# Department of Mathematics

## Faculty

### Chair

- Igor Rodnianski

### Associate Chair

- Zoltán Szabó

### Director of Undergraduate Studies

- Ana Menezes (acting)
- Zoltán Szabó

### Director of Graduate Studies

- Lue Pan
- Chenyang Xu

### Professor

- Michael Aizenman
- Noga M. Alon
- Manjul Bhargava
- Sun-Yung A. Chang
- Maria Chudnovsky
- Fernando Codá Marques
- Peter Constantin
- Mihalis Dafermos
- Zeev Dvir
- Charles L. Fefferman
- David Gabai
- June E. Huh
- Alexandru D. Ionescu
- Nicholas M. Katz
- Sergiu Klainerman
- János Kollár
- Emmy Murphy
- Assaf Naor
- Peter Steven Ozsváth
- John V. Pardon
- Igor Rodnianski
- Peter C. Sarnak
- Paul Seymour
- Amit Singer
- Christopher M. Skinner
- Allan M. Sly
- Zoltán Szabó
- Chenyang Xu
- Paul C. Yang
- Shou-Wu Zhang

### Assistant Professor

- Jonathan Hanselman
- Casey L. Kelleher
- Ana Menezes
- Evita Nestoridi
- Lue Pan
- Jacob Shapiro
- Jakub Witaszek
- Ian M. Zemke
- Ruobing Zhang

### Associated Faculty

- John P. Burgess, Philosophy
- René A. Carmona, Oper Res and Financial Eng
- Bernard Chazelle, Computer Science
- Hans P. Halvorson, Philosophy
- William A. Massey, Oper Res and Financial Eng
- Frans Pretorius, Physics
- Robert E. Tarjan, Computer Science
- Robert J. Vanderbei, Oper Res and Financial Eng
- Ramon van Handel, Oper Res and Financial Eng

### Instructor

- David Boozer
- Matija Bucic
- Alan Chang
- Jennifer Li
- Paul David Timothy William Minter
- Jean Pierre Mutanguha
- Laurel A. Ohm
- Sarah Peluse
- Semon Rezchikov
- Ravi Shankar
- Artane Siad
- Fan Wei
- Liyang Yang
- Andrew V Yarmola

### University Lecturer

- Jennifer M. Johnson

### Senior Lecturer

- Mark W. McConnell

### Lecturer

- Bjoern Bringmann
- Allen J. Fang
- Jonathan M. Fickenscher
- Tangli Ge
- Daniel Ginsberg
- Xiaoyu He
- Wei Ho
- Henry Theodore Horton
- Tatiana K. Howard
- Jef C. Laga
- Samuel Mundy
- Andrew O'Desky
- Eden Prywes
- Samuel Pérez-Ayala
- Hannah Schwartz
- John T. Sheridan
- Rita Teixeira da Costa
- David Villalobos

### Visiting Professor

- Bhargav B. Bhatt

### Visiting Lecturer with Rank of Professor

- Camillo De Lellis
- Jacob A. Lurie
- Akshay Venkatesh

## Program Information

## Information and Departmental Plan of Study

Most first-year students and sophomores interested in science, engineering, or finance take courses from the standard calculus and linear algebra sequence 103-104-201-202, which emphasizes concrete computations over more theoretical considerations. Note that 201 and 202 can be taken in either order.

Students who are not prepared to begin with 103 may take 100, an introduction to calculus with pre-calculus review.

Prospective economics majors can minimally fulfill their mathematics prerequisites with (100)-103-175. Note that 175 covers selected topics from 201, with biology and economics applications in mind. It is recommended that prospective math-track economics/finance majors take the standard sequence 103-104-201-202 instead of 175.

More mathematically inclined students, especially prospective physics majors, may opt to replace 201-202 with 203-204, for greater emphasis on theory and more challenging computational problems.

Prospective mathematics majors must take at least one course introducing formal mathematical argument and rigorous proofs. The recommended first-year sequence for prospective majors is 215-217-300. Prospective majors who already have substantial experience with university-level proof-based analysis courses may consider the accelerated sequence 216-218 instead. Other possible sequences for prospective majors include 214-204-203 and 203-204-215, although the latter two are relatively rare. Note that 203 and 204 can be taken in either order.

## Placement Guidelines

Students who need placement advice about 214, 215, or 216 should consult the Department of Mathematics home page (the "Undergraduate" tab has a section on placement) or contact Professor McConnell, the junior adviser.

Students with little or no background in calculus, but with strong pre-calculus skills, should take 103. Alternatively, 100 offers intensive pre-calculus review, along with an introduction to the main ideas of calculus, as preparation for 103. To succeed in 104 or 175, a background in differential calculus at the level of a 5 on the BC Advanced Placement Examination is advised. Students with a stronger calculus preparation in differential and integral calculus, as well as infinite series, may opt to start in 201. Students who possess, in addition, a particularly strong interest in mathematics as well as a SAT mathematics score of at least 750 may opt for 203 or 214 or 215 or 216 instead.

The placement workshops for incoming students at orientation are designed to help students consider all these issues in order to make a good initial self-placement in the mathematics curriculum, which can then be adjusted during drop/add if necessary.

## Advanced Placement

Incoming first-year students who report a score of 5 on the BC Advanced Placement Examination (or a 7 on the IC [higher level] math examination, or an A on the British A-level math exam) will receive one unit of advanced placement credit for MAT103. For more information, please consult the website of the Office of the Dean of the College.

One unit of advanced placement credit is provisionally granted when a student enrolls in MAT 104 or 175 in the fall term of their first year. Two units of advanced placement credit are provisionally granted when a student enrolls in MAT 201, 203, 215, or 216 in the fall term of their first year. Provisional credit will be converted to advanced placement credit upon successful completion of the relevant course.

## Prerequisites

Generally, either 215-217 or 216-218 or 203-204-215 are strongly recommended for admission to the department. Other paths involving 201, 202, and 214 are also possible. Prospective mathematics majors should consult with the department early and plan a program that includes as much of the 215-217-300 or 216-218 sequence as possible. Most majors begin taking courses at the 300-level by the second semester of sophomore year, in preparation for their junior independent work.

Some of our successful majors became interested in mathematics as a major after taking 103 or 104. Such students should consult with the junior adviser or the associate director of undergraduate studies as soon as possible.

Further information for prospective majors is available on the department home page.

## Program of Study

Students must complete four core requirements:

- one course in real analysis (e.g., 300 or 320 or 325 or 425 or 385)
- one course in complex analysis (e.g., 330 or 335)
- one course in algebra (e.g., 340 or 345)
- one course in geometry or topology (e.g., 355 or 365 or 560)

It is recommended that students complete some of these core requirements by the end of their sophomore year. Completing these core courses early allows for more options for junior and senior independent work.

Note: One course in discrete mathematics (e.g., 375, 377, or 378) can replace the geometry/topology core requirement, if desired.

**In addition to the four core requirements**, students must complete an additional four courses at the 300-level or higher, up to three of which may be cognate courses outside the mathematics department, with permission from the junior or senior advisers or the director of undergraduate studies. No more than two of the eight courses may be reading courses.

The departmental grade (the average grade of the eight departmental courses) together with grades and reports on independent work is the basis on which honors and prizes are awarded upon graduation.

Students should refer to Course Offerings to check which courses are offered in a given term. Programs of study in various fields of pure mathematics and applied mathematics are available. Appropriate plans of study may be arranged for students interested in specialized related fields such as physics, the biological sciences, finance, and computer science, for example. For students interested in these areas, a coherent program containing up to three courses in a cognate field may be approved.

## Independent Work

All departmental students engage in independent work, supervised by a member of the department chosen in consultation with a departmental adviser. The junior year independent work generally consists of participating actively in a junior seminar in both the fall and the spring semesters. Alternatively, a student may opt to replace one junior seminar with supervised reading in a special subject and then writing a paper based on that reading. The independent work in the senior year centers on writing a senior thesis. A substantial percentage of our majors work with faculty in other departments on their senior project.

## Senior Departmental Examination

Each senior takes an oral examination based on the senior thesis and the broader subfield to which it contributes. A departmental committee conducts the examination in May.

## Courses

### MAT 100 Calculus Foundations Fall/Spring QCR

Introduction to limits and derivatives as preparation for further courses in calculus. Fundamental functions (polynomials, rational functions, exponential, logarithmic, trigonometric) and their graphs will be also reviewed. Other topics include tangent and normal lines, linearization, computing area and rates of change. The emphasis will be on learning to think independently and creatively in the mathematical setting. Instructed by: Staff### MAT 102 Survey of Calculus Not offered this year QCR

One semester survey of the major concepts and computational techniques of calculus including limits, derivatives and integrals. Emphasis on basic examples and applications of calculus including approximation, differential equations, rates of change and error estimation for students who will take no further calculus. Prerequisites: MAT100 or equivalent. Restrictions: Cannot receive course credit for both MAT103 and MAT102. Provides adequate preparation for MAT175. Three classes. Instructed by: Staff### MAT 103 Calculus I Fall/Spring QCR

First semester of calculus. Topics include limits, continuity, the derivative, basic differentiation formulas and applications (curve-sketching, optimization, related rates), definite and indefinite integrals, the fundamental theorem of calculus. Prerequisite: MAT100 or equivalent. Instructed by: Staff### MAT 104 Calculus II Fall/Spring QCR

Continuation of MAT103. Topics include techniques of integration, arclength, area, volume, convergence of series and improper integrals, L'Hopital's rule, power series and Taylor's theorem, introduction to differential equations and complex numbers. Prerequisite: MAT103 or equivalent. Instructed by: Staff### MAT 175 Mathematics for Economics/Life Sciences Fall/Spring QCR

Survey of topics from multivariable calculus as preparation for future course work in economics or life sciences. Topics include basic techniques of integration, average value, vectors, partial derivatives, gradient, optimization of multivariable functions, and constrained optimization with Lagrange multipliers. Students preparing for math track econometrics and finance courses need MAT201/202 instead. Students who complete 175 can continue in 202 if they wish. Instructed by: Staff### MAT 191 An Integrated Introduction to Engineering, Mathematics, Physics (See EGR 191)

### MAT 192 An Integrated Introduction to Engineering, Mathematics, Physics (See EGR 192)

### MAT 199 Math Alive (See APC 199)

### MAT 201 Multivariable Calculus Fall/Spring QCR

Vectors in the plane and in space, vector functions and motion, surfaces, coordinate systems, functions of two or three variables and their derivatives, maxima and minima and applications, double and triple integrals, vector fields, and Stokes's theorem. Prerequisite: 104 or equivalent. Instructed by: Staff### MAT 202 Linear Algebra with Applications Fall/Spring QCR

Companion course to MAT201. Matrices, linear transformations, linear independence and dimension, bases and coordinates, determinants, orthogonal projection, least squares, eigenvectors and their applications to quadratic forms and dynamical systems. Prerequisite: MAT103 or equivalent. Instructed by: Staff### MAT 203 Advanced Vector Calculus Fall QCR

Vector spaces, limits, derivatives of vector-valued functions, Taylor's formula, Lagrange multipliers, double and triple integrals, change of coordinates, surface and line integrals, generalizations of the fundamental theorem of calculus to higher dimensions. More abstract than 201 but more concrete than 216/218. Recommended for prospective physics majors and others with a strong interest in applied mathematics. Prerequisite: MAT104 or equivalent. Instructed by: D. Gabai### MAT 204 Advanced Linear Algebra with Applications Spring QCR

Companion course to MAT203. Linear systems of equations, linear independence and dimension, linear transforms, determinants, (real and complex) eigenvectors and eigenvalues, orthogonality, spectral theorem, singular value decomposition, Jordan forms, other topics as time permits. More abstract than MAT202 but more concrete than MAT217. Recommended for prospective physics majors and others with a strong interest in applied mathematics. Prerequisite: MAT104 or equivalent. Instructed by: Staff### MAT 214 Numbers, Equations, and Proofs Fall QCR

An introduction to classical number theory to prepare for higher-level courses in the department. Topics include Pythagorean triples and sums of squares, unique factorization, Chinese remainder theorem, arithmetic of Gaussian integers, finite fields and cryptography, arithmetic functions, and quadratic reciprocity. There will be a topic from more advanced or more applied number theory such as p-adic numbers, cryptography, and Fermat's Last Theorem. This course is suitable both for students preparing to enter the mathematics department and for non-majors interested in exposure to higher mathematics. Instructed by: Staff### MAT 215 Single Variable Analysis with an Introduction to Proofs Fall/Spring QCR

An introduction to the mathematical discipline of analysis, to prepare for higher-level course work in the department. Topics include the rigorous epsilon-delta treatment of limits, convergence, and uniform convergence of sequences and series. Continuity, uniform continuity, and differentiability of functions. The Heine-Borel theorem, the Riemann integral, conditions for integrability of functions and term by term differentiation and integration of series of functions, Taylor's theorem. Instructed by: Staff### MAT 217 Honors Linear Algebra Spring QCR

A rigorous course in linear algebra with an emphasis on proof rather than applications. Topics include vector spaces, linear transformations, inner product spaces, determinants, eigenvalues, the Cayley-Hamilton theorem, Jordan form, the spectral theorem for normal transformations, bilinear and quadratic forms. Instructed by: Staff### MAT 218 Multivariable Analysis and Linear Algebra II Spring QCR

Continuation of Multivariable Analysis and Linear Algebra I (MAT 216) from the fall. A rigorous course in analysis with an emphasis on proof rather than applications. Topics include metric spaces, completeness, compactness, total derivatives, partial derivatives, inverse function theorem, implicit function theorem, Riemann integrals in several variables, Fubini. See the department website for details: http://www.math.princeton.edu. Instructed by: Staff### MAT 300 Multivariable Analysis I Fall QCR

Continuation of Single Variable Analysis (MAT215) and Honors Linear Algebra (MAT217) needed to prepare for further work in differential geometry, analysis and topology. Calculus on manifolds: Introduces the concept of differentiable manifold, develops the notions of vector fields and differential forms, Stokes' theorem and the de Rham complex. The basic existence theorem in ODEs is used to prove the Frobenius theorem on integrability of plane fields. The intent is to provide the preparation for the courses in differential geometry and topology. Instructed by: Staff### MAT 305 Mathematical Logic Not offered this year QCR

A development of logic from the mathematical viewpoint, including propositional and predicate calculus, consequence and deduction, truth and satisfaction, the Goedel completeness and incompleteness theorems. Applications to model theory, recursion theory, and set theory as time permits. Some underclass background in logic or in mathematics is recommended. Instructed by: Staff### MAT 306 Advanced Logic (See PHI 323)

### MAT 320 Introduction to Real Analysis Fall QCR

Introduction to real analysis, including the theory of Lebesgue measure and integration on the line and n-dimensional space and the theory of Fourier series. Prerequisite: MAT201 and MAT202 or equivalent. Instructed by: Staff### MAT 323 Topics in Mathematical Modeling (also
APC 323
) Not offered this year
QCR

Draws problems from the sciences and engineering for which mathematical models have been developed and analyzed to describe, understand and predict natural and man-made phenomena. Emphasizes model building strategies, analytical and computational methods, and how scientific problems motivate new mathematics. This interdisciplinary course in collaboration with Molecular Biology, Psychology and the Program in Neuroscience is directed toward upper class undergraduate students and first-year graduate students with knowledge of linear algebra and differential equations.
Instructed by: Staff ### MAT 325 Analysis I: Fourier Series and Partial Differential Equations Spring QCR

Basic facts about Fourier Series, Fourier Transformations, and applications to the classical partial differential equations will be covered. Also Fast Fourier Transforms, Finite Fourier Series, Dirichlet Characters, and applications to properties of primes. Prerequisites: 215, 218, or permission of instructor. Instructed by: Staff### MAT 330 Complex Analysis with Applications Spring QCR

The theory of functions of one complex variable, covering power series expansions, residues, contour integration, and conformal mapping. Although the theory will be given adequate treatment, the emphasis of this course is the use of complex analysis as a tool for solving problems. Prerequisite: MAT201 and MAT202 or equivalent. Instructed by: Staff### MAT 335 Analysis II: Complex Analysis Fall QCR

Study of functions of a complex variable, with emphasis on interrelations with other parts of mathematics. Cauchy's theorems, singularities, contour integration, power series, infinite products. The gamma and zeta functions and the prime number theorem. Elliptic functions, theta functions, Jacobi's triple product and combinatorics. An overall view of Special Functions via the hypergeometric series. This course is the second semester of a four-semester sequence, but may be taken independently of the other semesters. Instructed by: Staff### MAT 345 Algebra I Fall QCR

This course will cover the basics of symmetry and group theory, with applications. Topics include the fundamental theorem of finitely generated abelian groups, Sylow theorems, group actions, and the representation theory of finite groups, rings and modules. Instructed by: Staff### MAT 346 Algebra II Spring QCR

Continuation of MAT345. Further develop knowledge of algebraic structures by exploring examples that connect to higher mathematics. Instructed by: Staff### MAT 355 Introduction to Differential Geometry Spring QCR

Introduction to geometry of surfaces. Surfaces in Euclidean space, second fundamental form, minimal surfaces, geodesics, Gauss curvature, Gauss-Gonnet formula, uniformization of surfaces, elementary notions of contact geometry. Prerequisite: MAT218 or MAT300, or MAT203 or equivalent. Instructed by: F. Codá Marques### MAT 365 Topology Fall QCR

Introduction to point-set topology, the fundamental group, covering spaces, methods of calculation and applications. Prerequisite: MAT202 or 204 or 218 or equivalent. Instructed by: Z. Szabó### MAT 375 Introduction to Graph Theory (also
COS 342
) Spring
QCR

The fundamental theorems and algorithms of graph theory. Topics include: connectivity, matchings, graph coloring, planarity, the four-color theorem, extremal problems, network flows, and related algorithms. Prerequisite: MAT202 or 204 or 217 or equivalent.
Instructed by: P. Seymour ### MAT 377 Combinatorial Mathematics (also
APC 377
) Fall
QCR

Combinatorics is the study of enumeration and structure of discrete objects. These structures are widespread throughout mathematics, including geometry, topology and algebra, as well as computer science, physics and optimization. This course will give an introduction to modern techniques in the field, and how they relate to objects such as polytopes, permutations and hyperplane arrangements.
Instructed by: N. Alon ### MAT 378 Theory of Games Spring QCR

Games in extensive form, pure and behavioral strategies; normal form, mixed strategies, equilibrium points; coalitions, characteristic-function form, imputations, solution concepts; related topics and applications. Prerequisite: MAT202 or 204 or 217 or equivalent. MAT215 or equivalent is recommended. Instructed by: Staff### MAT 380 Probability and Stochastic Systems (See ORF 309)

### MAT 385 Probability Theory Spring QCR

Sequence of independent trials, applications to number theory and analysis, Monte Carlo method. Markov chains, ergodic theorem for Markov chains. Entropy and McMillan theorem. Random walks, recurrence and non-recurrence; connection with the linear difference equations. Strong laws of large numbers, random series and products. Weak convergence of probability measures, weak Helly theorems, Fourier transforms of distributions. Limit theorems of probability theory. Prerequisite: MAT203 or 218 or equivalent. Instructed by: A. Sly### MAT 391 Mathematics in Engineering I (See MAE 305)

### MAT 392 Mathematics in Engineering II (See MAE 306)

### MAT 393 Mathematical Programming Not offered this year QCR

Linear programs, duality, Dantzig's simplex method; theory of dual linear systems; matrix games, von Neumann's minimax theorem, simplex solution; algorithms for assignment, transport, flow; brief introduction to nonlinear programming. Instructed by: Staff### MAT 407 Theory of Computation (See COS 487)

### MAT 419 Topics in Number Theory Fall/Spring QCR

Topics introducing various aspects of number theory, including analytic and algebraic number theory, L-functions, and modular forms. See*Course Offerings*listing for topic details. Prerequisites: MAT 215, 345, 346 or equivalent. Instructed by: Staff

### MAT 425 Analysis III: Integration Theory and Hilbert Spaces Spring QCR

The theory of Lebesgue integration in n-dimensional space. Differentiation theory. Hilbert space theory and applications to Fourier Transforms, and partial differential equations. Introduction to fractals. This course is the third semester of a four-semester sequence, but may be taken independently of the other semesters. Prerequisites: MAT215 or 218 or equivalent. Instructed by: Staff### MAT 427 Ordinary Differential Equations Not offered this year QCR

Introduction to the study of ordinary differential equations; explicit solutions, general properties of solutions, and applications. Topics include explicit solutions of some non-linear equations in two variables by separation of variables and integrating factors, explicit solution of simultaneous linear equations with constant coefficients, explicit solution of some linear equations with variable forcing term by Laplace transform methods, geometric methods (description of the phase portrait), and the fundamental existence and uniqueness theorem. Instructed by: Staff### MAT 429 Topics in Analysis Not offered this year QCR

Introduction to incompressible fluid dynamics. The course will give an introduction to the mathematical theory of the Euler equations, the fundamental partial differential equation arising in the study of incompressible fluids. We will discuss several topics in analysis that emerge in the study of these equations: Lebesgue and Sobolev spaces, distribution theory, elliptic PDEs, singular integrals, and Fourier analysis. Content varies from year to year. See*Course Offerings*listing for topic details. Instructed by: Staff