 BP 002Math 0051 01RB Diff Eq R TA TA JR R - 3:00-3:50PM 32 BP 002Math 0051 01RC Diff Eq R TA TA LR R - 04:30PM-05:20PM 32 BP 002Math 0051 02 Diff Eq Eunice Kim C TWF - 9:30AM-10:20AM 80 Robinson 253Math 0051 02RA Diff Eq R TA TA AW W 8:30-9:20 32 BP 005Math 0051 02RB Diff Eq R TA TA KW W 4:30-5:20 32 BP 005Math 0051 02RC Diff Eq R TA TA MW W 6:00-6:50 32 BP 005Math 0061 01 Discrete Math David Smyth G+ MW - 01:30PM-02:45PM 34 BP 003CS/Math 0061 03 Discrete Math Karen Edwards E MWF - 10:30AM-11:20AM 60 Anderson 206

CS/Math 0061 04 Discrete Math Karen Edwards HTR - 1:30AM-2:20PM, F - 2:30PM - 3:20PM 60 Eaton 201

Math 0070 01 Linear Algebra Mary Glaser F TRF - 12:00-12:50PM 40 BP 101Math 0070 02 Linear Algebra Eunice Kim E+WF WF - 10:30AM-11:45AM 40 BP 101Math 0070 03 Linear Algebra Todd Quinto D+ TR - 10:30AM-11:45AM 40 BP 007Math 0070 04 Linear Algebra Zachary Faubion G+ MW - 01:30PM-02:45PM 40 BP 007Math 0087 01 Math Modeling Caleb Magruder C TWF - 09:30AM-10:20AM 34 BP 003

Math 0123 01 Data Analysis Christoph Börgers N+ TR - 6:00-7:15PM 40 BP 101Math 0128 01 Numerical Linear Algebra Christoph Börgers H+ TR - 1:30PM-02:45PM 40 BP 101

updated: 1/3/2019

Spring 2019 Mathematics Course Schedule

Math 0135 01 Analysis Zbigniew Nitecki G+ MW - 01:30PM-02:45PM 40Sci-Tech Ctr, Room 135

Math 0136 01 Analysis 2 Loring Tu G+ MW - 01:30PM-02:45PM 45 BP 002Math 0146 01 Abstract Algebra 2 Robert Kropholler D+ TR - 10:30AM-11:45AM 30 BP 005Math 0150 01 Point Set Genevieve Walsh R+ MW - 9:00AM-10:15AM 30 BP 007Math 0158 01 Complex Analysis George McNinch E+MW MW - 10:30AM-11:45AM 30 BP 003

Math 0162 01 Statistics James Murphy J+ TR - 03:00PM-04:15PM 40Sci-Tech Ctr, Room 136

Math 0213 01 Complex Analysis Fulton Gonzalez N+ TR - 6:00-7:15PM 20 BP 003Math 0216 01 Graduate Algebra 2 David Smyth I+ MW - 3:00-4:15PM 20 BP 005Math 0250 01 Algebraic Number Theory Michael Chou H+ TR - 1:30PM-02:45PM 20 BP 005Math 0250 02 Tomography Todd Quinto J+ TR - 3:00-4:15PM 20 BP 007Math 0252 01 PDEs II Xiaozhe Hu F+TR TR - 12:00-1:15PM 20 BP 005Math 0253 01 Numerical PDEs James Adler D+ TR - 10:30AM-11:45AM 20 BP 101Math 0292 01 Graduate Development SeminaJames Adler ARR F - 10:30AM-11:45AM 15 BP 117

updated: 1/3/2019

Math 61 Discrete MathematicsCourse Information

Spring 2019

BLOCK: G+, MW 1:30-2:45INSTRUCTOR: David SmythEMAIL: david.smyth@tufts.eduOFFICE: Bromfield-Pearson 217

PREREQUISITE: No formal prerequisites. Readiness to think deeply about challenging material is,however, essential.

TEXT: Mathematical Thinking: Problem-Solving and Proofs (2nd Edition) by John D’Angelo andDouglas West.

COURSE DESCRIPTION: This course has two primary goals. The first is to introduce you to thebasic conceptual building blocks - sets, functions, and relations - that lie at the foundation of allhigher mathematics and computer science. The second is to give you experience in developing,writing, and explaining mathematical proofs. You will practice these skills each week by writingup solutions to a range of interesting problems across discrete mathematics. In particular, we willcover the following topics.

1. Logical quantifiers, the language of mathematics

2. Induction

3. Functions, Bijections, and Cardinality.

4. Binomial Coefficients and Combinatorics.

5. Number Theory and Modular Arithmetic

6. Probability Theory and Game Theory

7. Graph Theory

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Math 87 Mathematical Modeling and CompuationCourse Information

Spring 2019

BLOCK: C, TWF 9:30-10:20INSTRUCTOR: Caleb MagruderEMAIL: caleb.magruder@tufts.eduOFFICE: Bromfield-Pearson 105OFFICE HOURS: (Fall 2018) W 10:30–11:45, Th 9:30–10:20

PREREQUISITE: Math 34 or 39, Math 70 or 72, or consent.

TEXT: None. (I will distribute notes.)

COURSE DESCRIPTION: This course is about using elementary mathematics and computing tosolve practical problems. Single-variable calculus is a prerequisite, as well as some basic linearalgebra skills; other mathematical and computational tools, such as elementary probability, ele-mentary combinatorics, and computing in MATLAB, will be introduced as they come up.

Mathematical modeling is an important area of study, where we consider how mathematicscan be used to model and solve problems in the real world. This class will be driven by studyingreal-world questions where mathematical models can be used as part of the decision-making pro-cess. Along the way, we’ll discover that many of these questions are best answered by combiningmathematical intuition with some computational experiments.

Some problems that we will study in this class include:

1. Managed use of natural resources. Consider a population of fish that has a natural growthrate, which is decreased by a certain amount of harvesting. How much harvesting should beallowed each year in order to maintain a sustainable population?

2. Optimal use of labor. Suppose you run a construction company that has fixed numbers oftradespeople, such as carpenters and plumbers. How should you decide what to build tomaximize your annual profits? What should you be willing to pay to increase your laborforce?

3. Spring networks. Structural modeling of mechanical networks can be used to estimate thedisplacement of bridges and buildings. How much does a bridge bend when loaded withvehicles? Can structural modeling anticipate resonant modes of vibration in a network? Is itpossible to detect structural defects of a bridge from displacement measurements alone?

Using basic mathematics and calculus, we will address some of these issues and others, such asdealing with the MBTA T system and penguins (separately of course...). This course will alsoserve as a good starting point for those interested in participating in the Mathematical Contest inModeling each Spring.

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Math 123 Mathematical Aspects of Data AnalysisCourse Information

Spring 2019

BLOCK: N+, TuTh 6:00–7:15INSTRUCTOR: Christoph BorgersEMAIL: cborgers@tufts.eduOFFICE: Bromfield-Pearson 215OFFICE HOURS: (Fall 2018) Tu, We 10:30–12:00 and by appointmentPHONE: (617) 627-2366

PREREQUISITE: Math 42 (Multivariate Calculus), Math 70 (Linear Algebra) or Math 72 (AbstractLinear Algebra), and willingness to do Matlab programming. Matlab experience is helpful, but notrequired.

TEXT: None. (I will distribute notes.)

COURSE DESCRIPTION: This class is about the computational analysis of large sets of multi-dimensional data, thought of as large clouds of points in Rd . We focus on fundamental ideas basedon Linear Algebra and Calculus, presented without assuming any prior background in Statistics.

1. The principal component analysis (PCA), a tool for representing high-dimensional data in alower-dimensional linear space while retaining the most important information.

2. k-means clustering, the simplest algorithm for detecting clusters in a cloud of points, andclustering using diffusion on weighted graphs.

3. Support vector machines, a class of methods for classifying points into two or more cate-gories (think cats and dogs, or cats, dogs, and rabbits), based on training examples. This isbased on optimization algorithms. (No background in optimization is assumed.) There is aninteresting connection between classification into multiple categories and the mathematicaltheory of multi-candidate elections.

4. Tensors, a generalization of the notions of vectors and matrices. The set of all real vectors ofdimension N is denoted by RN . The set of all real M×N-matrices is RM×N . The next step inthis development is RL×M×N , the set of real L×M×N-tensors. Think for instance of a videoconsisting of a temporal sequence of two-dimensional images. Each image might be thoughtof as a matrix, which makes the video a stack of matrices, a “third-order tensor”.

5. Most algorithms in this course have a linear flavor: PCA reduces to a lower-dimensionallinear space; clustering algorithms and support vector machines separate clusters by hy-perplanes. However, by first non-linearly mapping the data into a different space, then ap-plying the “linear” methods to the images, one can use PCA to reduce data more generally tolower-dimensional surfaces, or separate clouds of points by hypersurfaces. There is a sub-tlety here, known as the kernel trick, which allows doing the computation without evaluatingthe nonlinear map, and even without really knowing it. This trick seems like black magicuntil you understand its theoretical foundation, the Moore-Aronszajn theorem.

Matlab will be used throughout to illustrate all ideas discussed in the course.6

Math 128 Numerical Linear AlgebraCourse Information

Spring 2019

Block: H+ (TuTh 1:30–2:45)Instructor: Christoph BorgersEmail: cborgers@tufts.eduOffice: Bromfield-Pearson 215Office hours: (Fall 2018) TuWe 10:30–noon, and by appointmentPhone: (617) 627-2366

Prerequisites: Math 70 or 72, and willingness and ability to do significant Matlab pro-gramming with only a small amount of programming instruction.

Text: None required. I will distribute notes, and recommend various reading materialsfreely available online.

Course description: We will study the computational solution of two types of problemsin this course: (1) systems of equations: Ax = b, where b is a given vector, A is a givenrectangular matrix, and the vector x to be computed, and (2) eigenvalue problems: Ax = λx,where A is a given square matrix, and the vector x 6= 0 and the number λ are to becomputed. The importance of these problems could hardly be over-stated. Large linearsystems of equations are central parts of almost all computational problems in the physicalsciences and in engineering. Eigenvalue problems are the key to understanding the stabilityof physical systems. Even though these problems are so fundamental, their computationalsolution remains the subject of a large and active current research field.

Of course, linear systems of equations can in principle be solved using the row reductionmethod that you learned about in your linear algebra course, if not in high school. However,the systems that one must solve in practice are often gigantic: A system with a millionequations in equally many unknowns would be considered of moderate size. For systems ofthis size, row reduction is hopelessly inefficient, and one must invent faster methods, typicallyexploiting special properties of the systems to be solved.

General eigenvalue problems cannot be solved explicitly if the matrix A is larger than4×4. In fact, if one could solve all n×n eigenvalue problems, one could solve all polynomialequations of degree n (you will learn why in the course), but it is known that there is noexplicit formula for solving all polynomial equations of degree n > 4. One must thereforedevelop algorithms for computing approximations to the solutions of eigenvalue problems.

We will study the most fundamental algorithms for linear systems and eigenvalue prob-lems and their mathematical foundations. We will implement and test the algorithms inMatlab.

7

Math 135 Real Analysis ICourse Information

Spring 2019

Block: G+WF (Wed Fri 1:30–2:45)Instructor: Zbigniew NiteckiEmail: zbigniew.nitecki@tufts.eduOffice: Bromfield-Pearson 214Office hours: (Spring 2019)Mon. 11:30-12:00, Wed & Fri 11:30-12:45Phone: (617) 627-3843Prerequisites: Math 42 or 44, and 70, or consent.

Text: Real Mathematical Analysis, Second Edition by Charles C. Pugh, Springer, 2015.

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 32, 34, and 42, where the emphasis is onintuition and computation, the emphasis in real analysis is on justification and proofs.1

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).

1Students who have taken Math 44 will see much that is familiar, but in a much grander setting. Similarly,your proof-reading and-writing skills will be taken to a higher level.

8

Math 136 Real Analysis 2Course Information

Spring 2019

BLOCK: G+ (MW 1:30–2:45), Problem Session: Monday 7:30–8:30 p.m.INSTRUCTOR: Loring TuEMAIL: loring.tu@tufts.eduOFFICE: 206 Bromfield-Pearson HallOFFICE HOURS: (Fall 2018) Mon Wed 4:20–5:10 and most afternoonsPHONE: (617) 627-3262

PREREQUISITES: Math 135 or consent.

TEXT: Patrick Fitzpatrick, Advanced Calculus, 2nd edition, American Mathematical So-ciety, 2009. (ISBN 978-0-8218-4791-6)TEXT FOR FOURIER SERIES AND HILBERT SPACES (WILL BE ON RESERVE): Jerrold E. Mars-den and Michael J. Hoffman, Elementary Classical Analysis, 2nd edition, W. H. Freeman andCompany, 1993.

COURSE DESCRIPTION: In Math 136, you’ll learn fundamental modern mathematics thatis beautiful and useful in fields as diverse as physics and economics. You will apply themathematics you learned in Math 135 to lay the theoretical foundation of calculus andmodern analysis and hone your ability to precisely formulate mathematical ideas and toread and write proofs.

First, you will learn about the derivative of functions f : Rn→ Rm. This derivative is

defined as a linear transformation (or a matrix). You will learn the inverse and implicitfunction theorems, which are important for differential geometry and economics. Theinverse function theorem gives conditions under which one can locally invert a functionf : Rn

→ Rn (answer: (essentially) when the derivative matrix is invertible). We willdefine the Riemann integral for functions f : A → R where A is a bounded subset of Rn.This will allow us to define the volume of many sets in Rn. For example, the volume of theunit interval [0, 1] ⊂ R is one, as you might guess. However, the set of rational numbersin [0, 1] does not have volume. We define a beautiful generalization, sets of measure zero(such as Q ∩ [0, 1]), and use this concept to characterize the functions that are Riemannintegrable. Then we will learn convergence theorems for integrals and Fubini’s theorem.

Finally, we will learn Fourier Series and Hilbert Spaces. This will involve exciting con-cepts, such as infinite dimensional vector spaces and infinite bases. We anticipate usingthese concepts to solve the partial differential equation describing heat distribution in aninsulated rod. We anticipate proving that the solution becomes infinitely smooth as soonas the experiment starts and that the temperature of every point on the rod approachesthe average temperature as time increases.

There will be weekly problem sets, two tests during the term, and a cumulative finalexam.

9

Math 146 Abstract Algebra IICourse Information

Spring 2019

BLOCK: H+, TuTh 1:30–2:45INSTRUCTOR: Robert KrophollerEMAIL: robert.kropholler@tufts.eduOFFICE: SEC LL010OFFICE HOURS: (Fall 2018) Mo, We 11:00–12:00 and by appointment

PREREQUISITE: Math 145

TEXT: Online notes OPTIONAL TEXT:Abstract Algebra , 3rd ed., J. Beachy and W. Blair, Wave-land Press, 2006.

COURSE DESCRIPTION: This course is a continuation of Math 145. We will continue our studyof abstract algebraic structures such as groups, rings and fields in order to prove that there is noformula for solving a general polynomial equation of degree 5 or higher. The main ingredient isa beautiful connection between groups and the roots of a polynomial discovered by Galois in theearly 1800’s. Along the way we will see the formulas for the general cubic and quartic polynomialequation discovered by the Italians in the 1500’s.

There will be homework sets, a mid-term, and a final exam in the course.

10

Math 150-01 Point-Set TopologyCourse Information

Spring 2019

BLOCK: R+ (Monday, Wednesday 9:00-10:15 AM)INSTRUCTOR: Genevieve WalshEMAIL: genevieve.walsh@tufts.eduOFFICE: LL0111 SECOFFICE HOURS: (Fall 2018) Tuesday Thursday 9 - 10:30

PREREQUISITES: Some course involving proofs. For example: 70, 72, 61,63.

TEXT: Notes provided by the instructor.

COURSE DESCRIPTION: The subject of point-set topology is extremely useful in manyareas, particularly in analysis and topology, and some areas of group theory. It is alsovery fun, and a great way to learn to think rigorously. This class works both before 135or after. We will start by covering cardinality, the axiom of choice and general topology,including creating our own topologies. We will describe what a basis of a topology is, thesubspace topology, and the product topology. Then we will discuss various separationand covering properties, and investigate maps between topological spaces. Finally, wewill discuss notions of connectivity of a topological space. Throughout, specific exampleswill be emphasized. There are many pathological and interesting examples in the field.

This course is designed to get you to think mathematically for yourself. To that end,there will be a running list of definitions and theorems and the goal of the class is tojointly work through proving the theorems. Everyday, students from the class will presentproblems and the class will discuss them and decide if they are ready to proceed. Assuch, this course is a great preparation for business, economics, law and mathematics.This method of instruction is called Inquiry Based Learning, and it is a very fun, dfferent,and challenging way to deeply engage with the material. Please come with an open mindabout what you can accomplish.

There will be daily homework assignments, a midterm, and a final.

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Math 158 Complex AnalysisCourse Information

Spring 2019

BLOCK: E+MW; MW - 10:30AM-11:45AMINSTRUCTOR: George McNinchEMAIL: george.mcninch@tufts.eduOFFICE: Bromfield-Pearson 112OFFICE HOURS: (Fall 2018) Tu, We 11:00–12:00; Thu 1:00-2:00 and by appointmentPHONE: (617) 627-6210

PREREQUISITE: Math 42 (Multivariable Calculus) or equivalent, or consent of the instructor.

TEXT: Jerrold E. Marsden and Michael J. Hoffman, Basic Complex Analysis, 3rd ed., Freeman,1998. (ISBN-10: 071672877X)

COURSE DESCRIPTION:This course is an introduction to the theory of functions of a complex variable. The study of

such functions has wide-ranging applications; moreover, it is fascinating in its own right, withmany elegant and surprising results.

We will primarily explore the notion of an analytic function on a given (open) region in thecomplex plane. Quite simply, an analytic function is a differentiable function on such a region.In stark contrast to the situation for functions of a real variable, one knows that – without furtherassumptions – such a function may be described using power series, and is completely determinedby its behavior in an arbitrarily small open subset of the given region.

Topics to be explored in the study of analytic functions include Cauchy’s theorem and integralformula, Taylor and Laurent series, contour integration, the Maximum Principle, analytic continu-ation, harmonic functions, conformal mappings, and linear fractional transformations.

The material in the course will be developed rigorously, with plenty of theorem-proving, butalso with computational examples and problems. The course is appropriate for students majoringin mathematics, a science, or engineering.

I plan for the course to have a midterm, a final exam, and weekly homework sets.

12

Math 162 StatisticsCourse Information

Spring 2019

BLOCK: J+, TR - 03:00-04:15 or L+, TR - 04:30-05:45INSTRUCTOR: Ky TranEMAIL: ky.tran@tufts.eduOFFICE: Bromfield-Pearson 106OFFICE HOURS: Tue, Thu 1:20–2:50 and by appointmentPHONE: (617) 627-2357

PREREQUISITE: Math 161 or consent.

TEXT: Larsen, Richard J. and Marx, Morris L. An Introduction to Mathematical Statistics and ItsApplications, 6th edition, Pearson, ISBN-13: 978-0134114217

COURSE DESCRIPTION:In Math 161, we were introduced to several important probability distributions, including dis-

crete distributions (such as the Bernoulli, Geometric, Binomial, and Poisson distributions) as wellas continuous distributions (such as the Normal, Uniform, Exponential, and Gamma distributions).In Math 162, we will consider data samples which have been taken from these known distributionswhich are now familiar to us. We have seen that each of these distributions depends on at leastone parameter. For example, the Geometric distribution, which models the waiting time until thefirst success in a sequence of yes/no trials, depends on the success probability p. The Normaldistribution depends on two parameters: the mean and the standard deviation for the distribution.It is typically the case that when data is sampled from a particular distribution, the values for therelevant parameters are unknown. How do you estimate the values for these parameters based onyour sample data? Furthermore, since there are often several ways to estimate these parameters,what properties would you like these estimators to have? We will answer these questions (andmany other important questions) in Math 162.

In addition to parameter estimation, we will use sample data to construct confidence intervals(intervals which contain an unknown parameter for a probability distribution with high likelihood).We will also use sample data to test hypotheses related to the value of an unknown parameter in adistribution. We will conduct Goodness-of-Fit tests to verify whether sample data fits a particularprobability distribution, and we will also construct contingency tables to determine whether or nottwo data sets are independent from one another.

This course will have weekly homework assignments, two midterm exams, and a final exam.

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Math 213 Complex AnalysisCourse Information

Spring 2019

BLOCK: N+, Tuesdays and Thursdays 6:00–7:15 pmINSTRUCTOR: Fulton GonzalezEMAIL: fulton.gonzalez@tufts.eduOFFICE: Bromfield-Pearson 203OFFICE HOURS: (Fall 2018) Tuesdays and Thursdays, 12:00–1:30 pm or by appointmentPHONE: (617) 627-2368

PREREQUISITE: A graduate-level course in real analysis (e.g., Math 211 at Tufts), or the permis-sion of the instructor.

TEXT: Walter Rudin, Real and Complex Analysis, Third Edition, McGraw-Hill, 1987.

ADDITIONAL REFERENCES:Functions of a Complex Variable I, Second Edition, by John Conway, Springer, New York, 1978.A Guide to Complex Variables, by Steven Krantz, Mathematical Association of America, 2008.(Available online for free.)

COURSE DESCRIPTION: Complex function theory is a beautiful subject with many surprising anddeep results, and with connections to many other areas of mathematics, including number theory,representation theory, and harmonic analysis.

We’ll start at the beginning: Cauchy’s theorem and integral formula, the residue theorem, and theclassification of singularities. Then we’ll look at various properties of harmonic functions andthe Poisson kernel. We’ll examine consequences of the maximum modulus principle, includingPhragmen-Lindelof methods and interpolation theorems.

We will then explore two sister theorems: the Mittag-Leffler theorem, which posits the existenceof meromorphic function with preassigned poles, and the Weierstrass interpolation theorem (alongwith infinite products), which posits the existence of holomorphic functions with preassigned ze-ros.

Further topics will include normal families, conformality and the Riemann mapping theorem, an-alytic continuation and the monodromy theorem, Riemann surfaces and modular functions, valuedistribution theory, and Picard’s little and big theorems. Time permitting, we will also study sub-harmonicity and Hardy spaces.

14

Math 250-02 TomographyCourse Information

Spring 2019

BLOCK: J+, TuTh 3:00-4:15INSTRUCTOR: Todd QuintoEMAIL: todd.quinto@tufts.eduOFFICE: Bromfield-Pearson 204OFFICE HOURS: (Fall 2018) Tuesdays 1:00-2:00, Wednesdays 3:30-4:30, Fridays 1:30-2:30 andby appointmentPHONE: (617) 627-3402

PREREQUISITES: Math 211 or consent. If you have taken Math 136 and feel comfortable withRiemann integrals in R2 or Rn, please consider taking the course!

RECOMMENDED TEXTS:

• The Mathematics of Computerized Tomography, Frank Natterer, SIAM Classics in Mathe-matics, 2001.

• Mathematical Methods in Image Reconstruction, Frank Natterer and Frank Wubbeling, SIAMMonographs on Mathematical Modeling and Computation, 2001.

• Functional Analysis, Walter Rudin, McGraw-Hill Series in Higher Mathematics, 1973 (ed2., 1991).

COURSE DESCRIPTION: Tomography is the mathematics behind medical and industrial CAT scan-ners, as well as other imaging methods in medicine, industry, and science. In each case indirectdata, such as from X-rays, is used to find the internal structure of objects. Tomography has revo-lutionized diagnostic medicine; now doctors diagnose conditions such as stroke, epilepsy, tumors,and Alzheimer’s disease without doing exploratory surgery. Scientists use electron microscope to-mography to discover the subcellar and molecular structure of biological specimens, and engineersuse tomography to find structural defects in industrial nondestructive evaluation.

We will learn the mathematical analysis and applied mathematics that underlie the field. Youmight also do some programming in any language you like (e.g., C, C++, Fortran, Matlab, or ...).We will start with some functional analysis and then basic formulas, properties, and algorithmsfor the Radon line transform. We will discuss distributions, the Fourier transform, and SobolevSpaces.

We will cover microlocal analysis, a precise, powerful, and beautiful mathematical theory thatallows one to rigorously characterize singularities of functions. We will prove the microlocalproperties of the Radon transform. This will give us a paradigm to understand limitations intrinsicto each of the tomography problems we study. We will use microlocal analysis to describe someof the newest results characterizing artifacts and visible singularities in limited data X-ray CT.

If time, we plan to study other tomographic transforms, including those behind seismic imag-ing, synthetic aperture radar, and photoacoustic tomography.

I anticipate having weekly homework, a final test, and perhaps presentations at the end of thecourse.

15

Math 252 Partial Differential Equations IICourse Information

Spring 2019

BLOCK: D+, TuTh 10:30-11:45INSTRUCTOR: Xiaozhe HuEMAIL: xiaozhe.hu@tufts.eduOFFICE: Bromfield-Pearson 212OFFICE HOURS: (Fall 2018) Tuesday, Thursday 2:30–4:30pm and by appointmentPHONE: (617) 627-2356

PREREQUISITE: Math 251 or consent

TEXT: Partial Differential Equations, Second Edition by Lawrence C. Evans (American Mathe-matical Society, 2010)

COURSE DESCRIPTION: Partial differential equations (PDEs) are widely used to model a variatyof physical phenomena such as heat diffusion, sound, electostatics, fluid dynamics, and elasticity.Math 252 is a continuation of Math 251 (Partial Differential Equations I). In this course, we studyPDEs from the functional analysis point of view and its applications to both linear and nonliearPDEs.

We shall begin with a review of results in functional analysis relevant to PDEs, including dis-tribution and weak solutions. Much of this material will come from Chapter 5 in the textbooks byEvans. We will also include a survey of Lebesque spaces and Sobolev spaces and their applictionsto PDEs and variational principles.

We will then deploy these tools in the study of second-order elliptic equations, generalizingwhat we have learned about the Laplace and Poisson equations to cover the existence of weak so-lutions, the Lax-Milgram theorem, interior regularity and boundary regularity, and elliptic eigen-value problems. Much of this material will follow Chapter 6 in the textbook by Evans. We willnext turn our attention to second-order parabolic and hyperbolic equations. We will also investi-gate PDEs from the perspective of dynamical systems, including semigroup theory. This materialwill follow Chapter 7 in Evans.

We shall study various ways of representing the solutions of different types of PDEs and variousfeatures of these solutions. In particular, we will investigate the well-posedness of PDEs and theeixstence and uniqueness of the solutions. Meanwhile, we shall learn how to find solutions toPDEs via different methods, including seperation of variables, eigenfuction expansions, asymptoticmethods, Fourier analysis, ad finding Green’s functions.

The prerequisite is MA 251. There will be biweekly problem sets and a final exam.

16

MATHEMATICS MAJOR CHECKLIST

STUDENTS: Submit with the Advisement Report and any other major or minor checklists to the Student

Services Desk by the due date. See attached for instructions and policies. If substitutions are made, it is the

student’s responsibility to make sure the substitutions are acceptable to the Math Department.

Student Name: I.D.:

Other Major(s): Minors:

TEN COURSES TO BE DISTRIBUTED AS FOLLOWS:

I. Five courses required of all majors:

Semester/Year Grade Credit

1. Math 42 Calculus III OR

Math 44 Honors Calculus

2. Math 70 Linear Algebra OR

Math 72 Abstract Linear Algebra

3. Math 135 Real Analysis I

4. Math 145 Abstract Algebra I

5. Math 136 Real Analysis II OR

Math 146 Abstract Algebra II

II. Two additional 100-level math courses.

Course Number Course Title Semester/Year Grade Credit

III. Three additional mathematics courses numbered 50 or higher (in the new numbering scheme); up to

two of these courses may be replaced by courses in related fields including: Chemistry 133, 134; Computer

Science 15, 126, 160, 170; Economics 107, 154, 201, 202, 207; Electrical Engineering 18, 107, 108, 125;

Engineering Science 151, 152; Mechanical Engineering 137, 138, 150, 165, 166; Philosophy 33, 103, 114, 170;

Physics 12, 13 any course numbered above 30; Psychology 107, 108, 140.

Course Number Course Title Semester/Year Grade Credit

17

IV. If you have taken any other Mathematics courses that are not being used for the above requirements,

and you would like to have them considered for Latin Honors, please list them here:

Course Number Course Title Semester/Year Grade Credit

V. List any of the above courses that you are also using towards another major.

Course Number Course Title Semester/Year Grade Credit

Student’s Signature: Date:

I/We certify that completion of the above courses will satisfy all requirements for the Mathematics major.

Advisor’s Signature: Date:

Department Chair’s Signature: Date:

MATHEMATICS MAJOR INSTRUCTIONS AND POLICIES

• Ten courses required, beyond Calculus II.

• No major course may be taken Pass/Fail.

• If you have more than one major, please see Bulletin for rules on double-counting courses.

• If you have a minor, no more than two course credits used toward the minor may be used toward

foundation, distribution, major, or other minor requirements.

• Please see Departmental Website for complete major requirements and policies.18

APPLIED MATHEMATICS MAJOR CHECKLIST STUDENTS: Submit with the Advisement Report and any other major or minor checklists to the Student Services Desk by the due date. See attached for instructions and policies. If substitutions are made, it is the student’s responsibility to make sure the substitutions are acceptable to the Math Department.

Student Name: I.D.:

Other Major(s): Minors:

THIRTEEN COURSES TO BE DISTRIBUTED AS FOLLOWS:

I. Seven courses required of all majors: 1. Math 42 Calculus III

Math 44 Honors Calculus

2. Math 70 Linear Algebra

Math 72 Abstract Linear Algebra

3. Math 51 Differential Equations

Math 155 Nonlinear Dynamics & Chaos

4. Math 87 Mathematical Modeling

5. Math 158 Complex Variables

6. Math 135 Real Analysis I

7. Math 136 Real Analysis II

Semester/Year Grade Credit

OR

OR

OR

II. One of the following: Math 145 Abstract Algebra I

Math 61/Comp 61 Discrete Mathematics

Comp 15 Data Structures

Math/Comp 163 Computational Geometry

Semester/Year Grade Credit

III. One of the following three sequences: 1. Math 126 Numerical Analysis/

Math 128 Numerical Algebra

2. Math 151 Applications of Advanced Calculus/

Semester/Year Grade Credit

Math 152 Nonlinear Partial Differential Equations

3. Math 161 Probability /

Math 162 Statistics

19

IV. An additional course from the list below but not one of the courses chosen in section III:

Semester/Year Grade Credit

Math 126 Numerical Analysis

Math 128 Numerical Algebra

Math 151 Partial Differential Equations I

Math 152 Partial Differential Equations II

Math 161 Probability

Math 162 Statistics

IV. Two elective courses (Math courses numbered 61 or above are acceptable electives. With the approval of theMathematics Department, students may also choose as electives courses with strong mathematical content that are notlisted as Math courses.)Course Number Course Title Semester/Year Grade Credit

V. If you have taken any other Mathematics courses that are not being used for the above requirements, and youwould like to have them considered for Latin Honors, please list them here:Course Number Course Title Semester/Year Grade Credit

VI. List any of the above courses that you are also using towards another major.

Course Number Course Title Semester/Year Grade Credit

Student’s Signature: Date:

I/We certify that completion of the above courses will satisfy all requirements for the Applied Mathematics major.

Advisor’s Signature: Date:

Department Chair’s Signature: Date: 20

APPLIED MATHEMATICS MAJOR INSTRUCTIONS AND POLICIES

• Thirteen courses required, beyond Calculus II.

• No major course may be taken Pass/Fail.

• If you have more than one major, please see Bulletin for rules on double-counting courses.

• If you have a minor, no more than two course credits used toward the minor may be used toward

foundation, distribution, major, or other minor requirements.

• Please see Departmental Website for complete major requirements and policies.

21

MATHEMATICS MINOR CONCENTRATION CHECKLIST

In addition to this form, students must complete the “Declaration of Major(s)/Minor/Change of Advisor for Liberal Arts Students” form.

Name: I.D.#:

E-Mail Address: College and expected graduation semester/year:

Major(s):

Faculty Advisor for Minor (please print)

Please list courses by number. For transfer courses, list by title and add “T”. Indicate which courses are incomplete, in progress, or to be taken. Courses numbered under 100 will be renumbered starting in the Fall 2012 semester. Courses are listed here by their new number, with the old number in parentheses. Note: If substitutions are made for courses listed as “to be taken”, it is the student’s responsibility to make sure that the substitutions are acceptable.

Six courses distributed as follows:

I. Two courses required of all minors. (Check appropriate boxes.)If “in progress” or future semester, note semester.

Grade: Grade: 1. Math 42 (old: 13): Calculus III or ______ 2. Math 70 (old: 46): Linear Algebra or ______

Math 44 (old: 18): Honors Calculus ______ Math 72 (old: 54): Abstract Linear Algebra ______

II. Four additional math courses with course numbers Math 50 or higher (in the newnumbering scheme).These four courses must include Math 135: Real Analysis I or 145: Abstract Algebra (or both).

Note that Math 135 and 145 are typically only offered in the fall.

Grade: Grade: 1. __________________ ______ 3. __________________ ______ 2. __________________ ______ 4. __________________ ______

Student’s signature: __________________________________ Date: ____________

Advisor’s signature: __________________________________ Date: ____________

Note: It is the student’s responsibility to return completed, signed degree sheets to the Office of Student Services, Dowling Hall.

(form revised September 2, 2015)

22

Jobs and Careers

The Math Department encourages you go to discuss your career plans with your professors. All of

us would be happy to try and answer any questions you might have. Professor Quinto has built up

a collection of information on careers, summer opportunities, internships, and graduate schools

and his web site (http://equinto.math.tufts.edu) is a good source.

Career Services in Dowling Hall has information about writing cover letters, resumes and job-

hunting in general. They also organize on-campus interviews and networking sessions with

alumni. There are job fairs from time to time at various locations. Each January, for example,

there is a fair organized by the Actuarial Society of Greater New York.

On occasion, the Math Department organizes career talks, usually by recent Tufts graduates. In

the past we had talks on the careers in insurance, teaching, and accounting. Please let us know if

you have any suggestions.

The Math Society

The Math Society is a student run organization that involves mathematics beyond the classroom.

The club seeks to present mathematics in a new and interesting light through discussions,

presentations, and videos. The club is a resource for forming study groups and looking into career

options. You do not need to be a math major to join! See any of us about the details. Check out

https://www.facebook.com/tuftsmathsociety for more information.

The SIAM Student Chapter

Students in the Society for Industrial and Applied Mathematics (SIAM) student chapter organize

talks on applied mathematics by students, faculty and researchers in industry. It is a great way to

talk with other interested students about the range of applied math that’s going on at Tufts. You

do not need to be a math major to be involved, and undergraduates and graduate students from a

range of fields are members. Check out https://sites.google.com/site/tuftssiam/ for more

information.

23

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