Course Information Book

Department of Mathematics

Tufts University

Spring 2018

This booklet contains the complete schedule of courses offered by the Math Department in the Spring 2018

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 course*

Course schedule p. 2

Math 61 p. 3

Math 70 p. 4

Math 87 p. 5

Math 123 p. 6

Math 135 p. 7

Math 136 p. 8

Math 146 p. 9

Math 150-01 p. 10

Math 150-02 p. 11

Math 150-03 p. 12

Math 152 p. 13

Math 158 p. 14

Math 162 p. 15*

Math 168 p. 16

Math 212 p. 17

Math 216 p. 18

Math 218 p. 19

Math 228 p. 20

Math 250-01 p. 21

Math 250-02 p. 22

Mathematics Major Concentration Checklist p. 23

Applied Mathematics Major Concentration Checklist p. 24

Mathematics Minor Checklist p. 25

Jobs and Careers, Math Society, and SIAM p. 26

Block Schedule p. 27

Mathematics Course Schedule - Spring 2018 Subject Course Sect Title Instructor Name Block Time Block Description

MATH 0010 01 Quantitative Reasoning Glaser, Mary J+ TR 3:00-4:15PM

MATH 0016 01 Symmetry Kaufman, Gail D M - 09:30AM-10:20AM, TR - 10:30AM-11:20AM

MATH 0019 01 Math Of Social Choice Garant, Linda E+MW MW - 10:30AM-11:45AM

MATH 0019 02 Math Of Social Choice Duchin, Moon G+ MW - 01:30PM-02:45PM

MATH 0021 01 Introductory Statistics Garant, Linda G+ MW - 01:30PM-02:45PM

MATH 0021 02 Introductory Statistics Garmirian, Patricia C TWF - 09:30AM-10:20AM

MATH 0021 03 Introductory Statistics STAFF H TR - 01:30PM-02:20PM, F - 02:30PM-03:20PM

MATH 0032 01 Calculus I Chou, Michael D M - 09:30AM-10:20AM, TR - 10:30AM-11:20AM

MATH 0032 02 Calculus I Ruane, Kim* B TRF - 08:30AM-09:20AM

MATH 0034 01 Calculus II Faubion, Zachary G MWF 1:30-2:20PM

MATH 0034 02 Calculus II Hu, Xiaozhe* C TWF - 09:30AM-10:20AM

MATH 0034 03 Calculus II Lemke-Oliver, Robert F TRF 12:00-12:50PM

MATH 0042 01 Calculus III Adler, James* F TRF 12:00-12:50PM

MATH 0042 02 Calculus III Dyer, Jessica H TR 1:30-2:20, F 2:30-3:20

MATH 0051 01 Differential Equations McNinch, George E MWF 10:30-11:20

MATH 0051 02 Differential Equations Kim, Eunice C TWF - 09:30AM-10:20AM

MATH 0061 01 Discrete Mathematics Dyer, Jessica D M - 09:30AM-10:20AM, TR - 10:30AM-11:20AM

MATH 0061 02 Discrete Mathematics Faubion, Zachary I MW 3:00-3:50PM, F 3:30-4:20PM

MATH 0070 01 Linear Algebra Chou, Michael C TWF - 09:30AM-10:20AM

MATH 0070 02 Linear Algebra Magruder, Caleb I+ MW 3:00-4:15PM

MATH 0070 03 Linear Algebra Glaser, Mary F TRF - 12:00PM-12:50PM

MATH 0070 04 Linear Algebra Kim, Eunice J+ TR 3:00-4:15PM

MATH 0087 01 Mathematical Modeling Adler, James J+ TR 3:00-4:15PM

MATH 0123 01 Mathematical Aspects of Data Analysis Taylor, Kye Q+ TR 7:30-8:45PM

MATH 0135 01 Real Analysis I Hening, Alexandru F+TR TR - 12:00PM-01:15PM

MATH 0136 01 Real Analysis II Quinto, Todd E+MWF Mo, 1:30–2:45, WF 1:30–2:20

MATH 0136 01 Real Analysis II Tu, Loring G MW 3:00-3:50PM, F 3:30-4:20PM

MATH 0146 01 Abstract Algebra II Smyth, David I MW 3:00-3:50PM, F 3:30-4:20PM

MATH 0150 01 The Mathematics of Poverty and Inequality Boghosian, Bruce D+ TR - 10:30AM-11:45AM

MATH 0150 02 Elliptic Curves Lemke-Oliver, Robert H+ TR 1:30PM-02:45PM

MATH 0150 03 Constrained Optimization Magruder, Caleb K+ MW 4:30-5:45PM

MATH 0152 01 Partial Differential Equations II Boghosian, Bruce B+ TR 8:05-9:20AM

MATH 0158 01 Complex Variables Gonzalez, Fulton M+ MW 6:00-7:15PM

MATH 0162 01 Statistics Garmirian, Patricia I+ MW 3:00-4:15PM

MATH 0168 01 Algebraic Topology Kropholler, Robert G+ MW - 01:30PM-02:45PM

MATH 0212 01 Mathematical Biology and Stochastic Processes Hening, Alexandru 8 R 1:30-4:00PM

MATH 0216 01 Topics In Algebra Kropholler, Robert E+MW MW - 10:30AM-11:45AM

MATH 0218 01 Topics in Geometry and Topology Tu, Loring K+ MW 4:30-5:45PM

MATH 0228 01 Numerical Linear Algebra Borgers, Christoph ARR TR 9:00-10:15AM

MATH 0250 01 Graph Algorithms Hu, Xiaozhe F+TF TF - 12:00PM-01:15PM

MATH 0250 02 Galois Cohomology McNinch, George D M 9:30-10:20, TR 10:30-11:20

Discrete Math 0061, Course Description: In this course we will examine, think about, play with and get to know some of the objects and concepts that are at the core of advanced mathematics and computer science. Some specific examples of these concepts are sets, logic, proofs, relations and functions. We will also see these ideas in action by looking at combinatorics, cardinality and graphs (to list a few.)

The course is taught in the D and I blocks.

Math 70 Linear Algebra Spring 2018 Course Information

Block: C (TWF 9.30-10.20) Instructor: Michael Chou Email: Michael.Chou@tufts.edu Office: BP 105 Office: BP 105 Office hours: (Spring 2018)

Block: I+ (MW 3:00-4:15) Instructor: Caleb Magruder

Email: Caleb.Magruder@tufts.edu

Office: BP 105 Office hours: (Spring 2018)

Block: F (TRF 12:00-12:50) Instructor: Mary Glaser Email: Mary.Glaser@tufts.edu Office: BP 004 Office hours: (Spring 2018) Block: J+ (TR 3:00-4:15) Instructor: Eunice Kim Email: Eunice.Kim@tufts.edu Office: SEC LL010 Office hours: (Spring 2018)

Prerequisites: Math 34 or 39, or consent of instructor.

Text: Linear Algebra and Its Applications , 4th edition, by David Lay, Addison-Wesley

(Pearson), 2011.

Course description: Linear algebra begins the study of systems of linear equations in

several unknowns. The study of linear equations quickly leads to the important concepts of

vector spaces, dimension, linear transformations, eigenvectors and eigenvalues. These abstract

ideas enable effi- cient study of linear equations, and more importantly, they fit together in a

beautiful setting that provides a deeper understanding of these ideas.

Linear algebra arises everywhere in mathematics. It plays an important role in almost every upper level math course. It is also crucial in physics, chemistry, economics, biology, and

a range of other fields. Even when a problem involves nonlinear equations, as is often the case

in applications, linear systems still play a central role, since “linearizing” is a common

approach to non-linear problems.

This course introduces students to axiomatic mathematics and proofs as well as

fundamental mathematical ideas. Mathematics majors and minors are required to take linear

algebra (Math 70 or Math 72) and are urged to take it as early as possible, as it is a prerequisite for many upper- level mathematics courses. The course is also useful to majors

in computer science, economics, engineering, and the natural and social sciences.

The course will have two midterms and a final, as well as daily homework assignments.

Math 87 Mathematical Modeling and Computation Fall 2018 Course Information

Block: J+ (TR 3:00-4:15)

Instructor: James Adler

Email: James.Adler@tufts.edu

Office: TBA

Office hours: TBA

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

Text: none

Course description:

This course is about using elementary mathematics and computing to solve practical prob- lems. Single-

variable calculus is a prerequisite, as well as some basic linear algebra skills; other mathematical and

computational tools, such as elementary probability, elementary combinatorics, 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 driven by studying real-world

questions where mathematical models can be used as part of the decision-making process. Along the

way, we’ll discover that many of these questions are best answered by combining mathematical intuition

with some computational experiments.

Some problems that we will study in this class include:

1. The managed use of natural resources. Consider a population of fi that has a nat- ural growth

rate, which is decreased by a certain amount of harvesting. How much harvesting 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 fi numbers of

tradespeople, such as carpenters and plumbers. How should you decide what to build to

maximize your annual profit. What should you be willing to pay to increase your labor force?

3. Project scheduling. Think about scheduling a complex project, consisting of a large number of

tasks, some of which cannot be started until others have finished. What is the shortest total

amount of time needed to fi the project? How do delays in completion of some activities

affect the total completion time?

Using basic mathematics and calculus, we will address some of these issues and others, such as dealing

with the MBTA T system and penguins (separately of course...). This course will also serve as a good

starting point for those interested in participating in the Mathematical Contest in Modeling each Spring.

Math 123 Data AnalysisCourse Information

Spring 2018

BLOCK: Q+ (Tu, Th 7:30pm – 8:45pm)INSTRUCTOR: Kye TaylorEMAIL: kye.taylor@tufts.eduOFFICE: 106 Bromfield-Pearson Hall

PREREQUISITES: Math 42, Math 70/72, and the ability/willingness to program in Matlab or Python.

TEXT: None. I will distribute the slides that augment lectures.

COURSE DESCRIPTION: This class is about the computation analysis of large sets of high-dimensionaldata, thought of as large clouds of points in Rd. We focus on fundamental ideas based on LinearAlgebra, presented without assuming a background in Statistics. Topics include:

1. Data embeddings, including principal component analysis, a tool for representing high-dimensional data in lower-dimensional linear spaces.

2. Unsupervised learning algorithms including k-means clustering and spectral clustering

3. Support vector machines, a class of methods for classifying points into two ore more cate-gories based on training examples. Approaches for constructing models are primarily basedon optimization algorithms, though no background in optimization beyond multivariablecalculus is assumed. We will also discuss the connection between multi-class classificationand mutli-candidate elections.

4. 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 hyper-planes. However, by first non-linearly mapping the data into a different space, then applyingthe “linear” methods on the images, one can use PCA to reduce data more generally tolower-dimensional surfaces, or separate clouds of points using hypersurfaces. There is a sub-tlety here, known as the kernel trick, which allows doing the computation without evaluatingthe nonlinear map too often, or even without explicitly specifying it at all.

This course is suitable for upper level undergraduate and beginning graduate students fromMathematics, Engineering, Computer Science, or those with particular interest in data analysisand pattern classification. This course will consist of lectures and programming exercises.

Math 135 Real Analysis ICourse Information

Spring 2018

BLOCK: F+TR (TR 12:00 – 1:15)INSTRUCTOR: Alex HeningEMAIL: alexandru.hening@tufts.eduOFFICE: BP 201OFFICE HOURS:PHONE:

PREREQUISITES: Math 42 or 44, and 70 or 72, or consent.TEXT: Rudin, Principles of Mathematical Analysis, 3rd edition. We will cover chapters1-6 and parts of chapter 7.

COURSE DESCRIPTION:This course will be about the study of real functions, their derivatives and integrals. It

is a foundational course for mathematics, both pure and applied. While calculus is moreabout intuition and computation, the emphasis in real analysis is on rigorous proofs.

Our intuition does not always give the correct answer. That is why rigorous proofs arerequired in mathematics. Some of the mathematical concepts are not immediately easy tograsp, and our intuition can lead us astray. For example the set Z = {. . . ,−2,−1, 0, 1, 2, 3, . . . }of integers is a proper subset of the set of rationals Q =

{mn|m,n ∈ Z, n 6= 0

}. However,

as we will show during the course, there is a one-to-one correspondence (bijection) be-tween Z and Q. In this sense, the infinite sets Z and Q have the same ‘number’ of ele-ments. However, we will show there is no one-to-one correspondence between the realnumbers R and the rational numbers Q - even though both sets have an infinite numberof elements the set of real numbers is larger.

We will study metric spaces, compactness, connectedness, continuous mappings andmodes of convergence. We will also encounter results from calculus, such as the intermediate-value theorem, but in a more general setting that enlarges their applicability. By studyingthe fundamental ideas in a general setting, we will gain a deeper understanding of theideas and concepts involved.

The primary goal of this course is introducing the main concepts from real analysis. Asecondary goal is to get students used to reading and writing rigorous proofs.

There will be weekly homework, one midterm exam, and a final (and possibly a fewquizzes).

Math 136 Real Analysis 2Course Information

Spring 2018

Section 1: BLOCK E (Mo, We, Fr 10:30–11:45)INSTRUCTOR: Todd QuintoEMAIL: todd.quinto@tufts.eduOFFICE: 204 Bromfield-Pearson HallOFFICE HOURS: (Fall 2017) MWTh 1:30-2:30PHONE: (617) 627-3402Note: Section 1 will meet three days a week for 75 minutes each. The extended block onMonday (25 min.) is reserved for a problem session. The other class periods are extendedso you can play with the mathematics and conjecture and prove theorems yourself andin groups. Both sections will cover the same material and have the same tests and assign-ments.

Section 2: BLOCK: G+ (Mo, 1:30–2:45, We Fr 1:30–2:20)INSTRUCTOR: Loring TuEMAIL: loring.tu@tufts.eduOFFICE: 206 Bromfield-Pearson HallOFFICE HOURS: (Fall 2017) Mon 1:30–2:30 and most afternoonsPHONE: (617) 627-3262Note: Section 2 will meet three days a week for 50 minutes each. In addition, the extendedblock on Monday from 2:20 to 2:45 is reserved for a problem session. The extended blockon Wednesday from 2:20 to 2:45 may be used for a problem session if needed.

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 isdefined 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.

Math 146 Abstract Algebra II: Galois TheoryCourse Information

Spring 2018

BLOCK: I (MW 3:00–3:50, F 3:30-4:20)INSTRUCTOR: David SmythEMAIL: david.smyth@tufts.eduOFFICE: 217 Bromfield-Pearson HallOFFICE HOURS: (Spring 2018) MW 4:00-4:30, Th 2:30-4:30

PREREQUISITES: Math 145.

TEXT: There is no required textbook for the course, and lectures will be self-contained.If you own a copy of Fraleigh’s First Course in Abstract Algebra, the text for 145, that willserve as a useful secondary resource for most of the material that we cover.

COURSE DESCRIPTION: Galois theory is one of the crown jewels of modern mathemat-ics. Galois’ ideas (building on those of Abel, Legendre, and many others) exerted a majorinfluence on the course of modern mathematics, shifting emphasis away from explicit,messy algebraic calculations, and towards an understanding of the abstract structuresthat lay behind them. Using the techniques developed in this course, we will solve prob-lems which had been open for thousands of years (literally!). Here are two examples, onefrom geometry and one from algebra. A major unsolved problem from classical geometrywas how to trisect an angle using only ruler and compass. We’ll prove this is impossible.In a completely different direction, mathematicians in early modern europe spent a lot oftime hunting for generalizations of the quadratic formula to higher-degree polynomials.Again, we’ll see that this is impossible once the degree of the polynomial is at least five.

In more technical terms, topics to be covered include: symmetric functions, algebraicand transcendental field extensions, ruler and compass constructions, classification offinite fields, splitting fields, solvable groups, and the fundamental theorem of Galois the-ory.

Math 150-01 The Mathematics of Poverty and InequalityCourse Information

Fall 2016

BLOCK: D+ (Tue, Thu 10:30–11:45)INSTRUCTOR: Bruce BoghosianEMAIL: bruce.boghosian@tufts.eduOFFICE: 211 Bromfield-Pearson HallOFFICE HOURS: (Fall 2017) Tu 12–1:00, Th 12–1:00PHONE: (617) 627-3054TEXT: TBA, plus instructor’s notes, plus on-line material.PREREQUISITES: (i) Math 42 or 44 or consent is essential; (ii) MA 21 or MA 61 or some priorexposure to probability theory is important; (iii) MA 51 or some prior exposure to differentialequations would be helpful, though not essential; (iv) no prior experience in programming isassumed, but some exercises will be given that require the use of simple computational models,with ample instruction; (v) no prior background in economics will be assumed.

Today there are five people who have as much wealth as half the earth’s population. The enor-mous concentration of wealth and the unchecked growth of inequality have emerged as crucialsocial issues of our time. This course explores the extent to which mathematics can help shed lighton these matters.

We will begin by studying wealth distributions and and their quantification, including his-tograms, quantiles, the continuum approximation, and the Lorenz curve. Along the way, wewill learn about the abstract notion of a distribution due to Sobolev and Schwarz. These toolswill enable us to understand the history of empirical studies of wealth distribution, including theclassical observations of Pareto, Lorenz, Gini and Gibrat, as well as the more recent discovery ofwealth condensation by Bouchaud and Mezard which is thought to explain the origin of oligarchy.

We will next examine some of Amartya Sen’s ideas for quantifying inequality, and we willstudy various “static” inequality metrics in use today, including the Lorenz-Pareto, Gini, Atkin-son, Foster-Greer-Thorbecke and Sen indices. Along the way, we will learn about convexity,Jensen’s inequality, Holder means, and subgrid decomposibility.

Next we turn our attention to various ways of understanding wealth concentration. This willlead us to a study of random walks, the Wiener process, the Gambler’s Ruin problem, an intro-duction to Markov processes, an introduction to the Boltzmann and Fokker-Planck equations, andstrong and weak forms of conservation laws.

At this point, we will have the tools we need to study a class of agent-based dynamical mod-els, called asset-exchange models. We will show how simple models of agent transactions at themicroeconomic level can be used to derive dynamical equations for the wealth distribution. Wewill study the phenomenology exhibited by steady-state solutions to these equations in some de-tail, including work at the very state of the art of this subject. We will compare the results of thesemodels with empirical wealth distributions.

We will also explore what is known about the time-dependent behavior of these models, in-cluding implications for the growth of oligarchy, upward mobility, and time autocorrelation ofagent wealth. Open questions will be discussed.

The course will finish by examining the relationship between asset-exchange models and neo-classical microeconomics. Toward this end, we will review the latter subject, culminating in amathematical introduction to the General Equilibrium theory of Arrow and Debreu, upon whichmuch modern economic thinking is based.

Available databases for the study of wealth distribution will be surveyed, including that main-tained by the Federal Reserve and the U.S. Census Bureau, as well as available international data.

There will be weekly problem sets, at least one midterm and a final exam.

Math 150 Elliptic CurvesCourse Information

Fall 2017

BLOCK: H+ TTh (Tue, Thu 1:30–2:45)INSTRUCTOR: Robert Lemke OliverEMAIL: robert.lemke_oliver@tufts.eduOFFICE: Bromfield-Pearson 116OFFICE HOURS: (Fall 2017) Tue 12-1PHONE: (617) 627-0436

PREREQUISITES: Some exposure to proofs (e.g., Math 61, 63, 72, 135, or 145), or consent ofinstructor.

TEXT: Rational points on elliptic curves, 2nd ed., by Silverman and Tate. Springer UTMseries.

COURSE DESCRIPTION: An elliptic curve is defined remarkably simply – it is the set ofsolutions (x, y) to an equation of the form y2 = x3 + ax + b – so it is surprising that thereis room for any interesting behavior in their study. Indeed, if one is only concerned withthe real-valued solutions (i.e., x and y are real numbers), then almost all of the behaviorcan be understood using tools from calculus. However, if we require both x and y to berational numbers, then suddenly these methods do not apply. Indeed, the answers tovery basic questions are not obvious. Are there any solutions to the equation? Are thereinfinitely many solutions? Is there any structure to the solutions, or are they numericalaccidents? All of these questions require a deep and beautiful theory to answer, onewhich is of central importance in number theory. For example, the Birch and Swinnerton-Dyer conjecture, one of the seven Millenium Prize Problems, provides a unifying meansof determining whether the set of rational solutions is infinite.

Many classical problems in number theory can be understood via the theory of ellipticcurves. Fermat’s last theorem asserts that there are no non-zero integer solutions to theequation xn + yn = zn with n ≥ 3. Fermat famously didn’t write down a proof, and itwasn’t unti 1995 that Andrew Wiles provided a solution. Wiles’s proof uses crucially thetheory of elliptic curves. (More specifically, it amounted to showing that all elliptic curvesare “modular.” We will discuss what that word means, but not the ideas of Wiles’s proof.)

In this course, we will develop the basic theory of elliptic curves with an eye towardboth classical connections and recent developments (the theory of modular forms, ellip-tic curve cryptography, etc.). Grades will be based on weekly homework, a take-homemidterm, and a final presentation or exam, depending on enrollment. Assignments willhave a mix of computational and proof-based questions, so some familiarity with proofsis required.

Math 150-03 Constrained OptimizationCourse Information

Spring 2018

BLOCK: K+ (Mo, We 4:30–5:45)INSTRUCTOR: Caleb MagruderEMAIL: caleb.magruder@tufts.eduOFFICE: 105 Bromfield-Pearson HallOFFICE HOURS: (Spring 2018) Mo, We 10:30–12:00PHONE: (617) 627-1331

PREREQUISITES: Math 126 and 128 or consent. Programming ability in a language suchas C, C++, Fortran, Matlab, Python, etc.

TEXT: TBD

COURSE DESCRIPTION:Many problems in science, engineering, economics, and industry involve optimization

subject to constraints. Due to the wide and growing use of optimization in different areassuch as parameter estimation, model calibration, optimal control, and data assimilation,it is important for practitioners to understand optimization algorithms and their impacton science and engineering research and applications.

The course is a survey of constrained optimization algorithms and gives a compre-hensive description of the state-of-the-art techniques for solving continuous optimizationand inverse problems. The primary objective of the course is to develop an understand-ing of optimization algorithms and, more importantly, the capabilities and limitations ofeach. We will focus on convergence analysis, numerical stability, computational com-plexity, and, in particular, scalability of these algorithms. The topics include: theoryof constrained optimization (necessary and sufficient conditions for constrained opti-mization, Karush-Kuhn-Tucker conditions, etc.), optimization with implicit constraints,adjoint-based derivative computation, linear programming, penalty and barrier meth-ods, augmented Lagrangian, interior-point methods, sequential quadratic programming,optimal control, and more.

In-class examples, homework, and projects will feature some of the practical applica-tions of the algorithms to solve real-world problems as well as the theoretical justificationfor their use. Homework will consist of written problems as well as computer program-ming assignments in Matlab or Python.

This course is suitable for upper level undergraduates and beginning graduate stu-dents from Mathematics, Engineering, Operations Research, Computer Science, Biology,Chemistry, and Physics.

Math 152 Partial Differential Equations IICourse Information

Fall 2016

BLOCK: B+ (Tu, Th 8:05 – 9:20)INSTRUCTOR: Bruce BoghosianEMAIL: bruce.boghosian@tufts.eduOFFICE: 211 Bromfield-Pearson HallOFFICE HOURS: (Fall 2016) Tu 12:00 – 1:00, Th 12:00 – 1:00PHONE: (617) 627-3054

PREREQUISITES: Math 151 or consent.

TEXT: “An Introduction to Nonlinear Partial Differential Equations” by J. David Logan,Wiley, second edition, [link] .

COURSE DESCRIPTION:This course will substantially expand the repertoire of methods introduced in MA 151

to classify and analyze partial differential equations (PDEs), both linear and nonlinear.Emphasis will be on dynamical equations of hyperbolic and parabolic type, includingdiffusion, reaction-diffusion, linear and nonlinear wave, and hydrodynamic equations.

The course will begin with first-order quasilinear equations, the method of charac-teristics, complete integrals and envelope methods, asymptotic analysis methods, linearstability theory and similarity solutions. We will also discuss weak solutions to PDEs andjump conditions, and their application to shock propagation and traffic flow.

We will next turn to hyperbolic systems of equations, including those governing shallow-water waves and gas dynamics. We will learn about Riemann’s methodology for findingjump conditions for such systems, as well as hodograph and wavefront expansions.

This will be followed by an introduction to diffusion and reaction-diffusion equa-tions, using a variety of different methods of analysis and examples, including Burgers’equation and the Cole-Hopf transformation, the Kardar-Parisi-Zhang equation, and theKuramoto-Sivashinsky equation. The Korteweg-deVries equation will also be introduced,and used as an introduction to the method of inverse scattering.

We will finish with a study of hydrodynamic and magnetohydrodynamic equations,including an introduction to topological issues in hydrodynamics. In this part of thecourse, we will discuss the calculus of variations, and improve our understanding ofhow PDEs can often be derived from variational principles, and how those variationalprinciples can be used to establish existence and uniqueness of solutions.

Math 158 Complex VariablesCourse Information

Spring 2018

Block: M+ (Mon, Wed 6:00–7:15 p.m.)Instructor: Fulton GonzalezEmail: fulton.gonzalez@tufts.eduOffice: Bromfield-Pearson 203Office hours: (Fall 2017) Mon 1:30 –2:30 p.m., Fri 11:30 a.m. –1:30 p.m.Phone: (617) 627-2368

Prerequisite: Math 34 or consent

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

Course description: This course is an introduction to the theory of complex functions,a subject with wide-ranging applications and one which is fascinating in its own right, withmany elegant and surprising results.

We will primarily explore the notion of an analytic function on the complex plane (or a givenregion in it), and study many of its implications. Quite simply, an analytic function is adifferentiable function on an open set, and yet such a function is completely determined byits behavior in an arbitrarily small open subset of the plane.

The principal topics in our study of analytic functions include Cauchy’s theorem and integralformula, Taylor and Laurent series, contour integration, the Maximum Principle, analyticcontinuation, harmonic functions, conformal mappings, and linear fractional transforma-tions. We will also explore some of the many applications of complex analysis, such astwo-dimensional potential theory, including the Poisson transform, Riemann surfaces, andpossibly Fourier series or analytic number theory.

The course will be taught rigorously, with plenty of theorem-proving, but also with compu-tational examples and problems. It is appropriate for mathematics, science, and engineeringmajors.

Math 168 Algebraic TopologyCourse Information

Spring 2018

BLOCK: G+ (Mo, Wed 13:30–14:45)INSTRUCTOR: Robert KrophollerEMAIL: robert.kropholler@tufts.eduOFFICE: SEC 010OFFICE HOURS: (Fall 2017) Wed 10 – 12PHONE: (617) 627-2363

PREREQUISITES: Basic knowledge of groups, at least 1 course above the 100 level or con-sent.

TEXT: Algebraic Topology by Allen Hatcher.Available at https://www.math.cornell.edu/˜hatcher/AT/AT.pdf

COURSE DESCRIPTION: Topology is the subject of understanding topological spaces up tocontinuous deformations and stretching. However, given two topological spaces, it canbe tricky to under- stand whether they are the same. For instance, it is often said that atopologist cant tell the difference between a donut and a coffee mug, yet we can tell thedifference between a sphere and a donut.

Many topological spaces can be differentiated from one another by algebraic invari-ants. These come in many flavours, such as homotopy (πn(X, b)), homology (Hn(X)), andcohomology (Hn(X)). I will discuss the first of these and focus primarily on π1(X, b), alsoknown as, the fundamental group. I will start by recalling the definitions from topologyrequired for the course. I will then define simplicial and cellular complexes, which forma large class of topological spaces. These will be used to give concrete examples later inthe course.

I will define homotopy and the fundamental group π1(X, b), with some examples be-ing discussed. We will then study groups via group presentations, which will allow usto define one of the key theorems: understanding the fundamental group of a space bybreaking it into smaller spaces (the Seifert-van Kampen Theorem). The remainder of thecourse will consider covering spaces which prove to be very useful throughout the subjectof algebraic topology. If time permits, I will touch on the higher homotopy groups, n(X,b), and how they can be used to classify topological spaces.

Math 212 Topics in Analysis: Mathematical Biology andStochastic ProcessesCourse Information

Spring 2018

BLOCK: Thursday 1:30-4:00INSTRUCTOR: Alex HeningEMAIL: alexandru.hening@tufts.eduOFFICE:OFFICE HOURS:PHONE:

PREREQUISITES: Math 135 or consent of instructor.TEXT: We will not use a specific textbook.

COURSE DESCRIPTION:The course will start with an introduction to probability theory and stochastic pro-

cesses: measures, measurable functions and random variables, integration, norms andinequalities, ergodic theory, Markov processes in discrete and continuous time, stochasticintegrals, Brownian motion and stochastic differential equations.

Once we have covered the basics we will look at applications of stochastic processesin mathematical ecology. All ecological systems are inherently stochastic - this happensbecause of the randomness of the environmental fluctuations. We will look in depth atstochastic Lotka-Volterra systems (both for competition and predator-prey interactions),the dynamics of populations living in patchy environments, evolutionary stable strategies(ESS), harvesting of populations in stochastic environments and other examples.

The course grade will be based on the presentation of a paper.

Math 216 Algebra IICourse Information

Spring 2018

BLOCK: E+ (Mo, Wed 10:30–11:45)INSTRUCTOR: Robert KrophollerEMAIL: robert.kropholler@tufts.eduOFFICE: 202 Bromfield-Pearson HallOFFICE HOURS: (Fall 2017) Wed 10 – 12PHONE: (617) 627-2363

PREREQUISITES: Math 215 or consent.

TEXT: Abstract Algebra, 3rd edition by David S. Dummit and Richard M. Foote, Wiley &Sons, 2004, New York. (ISBN 0-471-43334-9)

COURSE DESCRIPTION:This course is a continuation of Math 215 where we studied Groups, Rings, and Fields.

In this sequel, we will continue our study of these topics starting where we left off in 215.The first several weeks of the course will be used to finish up Field Theory and GaloisTheory and then we will switch topics entirely.

The material will then change to the study of infinite groups and questions in decid-ability. We will start by studying free groups and group presentations. These are the keyobjects in combinatorial group theory. We will become familiar with these objects givingseveral examples of familiar groups in this new light.

We will then move onto decision theory in groups. A decision problem requires somefinite way of describing the input object, for groups we will use finite presentations. In1911, Dehn asked three question about finite presentations of infinite groups.

1. The Word Problem Given a group G, is there an algorithm which will determinewhether or not a given element in G is equivalent to the identity element?

2. The Conjugacy Problem Given a group G, is there an algorithm which will deter-mine if two words are conjugate in G?

3. The Isomorphism Problem Given two finite presentations G1 and G2, is there analgorithm which will determine if G1 is isomorphic to G2?

We shall see that there exists groups where there is no algorithm for the word andconjugacy problems. Moreover using this we will see that the isomorphism problem isunsolvable by showing that one cannot even decide if a group is the trivial group. Finallywe will move onto looking at other problems and seeing how stable these problems areunder passing to and from finite index subgroups.

Math 218 Differential GeometryCourse Information

Spring 2018

BLOCK: K+, Mon Wed 4:30–5:45 p.m.INSTRUCTOR: Loring TuEMAIL: loring.tu@tufts.eduOFFICE: Bromfield-Pearson 206OFFICE HOURS: (Fall 2017) Mon 1:00 to 2:30, most afternoonsPHONE: (617) 627-3262

PREREQUISITES: Math 217 or consent

TEXT: Loring Tu, Differential Geometry: Connections, Curvature, and Characteristic Classes,Springer, 2017. ISBN 978-3-319-55082-4.

Loring Tu, An Introduction to Manifolds, 2nd edition, Springer, 2011.

COURSE DESCRIPTION:Manifolds are higher-dimensional generalizations of smooth curves and surfaces. Math

217 deals with the differential topology of manifolds. This course is a continuation inwhich the emphasis is on the differential geometry of manifolds. Geometry differs fromtopology in the presence of a metric, which one can think of as a ruler at each point of themanifold. With a metric, one can measure distances, angles, areas, volumes, and curva-ture.

Two of the main theorems of classical differential geometry are Gauss’s TheoremaEgregium and the Gauss-Bonnet theorem. The Theorema Egregium states that the Gaus-sian curvature, which is defined in terms of the embedding of a surface in space, is infact independent of the embedding and dependent only on the metric. The Gauss-Bonnettheorem asserts that the integral of the Gaussian curvature, which a priori depends onthe metric, is a topological invariant and therefore independent of the metric. The Gauss-Bonnet theorem, suitably generalized to higher-dimensions, therefore gives a very im-portant relation between the differential geometry and the differential topology of a man-ifold.

The generalization of the Gauss-Bonnet theorem is best understood in terms of char-acteristic classes, which are cohomology classes constructed from the curvature form on avector bundle. The characteristic classes, when integrated over a manifold, produce newtopological invariants called characteristic numbers, of which the Euler characteristic is aspecial case.

We will cover Gauss’s theory of surfaces, Gauss’s Theorema Egregium, Riemannianmanifolds, connections, curvature, geodesics, Gauss-Bonnet theorem, de Rham cohomol-ogy, and characteristic classes of vector bundles.

Math 228 Numerical Linear AlgebraCourse Information

Spring 2018

BLOCK: Tu, Th 9–10:15 (not a regular Tufts block)INSTRUCTOR: Christoph BorgersEMAIL: christoph.borgers@tufts.eduOFFICE: 215 Bromfield-Pearson HallOFFICE HOURS: (Fall 2017) Mo 4–6, Tu 4–6PHONE: (617) 627-2366

PREREQUISITES: Linear algebra, and experience programming in a language such as Mat-lab, C++, Python, etc. If you are are not a graduate student and are interested in takingthis course, please speak to the instructor.

TEXT: Lloyd N. Trefethen and David Bau, Numerical Linear ALgebra, SIAM (1997).

COURSE DESCRIPTION:

This is an introduction to numerical linear algebra, emphasizing fundamental mathe-matical ideas. Topics include, in roughly the order in which they appear in the course:

• norms, stability of problems (conditioning), stability of algorithms

• least squares problems, QR decomposition, singular value decomposition

• Gaussian elimination and its stability properties

• power method and inverse iteration for calculating eigenvalues and eigenvectors

• QR algorithm for calculating eigenvalues and eigenvectors

• Krylov subspace methods: Arnoldi iteration, GMRES, the Lanczos method, conju-gate gradients, preconditioning

• simple linear iterative methods

• multigrid methods

Math 250 Graph AlgorithmsCourse Information

Spring 2018

BLOCK: F+ (Tuesday, Friday 12:00–1:15 pm)INSTRUCTOR: Xiaozhe HuEMAIL: Xiaozhe.Hu@tufts.eduOFFICE: 212 Bromfield-Pearson HallOFFICE HOURS: (Fall 2017) Tuesday 10:00 am–12:00pm, Thursday 3:00–5:00 pm, or by ap-pointmentPHONE: (617) 627-2356

PREREQUISITES: Math 128 or 228, or consent. Some programming ability in a languagesuch as C, C++, Fortran, Matlab, etc.

TEXT: TBD

COURSE DESCRIPTION:Graphs, which are mathematical structures used to model pairwise relations between

objects, are among the most fundamental data structures in computer science. Graphsare widely used to model many types of relations and processes in physical, biological,social, and information systems. Graph algorithms are the algorithms that operate on thegraphs and become more and more important because of their applications to modernlife, for example, the power grid, computer networks, machine learning, social network,and computational biology.

The course will focus on the development of graph algorithms and the primary ob-jective of the course is to understand the construction of graph algorithms, applicationof them, and more importantly, the applicability and limits of their usages. We will lookat graph related problems and motivate them by practical interpretations. We will usean algorithmic point of view and mainly interested in how to find an (nearly) optimalalgorithm as efficient as possible. We will discuss the theoretical foundations of graphalgorithms and also discuss their computational complexity, storage, and applications.Moreover, we will discuss graph algorithms in the language of linear algebra wheneverpossible. The topics include: basic graph theory, shortest path, spanning tree, connectiv-ity, coloring, matching, spectral of graphs, and clustering. In-class example, homework,and projects will feature some of the practical applications of these algorithms to solvereal-world problems. Homework problems will consist written problems as well as com-puter programming assignments in Matlab (or your favorite programming language).

Math 250 Topics in Algebra: Galois CohomologyCourse Information

Spring 2018

BLOCK: D-block (M 9:30-10:20, TR 10:30-11:20)INSTRUCTOR: George McNinchEMAIL: george.mcninch@tufts.eduOFFICE: 112 Bromfield-Pearson HallOFFICE HOURS: McNinch is on sabbatical Fall 2017; discussion possible by appointmentPHONE: (617) 627-6210

PREREQUISITES: Math 215..

TEXT: The course will follow (some part of) the monograph “Central Simple Algebrasand Galois Cohomology” by P. Gille and T. Szamuely, Cambridge University Press. Youare not required to purchase the book. I note that Cambridge University Press sells ane-book version.

EXPECTATIONS: The course will have weekly problem sets.

COURSE DESCRIPTION: Let k be a field, and let A be central simple algebra (CSA) overk – i.e. A is a (possibly non-commutative) k-algebra which is central (i.e. the center ofA coincides with k) and simple (i.e. the only non-zero two-sided ideal of A is A itself)and for which dimk A < ∞. According to a theorem of Wedderburn, A ' Matn×n(D)for some k-division algebra D. One says that A and D are Brauer equivalent; the tensorproduct equips the collection Br(k) of Brauer classes of CSA’s over k with the structure ofan (abelian) group, called the Brauer group of k.

A key feature is that when k is separably closed, a CSA is always “split” - i.e. isisomorphic to Matn×n(k) for some n. Let Γ = Gal(ksep/k) denotes the absolute Galoisgroup of k (where ksep is a separable closure of k). If A is a CSA over any field k, one mayview A as a “twisted form” (or “Γ-twisted form”) of Matd×d(k) where dimk A = d2.

The course will introduce Wedderburn’s theorem, and it will then explain the abovepoint-of-view by introducing the notion of galois descent. We will discuss the cohomologyof groups, and we will see that Br(k) identifies with the cohomology group H2(Γ, k×sep).

We will see other mathematical objects which can be classified by analogous Galoiscohomological machinery, include quadratic forms and Severi-Brauer varieties.

As a piece of propaganda to conclude this course summary, let me quickly try to con-vince you that CSA’s encode interesting information about a field k. A CSA Q of dimen-sion 4 over k is called a quaternion algebra (the split case being Q = Mat2×2(k)). Thereis a (quadratic) norm mapping N : Q → k which determines a quadratic form on the 3dimensional space Q+ of elements in Q of trace 0; the zero-locus of this quadratic formdetermines a smooth conic C in the projective space P(Q+) = P2. A theorem of Witt nowasserts that Q1 ' Q2 if and only if C1 and C2 are birationally equivalent (over k). (ThisTheorem is proved in the first chapter of [Gille-Szamuely]).