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## MATH 19:Calculus

Introduction to differential calculus of functions of one variable. Topics: review of elementary functions including exponentials and logarithms, limits, rates of change, the derivative, and applications. Math 19, 20, and 21 cover the same material as Math 41 and 42, but in three quarters rather than two. Prerequisites: precalculus, including trigonometry, advanced algebra, and analysis of elementary functions.
Terms: Aut, Win, Sum | Units: 3 | UG Reqs: GER:DB-Math, WAY-FR | Grading: Letter or Credit/No Credit

## MATH 20:Calculus

Continuation of 19. Applications of differential calculus; introduction to integral calculus of functions of one variable, including: the definite integral, methods of symbolic and numerical integration, applications of the definite integral. Prerequisites: 19 or equivalent.
Terms: Win, Spr | Units: 3 | UG Reqs: GER:DB-Math, WAY-FR | Grading: Letter or Credit/No Credit

## MATH 21:Calculus

Continuation of 20. Applications of integral calculus, introduction to differential equations, infinite series. Prerequisite: 20 or equivalent.
Terms: Spr | Units: 4 | UG Reqs: GER:DB-Math, WAY-FR | Grading: Letter or Credit/No Credit

## MATH 41:Calculus (accelerated)

Introduction to differential and integral calculus of functions of one variable. Topics: limits, rates of change, the derivative and applications, introduction to the definite integral and integration. Math 41 and 42 cover the same material as Math 19-20-21, but in two quarters rather than three. Prerequisites: trigonometry, advanced algebra, and analysis of elementary functions, including exponentials and logarithms.
Terms: Aut | Units: 5 | UG Reqs: GER:DB-Math, WAY-FR | Grading: Letter or Credit/No Credit

## MATH 41A:Calculus ACE

Students attend MATH 41 lectures with different recitation sessions, four hours instead of two, emphasizing engineering applications. Prerequisite: application; see http://soe.stanford.edu/edp/programs/ace.html.
Terms: Aut | Units: 6 | UG Reqs: GER:DB-Math, WAY-FR | Grading: Letter (ABCD/NP)

## MATH 42:Calculus (Accelerated)

Continuation of 41. Methods of symbolic and numerical integration, applications of the definite integral, introduction to differential equations, infinite series. Prerequisite: 41 or equivalent.
Terms: Aut, Win | Units: 5 | UG Reqs: GER:DB-Math, WAY-FR | Grading: Letter or Credit/No Credit

## MATH 42A:Calculus ACE

Students attend MATH 42 lectures with different recitation sessions, four hours instead of two, emphasizing engineering applications. Prerequisite: application; see http://soe.stanford.edu/edp/programs/ace.html.
Terms: Aut, Win | Units: 6 | UG Reqs: GER:DB-Math, WAY-FR | Grading: Letter (ABCD/NP)

## MATH 51:Linear Algebra and Differential Calculus of Several Variables

Geometry and algebra of vectors, systems of linear equations, matrices and linear transformations, diagonalization and eigenvectors, vector valued functions and functions of several variables, parametric curves, partial derivatives and gradients, the derivative as a matrix, chain rule in several variables, constrained and unconstrained optimization. Prerequisite: 21, or 42, or a score of 4 on the BC Advanced Placement exam or 5 on the AB Advanced Placement exam, or consent of instructor.
Terms: Aut, Win, Spr, Sum | Units: 5 | UG Reqs: GER:DB-Math, WAY-FR | Grading: Letter or Credit/No Credit

## MATH 51A:Linear Algebra and Differential Calculus of Several Variables, ACE

Students attend MATH 51 lectures with different recitation sessions: four hours per week instead of two, emphasizing engineering applications. Prerequisite: application; see http://soe.stanford.edu/edp/programs/ace.html.
Terms: Aut, Win, Spr | Units: 6 | UG Reqs: GER:DB-Math, WAY-FR | Grading: Letter (ABCD/NP)

## MATH 51H:Honors Multivariable Mathematics

For prospective Mathematics majors in the honors program and students from other areas of science or engineering who have a strong mathematics background. Three quarter sequence covers the material of 51, 52, 53, and additional advanced calculus and ordinary and partial differential equations. Unified treatment of multivariable calculus, linear algebra, and differential equations with a different order of topics and emphasis from standard courses. Students should know one-variable calculus and have an interest in a theoretical approach to the subject. Prerequisite: score of 5 on BC Advanced Placement exam, or consent of instructor.
Terms: Aut | Units: 5 | UG Reqs: GER:DB-Math, WAY-FR | Grading: Letter (ABCD/NP)
Instructors: ; Simon, L. (PI)

## MATH 52:Integral Calculus of Several Variables

Iterated integrals, line and surface integrals, vector analysis with applications to vector potentials and conservative vector fields, physical interpretations. Divergence theorem and the theorems of Green, Gauss, and Stokes. Prerequisite: 51 and 42 or equivalents.
Terms: Aut, Win, Spr | Units: 5 | UG Reqs: GER:DB-Math, WAY-FR | Grading: Letter or Credit/No Credit

## MATH 52H:Honors Multivariable Mathematics

Continuation of 51H. Prerequisite: 51H.
Terms: Win | Units: 5 | UG Reqs: GER:DB-Math, WAY-FR | Grading: Letter (ABCD/NP)
Instructors: ; Eliashberg, Y. (PI)

## MATH 53:Ordinary Differential Equations with Linear Algebra

Ordinary differential equations and initial value problems, systems of linear differential equations with constant coefficients, applications of second-order equations to oscillations, matrix exponentials, Laplace transforms, stability of non-linear systems and phase plane analysis, numerical methods. Prerequisite: 51 and 42 or equivalents.
Terms: Aut, Win, Spr, Sum | Units: 5 | UG Reqs: GER:DB-Math, WAY-FR | Grading: Letter or Credit/No Credit

## MATH 53H:Honors Multivariable Mathematics

Continuation of 52H. Prerequisite: 52H.
Terms: Spr | Units: 5 | UG Reqs: GER:DB-Math, WAY-FR | Grading: Letter (ABCD/NP)
Instructors: ; Eliashberg, Y. (PI)

## MATH 80Q:Capillary Surfaces: Explored and Unexplored Territory

Preference to sophomores. Capillary surfaces: the interfaces between fluids that are adjacent to each other and do not mix. Recently discovered phenomena, predicted mathematically and subsequently confirmed by experiments, some done in space shuttles. Interested students may participate in ongoing investigations with affinity between mathematics and physics.
Terms: Spr | Units: 3 | UG Reqs: WAY-FR, WAY-SMA | Grading: Letter (ABCD/NP)
Instructors: ; Finn, R. (PI)

## MATH 87Q:Mathematics of Knots, Braids, Links, and Tangles

Preference to sophomores. Types of knots and how knots can be distinguished from one another by means of numerical or polynomial invariants. The geometry and algebra of braids, including their relationships to knots. Topology of surfaces. Brief summary of applications to biology, chemistry, and physics.
Terms: Win | Units: 3 | UG Reqs: WAY-FR | Grading: Letter (ABCD/NP)
Instructors: ; Wieczorek, W. (PI)

## MATH 104:Applied Matrix Theory

Linear algebra for applications in science and engineering: orthogonality, projections, the four fundamental subspaces of a matrix, spectral theory for symmetric matrices, the singular value decomposition, the QR decomposition, least-squares, the condition number of a matrix, algorithms for solving linear systems. Prerequisites: MATH 51 and MATH 52 or 53.
Terms: Win, Spr, Sum | Units: 3 | UG Reqs: GER:DB-Math | Grading: Letter or Credit/No Credit

## MATH 106:Functions of a Complex Variable

Complex numbers, analytic functions, Cauchy-Riemann equations, complex integration, Cauchy integral formula, residues, elementary conformal mappings. Prerequisite: 52.
Terms: Spr | Units: 3 | UG Reqs: GER:DB-Math | Grading: Letter or Credit/No Credit
Instructors: ; Yang, T. (PI)

## MATH 108:Introduction to Combinatorics and Its Applications

Topics: graphs, trees (Cayley's Theorem, application to phylogony), eigenvalues, basic enumeration (permutations, Stirling and Bell numbers), recurrences, generating functions, basic asymptotics. Prerequisites: 51 or equivalent.
Terms: Spr | Units: 3 | UG Reqs: GER:DB-Math | Grading: Letter or Credit/No Credit
Instructors: ; Hicks, A. (PI)

## MATH 109:Applied Group Theory

Applications of the theory of groups. Topics: elements of group theory, groups of symmetries, matrix groups, group actions, and applications to combinatorics and computing. Applications: rotational symmetry groups, the study of the Platonic solids, crystallographic groups and their applications in chemistry and physics. WIM.
Terms: Win | Units: 3 | UG Reqs: GER:DB-Math, WAY-FR | Grading: Letter or Credit/No Credit
Instructors: ; Bump, D. (PI)

## MATH 110:Applied Number Theory and Field Theory

Number theory and its applications to modern cryptography. Topics: congruences, finite fields, primality testing and factorization, public key cryptography, error correcting codes, and elliptic curves, emphasizing algorithms. WIM.
Terms: Spr | Units: 3 | UG Reqs: GER:DB-Math, WAY-FR | Grading: Letter or Credit/No Credit
Instructors: ; Kane, D. (PI)

## MATH 111:Computational Commutative Algebra

Introduction to the theory of commutative rings, ideals, and modules. Systems of polynomial equations in several variables from the algorithmic viewpoint. Groebner bases, Buchberger's algorithm, elimination theory. Applications to algebraic geometry and to geometric problems.
Terms: not given this year | Units: 3 | UG Reqs: GER:DB-Math | Grading: Letter or Credit/No Credit

## MATH 113:Linear Algebra and Matrix Theory

Algebraic properties of matrices and their interpretation in geometric terms. The relationship between the algebraic and geometric points of view and matters fundamental to the study and solution of linear equations. Topics: linear equations, vector spaces, linear dependence, bases and coordinate systems; linear transformations and matrices; similarity; eigenvectors and eigenvalues; diagonalization.
Terms: Aut, Win, Spr | Units: 3 | UG Reqs: GER:DB-Math, WAY-FR | Grading: Letter or Credit/No Credit

## MATH 114:Introduction to Scientific Computing (CME 108)

Introduction to Scientific Computing Numerical computation for mathematical, computational, physical sciences and engineering: error analysis, floating-point arithmetic, nonlinear equations, numerical solution of systems of algebraic equations, banded matrices, least squares, unconstrained optimization, polynomial interpolation, numerical differentiation and integration, numerical solution of ordinary differential equations, truncation error, numerical stability for time dependent problems and stiffness.nImplementation of numerical methods in MATLAB programming assignments.nPrerequisites: MATH 51, 52, 53; prior programming experience (MATLAB or other language at level of CS 106A or higher).
Terms: Win, Sum | Units: 3-4 | Grading: Letter or Credit/No Credit
Instructors: ; Sing Long Collao, C. (PI)

## MATH 115:Functions of a Real Variable

The development of real analysis in Euclidean space: sequences and series, limits, continuous functions, derivatives, integrals. Basic point set topology. Honors math majors and students who intend to do graduate work in mathematics should take 171. Prerequisite: 51.
Terms: Aut, Spr | Units: 3 | UG Reqs: GER:DB-Math | Grading: Letter or Credit/No Credit
Instructors: ; Maximo, D. (PI); Yang, T. (PI)

## MATH 116:Complex Analysis

Analytic functions, Cauchy integral formula, power series and Laurent series, calculus of residues and applications, conformal mapping, analytic continuation, introduction to Riemann surfaces, Fourier series and integrals. Prerequisites: 52, and 115 or 171.
Terms: Win | Units: 3 | UG Reqs: GER:DB-Math | Grading: Letter or Credit/No Credit
Instructors: ; Li, J. (PI)

## MATH 118:Mathematics of Computation

Notions of analysis and algorithms central to modern scientific computing: continuous and discrete Fourier expansions, the fast Fourier transform, orthogonal polynomials, interpolation, quadrature, numerical differentiation, analysis and discretization of initial-value and boundary-value ODE, finite and spectral elements. Prerequisites: MATH 51 and 53.
Terms: Aut | Units: 3 | UG Reqs: GER:DB-Math | Grading: Letter or Credit/No Credit
Instructors: ; Ying, L. (PI)

## MATH 120:Groups and Rings

Groups acting on sets, examples of finite groups, Sylow theorems, solvable and simple groups. Fields, rings, and ideals; polynomial rings over a field; PID and non-PID. Unique factorization domains. WIM.
Terms: Aut, Spr | Units: 3 | UG Reqs: GER:DB-Math, WAY-FR | Grading: Letter or Credit/No Credit
Instructors: ; Li, Z. (PI); Luu, M. (PI)

## MATH 121:Galois Theory

Field of fractions, splitting fields, separability, finite fields. Galois groups, Galois correspondence, examples and applications. Prerequisite: Math 120 and (also recommended) 113.
Terms: Win | Units: 3 | UG Reqs: GER:DB-Math | Grading: Letter or Credit/No Credit

## MATH 122:Modules and Group Representations

Modules over PID. Tensor algebra. Group representations and group rings. Maschke's theorem and character theory. Character tables, construction of representations. Prerequisite: Math 120. Also recommended: 113.
Terms: Spr | Units: 3 | Grading: Letter or Credit/No Credit
Instructors: ; Venkatesh, A. (PI)

## MATH 131P:Partial Differential Equations I

An introduction to PDE; particularly suitable for non-Math majors. Topics include physical examples of PDE's, method of characteristics, D'Alembert's formula, maximum principles, heat kernel, Duhamel's principle, separation of variables, Fourier series, Harmonic functions, Bessel functions, spherical harmonics. Students who have taken MATH 171 should consider taking MATH 173 rather than 131p. Prerequisite: 53.
Terms: Aut, Win | Units: 3 | UG Reqs: GER:DB-Math | Grading: Letter or Credit/No Credit

## MATH 136:Stochastic Processes (STATS 219)

Introduction to measure theory, Lp spaces and Hilbert spaces. Random variables, expectation, conditional expectation, conditional distribution. Uniform integrability, almost sure and Lp convergence. Stochastic processes: definition, stationarity, sample path continuity. Examples: random walk, Markov chains, Gaussian processes, Poisson processes, Martingales. Construction and basic properties of Brownian motion. Prerequisite: STATS 116 or MATH 151 or equivalent. Recommended: MATH 115 or equivalent.
Terms: Aut | Units: 3 | UG Reqs: GER:DB-Math | Grading: Letter or Credit/No Credit
Instructors: ; Dembo, A. (PI); Zheng, T. (PI)

## MATH 147:Differential Topology

Smooth manifolds, transversality, Sards' theorem, embeddings, degree of a map, Borsuk-Ulam theorem, Hopf degree theorem, Jordan curve theorem. Prerequisite: 115 or 171.
Terms: Spr | Units: 3 | UG Reqs: GER:DB-Math | Grading: Letter or Credit/No Credit
Instructors: ; Berwick-Evans, D. (PI)

## MATH 149:Applied Algebraic Topology

Introduction to algebraic topology and its applications, in particular persistent homology as a tool for shape and pattern recognition from high dimensional data sets, with examples analyzed using state-of-the-art software. Prerequisite: linear algebra.
Terms: Win | Units: 3 | Grading: Letter or Credit/No Credit
Instructors: ; Carlsson, G. (PI)

## MATH 151:Introduction to Probability Theory

Counting; axioms of probability; conditioning and independence; expectation and variance; discrete and continuous random variables and distributions; joint distributions and dependence; central limit theorem and laws of large numbers. Prerequisite: 52 or consent of instructor.
Terms: Win | Units: 3 | UG Reqs: GER:DB-Math | Grading: Letter or Credit/No Credit
Instructors: ; Dembo, A. (PI)

## MATH 155:Analytic Number Theory

Topics in analytic number theory such as the distribution of prime numbers, the prime number theorem, twin primes and Goldbach's conjecture, the theory of quadratic forms, Dirichlet's class number formula, Dirichlet's theorem on primes in arithmetic progressions, and the fifteen theorem. Prerequisite: 152, or familiarity with the Euclidean algorithm, congruences, residue classes and reduced residue classes, primitive roots, and quadratic reciprocity.
Terms: Spr | Units: 3 | UG Reqs: GER:DB-Math | Grading: Letter or Credit/No Credit
Instructors: ; Soundararajan, K. (PI)

## MATH 159:Discrete Probabilistic Methods

Modern discrete probabilistic methods suitable for analyzing discrete structures of the type arising in number theory, graph theory, combinatorics, computer science, information theory and molecular sequence analysis. Prerequisite: STATS 116/MATH 151 or equivalent.
Terms: Win | Units: 3 | Grading: Letter or Credit/No Credit
Instructors: ; Dembo, A. (PI)

## MATH 16:Mathematics and Statistics in the Real World (STATS 90)

Introduction to non-calculus applications of mathematical ideas and principles in real-world problems. Topics include probability and counting, basic statistical concepts, geometric series. Applications include insurance, gambler's ruin, false positives in disease testing, present value of money, and mortgages. No knowledge of calculus required. Enrollment limited to students who do not have Stanford credit for a high school or college course in calculus or statistics.
Terms: not given this year | Units: 3 | UG Reqs: GER:DB-Math | Grading: Letter or Credit/No Credit

## MATH 161:Set Theory

Informal and axiomatic set theory: sets, relations, functions, and set-theoretical operations. The Zermelo-Fraenkel axiom system and the special role of the axiom of choice and its various equivalents. Well-orderings and ordinal numbers; transfinite induction and transfinite recursion. Equinumerosity and cardinal numbers; Cantor's Alephs and cardinal arithmetic. Open problems in set theory. Prerequisite: students should be comfortable doing proofs.
Terms: Aut | Units: 3 | UG Reqs: GER:DB-Math | Grading: Letter or Credit/No Credit
Instructors: ; Sommer, R. (PI)

## MATH 163:The Greek Invention of Mathematics

How was mathematics invented? A survey of the main creative ideas of ancient Greek mathematics. Among the issues explored are the axiomatic system of Euclid's Elements, the origins of the calculus in Greek measurements of solids and surfaces, and Archimedes' creation of mathematical physics. We will provide proofs of ancient theorems, and also learn how such theorems are even known today thanks to the recovery of ancient manuscripts.
Terms: not given this year | Units: 3-5 | UG Reqs: GER:DB-Hum | Grading: Letter or Credit/No Credit

## MATH 171:Fundamental Concepts of Analysis

Recommended for Mathematics majors and required of honors Mathematics majors. Similar to 115 but altered content and more theoretical orientation. Properties of Riemann integrals, continuous functions and convergence in metric spaces; compact metric spaces, basic point set topology. Prerequisite: 51H or 115 or consent of the instructor. WIM
Terms: Aut, Spr | Units: 3 | UG Reqs: GER:DB-Math, WAY-FR | Grading: Letter or Credit/No Credit

## MATH 173:Theory of Partial Differential Equations

A rigorous introduction to PDE accessible to advanced undergraduates. Elliptic, parabolic, and hyperbolic equations in many space dimensions including basic properties of solutions such as maximum principles, causality, and conservation laws. Methods include the Fourier transform as well as more classical methods. The Lebesgue integral will be used throughout, but a summary of its properties will be provided to make the course accessible to students who have not had 172 or 205A. Prerequisite: 171 or equivalent.
Terms: Spr | Units: 3 | Grading: Letter or Credit/No Credit
Instructors: ; Ignatova, M. (PI)

## MATH 175:Elementary Functional Analysis

Linear operators on Hilbert space. Spectral theory of compact operators; applications to integral equations. Elements of Banach space theory. Prerequisite: 115 or 171.
Terms: Spr | Units: 3 | UG Reqs: GER:DB-Math | Grading: Letter or Credit/No Credit
Instructors: ; Menz, G. (PI)

## MATH 193:Polya Problem Solving Seminar

Topics in mathematics and problem solving strategies with an eye towards the Putnam Competition. Topics may include parity, the pigeonhole principle, number theory, recurrence, generating functions, and probability. Students present solutions to the class. Open to anyone with an interest in mathematics.
Terms: Aut | Units: 1 | Repeatable for credit | Grading: Satisfactory/No Credit
Instructors: ; Soundararajan, K. (PI)

## MATH 199:Independent Work

Undergraduates pursue a reading program; topics limited to those not in regular department course offerings. Credit can fulfill the elective requirement for math majors. Approval of Undergraduate Affairs Committee is required to use credit for honors majors area requirement. Contact department student services specialist to enroll.
Terms: Aut, Win, Spr, Sum | Units: 1-3 | Repeatable for credit | Grading: Letter or Credit/No Credit

## MATH 210A:Modern Algebra I

Basic commutative ring and module theory, tensor algebra, homological constructions, linear and multilinear algebra, introduction to representation theory. Prerequisite: 122 or equivalent.
Terms: Aut | Units: 3 | Grading: Letter or Credit/No Credit
Instructors: ; Yun, Z. (PI)

## MATH 210B:Modern Algebra II

Continuation of 210A. Topics in Galois theory, commutative algebra, and algebraic geometry. Prerequisites: 210A, and 121 or equivalent.
Terms: Win | Units: 3 | Grading: Letter or Credit/No Credit

## MATH 210C:Lie Theory

Topics in Lie groups, Lie algebras, and/or representation theory. Prerequisite: math 210B. May be repeated for credit.
Terms: Spr | Units: 3 | Repeatable for credit | Grading: Letter or Credit/No Credit
Instructors: ; Venkatesh, A. (PI)

## MATH 215B:Complex Analysis, Geometry, and Topology

Topics: fundamental group and covering spaces, homology, cohomology, products, basic homotopy theory, and applications. Prerequisites: 113, 120, and 171, or equivalent; 215A is not a prerequisite for 215B.
Terms: Win | Units: 3 | Grading: Letter or Credit/No Credit
Instructors: ; Galatius, S. (PI)

## MATH 215C:Complex Analysis, Geometry, and Topology

Differentiable manifolds, transversality, degree of a mapping, vector fields, intersection theory, and Poincare duality. Differential forms and the DeRham theorem. Prerequisite: 215B or equivalent.
Terms: Spr | Units: 3 | Grading: Letter or Credit/No Credit
Instructors: ; Carlsson, G. (PI)

## MATH 217A:Differential Geometry

Smooth manifolds and submanifolds, tensors and forms, Lie and exterior derivative, DeRham cohomology, distributions and the Frobenius theorem, vector bundles, connection theory, parallel transport and curvature, affine connections, geodesics and the exponential map, connections on the principal frame bundle. Prerequisite: 215C or equivalent.
Terms: Spr | Units: 3 | Grading: Letter or Credit/No Credit
Instructors: ; Mazzeo, R. (PI)

## MATH 226:Numerical Solution of Partial Differential Equations (CME 306)

Hyperbolic partial differential equations: stability, convergence and qualitative properties; nonlinear hyperbolic equations and systems; combined solution methods from elliptic, parabolic, and hyperbolic problems. Examples include: Burger's equation, Euler equations for compressible flow, Navier-Stokes equations for incompressible flow. Prerequisites: MATH 220A or CME 302.
Terms: Spr | Units: 3 | Grading: Letter or Credit/No Credit
Instructors: ; Ying, L. (PI)

## MATH 228:Stochastic Methods in Engineering (CME 308)

Review of basic probability; Monte Carlo simulation; state space models and time series; parameter estimation, prediction, and filtering; Markov chains and processes; stochastic control; and stochastic differential equations. Examples from various engineering disciplines. Prerequisites: exposure to probability; background in real variables and analysis.
Terms: Spr | Units: 3 | Grading: Letter or Credit/No Credit
Instructors: ; Papanicolaou, G. (PI)

## MATH 230A:Theory of Probability (STATS 310A)

Mathematical tools: sigma algebras, measure theory, connections between coin tossing and Lebesgue measure, basic convergence theorems. Probability: independence, Borel-Cantelli lemmas, almost sure and Lp convergence, weak and strong laws of large numbers. Large deviations. Weak convergence; central limit theorems; Poisson convergence; Stein's method. Prerequisites: 116, MATH 171.
Terms: Aut | Units: 2-4 | Grading: Letter or Credit/No Credit

## MATH 230B:Theory of Probability (STATS 310B)

Conditional expectations, discrete time martingales, stopping times, uniform integrability, applications to 0-1 laws, Radon-Nikodym Theorem, ruin problems, etc. Other topics as time allows selected from (i) local limit theorems, (ii) renewal theory, (iii) discrete time Markov chains, (iv) random walk theory,nn(v) ergodic theory. Prerequisite: 310A or MATH 230A.
Terms: Win | Units: 2-3 | Grading: Letter or Credit/No Credit
Instructors: ; Montanari, A. (PI)

## MATH 230C:Theory of Probability (STATS 310C)

Continuous time stochastic processes: martingales, Brownian motion, stationary independent increments, Markov jump processes and Gaussian processes. Invariance principle, random walks, LIL and functional CLT. Markov and strong Markov property. Infinitely divisible laws. Some ergodic theory. Prerequisite: 310B or MATH 230B.
Terms: Spr | Units: 2-4 | Grading: Letter or Credit/No Credit
Instructors: ; Chatterjee, S. (PI)

## MATH 236:Introduction to Stochastic Differential Equations

Brownian motion, stochastic integrals, and diffusions as solutions of stochastic differential equations. Functionals of diffusions and their connection with partial differential equations. Random walk approximation of diffusions. Prerequisite: 136 or equivalent and differential equations.
Terms: Win | Units: 3 | Grading: Letter or Credit/No Credit
Instructors: ; Papanicolaou, G. (PI)

## MATH 238:Mathematical Finance (STATS 250)

Stochastic models of financial markets. Forward and futures contracts. European options and equivalent martingale measures. Hedging strategies and management of risk. Term structure models and interest rate derivatives. Optimal stopping and American options. Corequisites: MATH 236 and 227 or equivalent.
Terms: Win | Units: 3 | Grading: Letter or Credit/No Credit
Instructors: ; Papanicolaou, G. (PI)

## MATH 257C:Symplectic Geometry and Topology

Continuation of 257B. May be repeated for credit.
Terms: Win | Units: 3 | Grading: Letter or Credit/No Credit
Instructors: ; Ionel, E. (PI)

## MATH 263A:Infinite-dimensional Lie Algebras

Basics of Kac-Moody Lie algebras, which include both finite dimensional semisimple Lie algebras and their infinite-dimensional analogs, up to the Kac-Weyl character formula and Macdonald identities, and the Boson-Fermion correspondence. May be repeated for credit. Prerequisite: 210 or equivalent.
Terms: Win | Units: 3 | Repeatable for credit | Grading: Letter or Credit/No Credit
Instructors: ; Bump, D. (PI)

## MATH 263B:Quantum Groups and the Yang-Baxter Equation

Two classes of phenomena in mathematical physics, namely the solvable lattice models in statistical physics and Heisenberg spin chains lead to the same identity, namely the Yang-Baxter equation. Quasitriangular Hopf algebras (quantum groups), "braided monoidal category," such as Kuperberg's proof of the alternating sign conjecture, deformations of the Weyl character formula, and knot invariants such as the Jones polynomial. May be repeated for credit.
Terms: Spr | Units: 3 | Repeatable for credit | Grading: Letter or Credit/No Credit
Instructors: ; Bump, D. (PI)

## MATH 391:Research Seminar in Logic and the Foundations of Mathematics (PHIL 391)

Contemporary work. May be repeated a total of three times for credit. Math 391 students attend the logic colloquium in 380-381T.
Terms: Aut, Win, Spr | Units: 1-3 | Repeatable for credit | Grading: Letter or Credit/No Credit

## MATH 51M:Introduction to MATLAB for Multivariable Mathematics

Corequisite: MATH 51.
Terms: Aut | Units: 1 | Grading: Letter or Credit/No Credit
Instructors: ; Diao, P. (PI)

## MATH 70SI:The Game of Go: Strategy, Theory, and History

Strategy and mathematical theories of the game of Go, with guest appearance by a professional Go player.
Terms: not given this year | Units: 1 | Grading: Satisfactory/No Credit

## MATH 78SI:Speedcubing: HIstory, Theory, and Practice

History of the Rubik's cube; the current cubing community; basic mathematical theory; concepts to improve speed solving skill. Prior ability to solve cube not required.
Terms: not given this year | Units: 1 | Grading: Satisfactory/No Credit

## MATH 88Q:The Mathematics of the Rubik's Cube

Preference to sophomores. Group theory through topics that can be illustrated with the Rubik's cube: subgroups, homomorphisms and quotient groups, the symmetric and alternating groups, conjugation, commutators, and Sylow subgroups.
Terms: not given this year | Units: 3 | Grading: Letter (ABCD/NP)

## MATH 100:Mathematics for Elementary School Teachers

Mathematics and pedagogical strategies. Core mathematical content in grades K-6, classroom presentation, how to handle student errors, and mathematical issues that come up during instruction.
Terms: not given this year | Units: 4 | Grading: Letter or Credit/No Credit

## MATH 132:Partial Differential Equations II

Laplace's equation and properties of harmonic functions. Green's functions. Distributions and Fourier transforms. Eigenvalue problems and generalized Fourier series. Numerical solutions. Prerequisite: 131P.
Terms: not given this year | Units: 3 | UG Reqs: GER:DB-Math | Grading: Letter or Credit/No Credit

## MATH 137:Mathematical Methods of Classical Mechanics

Newtonian mechanics. Lagrangian formalism. E. Noether's theorem. Oscillations. Rigid bodies. Introduction to symplectic geometry. Hamiltonian formalism. Legendre transform. Variational principles. Geometric optics. Introduction to the theory of integrable systems. Prerequisites: 51, 52, 53, or 51H, 52H, 53H.
Terms: not given this year | Units: 3 | UG Reqs: GER:DB-Math | Grading: Letter or Credit/No Credit

## MATH 138:Celestial Mechanics

Mathematically rigorous introduction to the classical N-body problem: the motion of N particles evolving according to Newton's law. Topics include: the Kepler problem and its symmetries; other central force problems; conservation theorems; variational methods; Hamilton-Jacobi theory; the role of equilibrium points and stability; and symplectic methods. Prerequisites: 53, and 115 or 171.
Terms: not given this year | Units: 3 | UG Reqs: GER:DB-Math | Grading: Letter or Credit/No Credit

## MATH 143:Differential Geometry

Geometry of curves and surfaces in three-space and higher dimensional manifolds. Parallel transport, curvature, and geodesics. Surfaces with constant curvature. Minimal surfaces.
Terms: not given this year | Units: 3 | UG Reqs: GER:DB-Math | Grading: Letter or Credit/No Credit

## MATH 144:Riemannian Geometry

Smooth manifolds, tensor fields, geometry of Riemannian and Lorentz metrics, the Levi-Civita connection and curvature tensor, Ricci curvature, scalar curvature, and Einstein manifolds, spaces of constant curvature. Prerequisites: Math 51, 52, and 53.
Terms: Win | Units: 3 | Grading: Letter or Credit/No Credit
Instructors: ; Schoen, R. (PI)

## MATH 145:Algebraic Geometry

Hilbert's nullstellensatz, complex affine and projective curves, Bezout's theorem, the degree/genus formula, blow-up, Riemann-Roch theorem. Prerequisites: 120, and 121 or knowledge of fraction fields. Recommended: familiarity with surfaces equivalent to 143, 146, 147, or 148.
Terms: not given this year | Units: 3 | UG Reqs: GER:DB-Math | Grading: Letter or Credit/No Credit

## MATH 146:Analysis on Manifolds

Differentiable manifolds, tangent space, submanifolds, implicit function theorem, differential forms, vector and tensor fields. Frobenius' theorem, DeRham theory. Prerequisite: 52 or 52H.
Terms: not given this year | Units: 3 | UG Reqs: GER:DB-Math | Grading: Letter or Credit/No Credit

## MATH 148:Algebraic Topology

Fundamental group, covering spaces, Euler characteristic, homology, classification of surfaces, knots. Prerequisite: 109 or 120.
Terms: not given this year | Units: 3 | UG Reqs: GER:DB-Math | Grading: Letter or Credit/No Credit

## MATH 152:Elementary Theory of Numbers

Euclid's algorithm, fundamental theorems on divisibility; prime numbers; congruence of numbers; theorems of Fermat, Euler, Wilson; congruences of first and higher degrees; quadratic residues; introduction to the theory of binary quadratic forms; quadratic reciprocity; partitions.
Terms: Win | Units: 3 | UG Reqs: GER:DB-Math | Grading: Letter or Credit/No Credit
Instructors: ; Lucianovic, M. (PI)

## MATH 154:Algebraic Number Theory

Properties of number fields and Dedekind domains, quadratic and cyclotomic fields, applications to some classical Diophantine equations; introduction to elliptic curves. Prerequisites: 120 and 121, especially modules over principal ideal domains and Galois theory of finite fields.
Terms: alternate years, given next year | Units: 3 | UG Reqs: GER:DB-Math | Grading: Letter or Credit/No Credit

## MATH 162:Philosophy of Mathematics (PHIL 162, PHIL 262)

(Graduate students register for PHIL 262.) 20th-century approaches to the foundations and philosophy of mathematics. The background in mathematics, set theory, and logic. Schools and programs of logicism, predicativism, platonism, formalism, and constructivism. Readings from leading thinkers. Prerequisite: PHIL151 or consent of instructor.
Terms: not given this year | Units: 4 | UG Reqs: GER:DB-Math | Grading: Letter or Credit/No Credit

## MATH 172:Lebesgue Integration and Fourier Analysis

Similar to 205A, but for undergraduate Math majors and graduate students in other disciplines. Topics include Lebesgue measure on Euclidean space, Lebesgue integration, L^p spaces, the Fourier transform, the Hardy-Littlewood maximal function and Lebesgue differentiation. Prerequisite: 171 or consent of instructor.
Terms: Win | Units: 3 | UG Reqs: GER:DB-Math | Grading: Letter or Credit/No Credit
Instructors: ; Vasy, A. (PI)

## MATH 174:Calculus of Variations

An introductory course emphasizing the historical development of the theory, its connections to physics and mechanics, its independent mathematical interest, and its contacts with daily life experience. Applications to minimal surfaces and to capillary surface interfaces. Prerequisites: Math 171 or equivalent.
Terms: not given this year | Units: 3 | Grading: Letter or Credit/No Credit

## MATH 180:Introduction to Financial Mathematics

Financial derivatives: contracts and options. Hedging and risk management. Arbitrage, interest rate, and discounted value. Geometric random walk and Brownian motion as models of risky assets. Initial boundary value problems for the heat and related partial differential equations. Self-financing replicating portfolio. Black-Scholes pricing of European options. Dividends. Implied volatility. Optimal stopping and American options. Prerequisite: 53. Corequisites: 131, 151 or STATS 116.
Terms: not given this year | Units: 3 | UG Reqs: GER:DB-Math | Grading: Letter or Credit/No Credit

Weekly lectures by different experts on topics in pure and applied mathematics that go beyond the standard curriculum. May be repeated for credit for up to 3 units. Does not count toward the math major or minor.
Terms: Aut, Win, Spr | Units: 1 | Repeatable for credit | Grading: Satisfactory/No Credit
Instructors: ; Bump, D. (PI); Luu, M. (PI)

## MATH 197:Senior Honors Thesis

(Staff) May be repeated 3 times for a max of 9 units.
Terms: Aut, Win, Spr | Units: 1-6 | Repeatable for credit | Grading: Letter or Credit/No Credit

## MATH 198:Practical Training

Only for students majoring in mathematics. Students obtain employment in a relevant industrial or research activity to enhance their professional experience. Students submit a concise report detailing work activities, problems worked on, and key results. May be repeated for credit up to 3 units. Prerequisite: qualified offer of employment and consent of instructor. Prior approval by Math Department is required; you must contact the Math Department's Student Services staff for instructions before being granted permission to enroll.
Terms: Aut, Win, Spr, Sum | Units: 1 | Repeatable for credit | Grading: Satisfactory/No Credit

## MATH 205A:Real Analysis

Basic measure theory and the theory of Lebesgue integration. Prerequisite: 171 or equivalent.
Terms: Aut | Units: 3 | Grading: Letter or Credit/No Credit
Instructors: ; Simon, L. (PI)

## MATH 205B:Real Analysis

Point set topology, basic functional analysis, Fourier series, and Fourier transform. Prerequisites: 171 and 205A or equivalent.
Terms: Win | Units: 3 | Grading: Letter or Credit/No Credit
Instructors: ; Mazzeo, R. (PI)

## MATH 215A:Complex Analysis, Geometry, and Topology

Analytic functions, complex integration, Cauchy's theorem, residue theorem, argument principle, conformal mappings, Riemann mapping theorem, Picard's theorem, elliptic functions, analytic continuation and Riemann surfaces.
Terms: Aut | Units: 3 | Grading: Letter or Credit/No Credit
Instructors: ; Ryzhik, L. (PI)

## MATH 216A:Introduction to Algebraic Geometry

Algebraic curves, algebraic varieties, sheaves, cohomology, Riemann-Roch theorem. Classification of algebraic surfaces, moduli spaces, deformation theory and obstruction theory, the notion of schemes. May be repeated for credit. Prerequisites: 210ABC or equivalent.
Terms: Aut | Units: 3 | Repeatable for credit | Grading: Letter or Credit/No Credit
Instructors: ; Li, Z. (PI)

## MATH 216B:Introduction to Algebraic Geometry

Continuation of 216A. May be repeated for credit.
Terms: Win | Units: 3 | Repeatable for credit | Grading: Letter or Credit/No Credit
Instructors: ; Vakil, R. (PI)

## MATH 216C:Introduction to Algebraic Geometry

Continuation of 216B. May be repeated for credit.
Terms: Spr | Units: 3 | Repeatable for credit | Grading: Letter or Credit/No Credit
Instructors: ; Vakil, R. (PI)

## MATH 220:Partial Differential Equations of Applied Mathematics (CME 303)

First-order partial differential equations; method of characteristics; weak solutions; elliptic, parabolic, and hyperbolic equations; Fourier transform; Fourier series; and eigenvalue problems. Prerequisite: foundation in multivariable calculus and ordinary differential equations.
Terms: Aut | Units: 3 | Grading: Letter or Credit/No Credit
Instructors: ; Ryzhik, L. (PI)

## MATH 221A:Mathematical Methods of Imaging (CME 321A)

Image denoising and deblurring with optimization and partial differential equations methods. Imaging functionals based on total variation and l-1 minimization. Fast algorithms and their implementation.
Terms: Win | Units: 3 | Grading: Letter or Credit/No Credit
Instructors: ; Ryzhik, L. (PI)

## MATH 221B:Mathematical Methods of Imaging (CME 321B)

Array imaging using Kirchhoff migration and beamforming, resolution theory for broad and narrow band array imaging in homogeneous media, topics in high-frequency, variable background imaging with velocity estimation, interferometric imaging methods, the role of noise and inhomogeneities, and variational problems that arise in optimizing the performance of array imaging algorithms.
Terms: Spr | Units: 3 | Grading: Letter or Credit/No Credit
Instructors: ; Papanicolaou, G. (PI)

## MATH 222:Computational Methods for Fronts, Interfaces, and Waves

High-order methods for multidimensional systems of conservation laws and Hamilton-Jacobi equations (central schemes, discontinuous Galerkin methods, relaxation methods). Level set methods and fast marching methods. Computation of multi-valued solutions. Multi-scale analysis, including wavelet-based methods. Boundary schemes (perfectly matched layers). Examples from (but not limited to) geometrical optics, transport equations, reaction-diffusion equations, imaging, and signal processing.
Terms: not given this year | Units: 3 | Grading: Letter or Credit/No Credit

## MATH 224:Topics in Mathematical Biology

Mathematical models for biological processes based on ordinary and partial differential equations. Topics: population and infectious diseases dynamics, biological oscillators, reaction diffusion models, biological waves, and pattern formation. Prerequisites: 53 and 131, or equivalents.
Terms: not given this year | Units: 3 | Grading: Letter or Credit/No Credit

## MATH 227:Partial Differential Equations and Diffusion Processes

Parabolic and elliptic partial differential equations and their relation to diffusion processes. First order equations and optimal control. Emphasis is on applications to mathematical finance. Prerequisites: MATH 131 and MATH 136/STATS 219, or equivalents.
Terms: not given this year | Units: 3 | Grading: Letter or Credit/No Credit

## MATH 231A:An Introduction to Random Matrix Theory (STATS 351A)

Patterns in the eigenvalue distribution of typical large matrices, which also show up in physics (energy distribution in scattering experiments), combinatorics (length of longest increasing subsequence), first passage percolation and number theory (zeros of the zeta function). Classical compact ensembles (random orthogonal matrices). The tools of determinental point processes.
Terms: not given this year | Units: 3 | Grading: Letter (ABCD/NP)

## MATH 231C:Free Probability

Background from operator theory, addition and multiplication theorems for operators, spectral properties of infinite-dimensional operators, the free additive and multiplicative convolutions of probability measures and their classical counterparts, asymptotic freeness of large random matrices, and free entropy and free dimension. Prerequisite: STATS 310B or equivalent.
Terms: not given this year | Units: 3 | Grading: Letter or Credit/No Credit

## MATH 232:Topics in Probability: Percolation Theory

An introduction to some of the most important theorems and open problems in percolation theory. Topics include some of the difficult early breakthroughs of Kesten, Menshikov, Aizenman and others, and recent fields-medal winning works of Schramm, Lawler, Werner and Smirnov. Prerequisites: graduate-level probability.
Terms: not given this year | Units: 3 | Repeatable for credit | Grading: Letter or Credit/No Credit

## MATH 233:Probabilistic Methods in Analysis

Proofs and constructions in analysis obtained from basic results in Probability Theory and a 'probabilistic way of thinking.' Topics: Rademacher functions, Gaussian processes, entropy.
Terms: not given this year | Units: 3 | Repeatable for credit | Grading: Letter or Credit/No Credit

## MATH 234:Large Deviations Theory (STATS 374)

Combinatorial estimates and the method of types. Large deviation probabilities for partial sums and for empirical distributions, Cramer's and Sanov's theorems and their Markov extensions. Applications in statistics, information theory, and statistical mechanics. Prerequisite: MATH 230A or STATS 310.
Terms: not given this year | Units: 3 | Grading: Letter or Credit/No Credit

## MATH 239:Computation and Simulation in Finance

Monte Carlo, finite difference, tree, and transform methods for the numerical solution of partial differential equations in finance. Emphasis is on derivative security pricing. Prerequisite: 238 or equivalent.
Terms: not given this year | Units: 3 | Grading: Letter or Credit/No Credit

## MATH 243:Functions of Several Complex Variables

Terms: not given this year | Units: 3 | Repeatable for credit | Grading: Letter or Credit/No Credit

## MATH 244:Riemann Surfaces

Compact Riemann surfaces and algebraic curves; cohomology of sheaves; Serre duality; Riemann-Roch theorem and application; Jacobians; Abel's theorem. May be repeated for credit.
Terms: not given this year | Units: 3 | Repeatable for credit | Grading: Letter or Credit/No Credit

## MATH 245A:Topics in Algebraic Geometry: Moduli Theory

Topics in the study of moduli spaces: Basic of algebraic surfaces, Hodge structure of surfaces, moduli of K3 surfaces, cycles and rational curves in K3 surfaces, Torelli for K3 surfaces.
Terms: Win | Units: 3 | Repeatable for credit | Grading: Letter or Credit/No Credit
Instructors: ; Li, J. (PI)

## MATH 245B:Topics in Algebraic Geometry: Intersection Theory

Topics such as intersection theory on surfaces, toric varieties, and homogeneous spaces; numerical criteria for positivity; Chow groups and rings. May be repeated for credit.
Terms: not given this year | Units: 3 | Repeatable for credit | Grading: Letter or Credit/No Credit

## MATH 245C:Topics in Algebraic Geometry: Alterations

Terms: not given this year | Units: 3 | Repeatable for credit | Grading: Letter or Credit/No Credit

## MATH 247:Topics in Group Theory

Topics include the Burnside basis theorem, classification of p-groups, regular and powerful groups, Sylow theorems, the Frattini argument, nilpotent groups, solvable groups, theorems of P. Hall, group cohomology, and the Schur-Zassenhaus theorem. The classical groups and introduction to the classification of finite simple groups and its applications. May be repeated for credit.
Terms: not given this year | Units: 3 | Repeatable for credit | Grading: Letter or Credit/No Credit

## MATH 248:Ergodic Theory and Szemeredi's Theorem

An introduction to ergodic theory leading to (and proving) Szemeredi's theorem and its multidimensional extension. Prerequisite: 205a and some knowledge of Hilbert spaces.
Terms: not given this year | Units: 3 | Repeatable for credit | Grading: Letter or Credit/No Credit

## MATH 248A:Algebraic Number Theory

Structure theory and Galois theory of local and global fields, finiteness theorems for class numbers and units, adelic techniques. Prerequisites: MATH 210A,B.
Terms: not given this year | Units: 3 | Repeatable for credit | Grading: Letter or Credit/No Credit

## MATH 249A:Topics in number theory

Terms: Aut | Units: 3 | Repeatable for credit | Grading: Letter or Credit/No Credit
Instructors: ; Soundararajan, K. (PI)

## MATH 249B:Topics in Number Theory

Terms: Win | Units: 3 | Repeatable for credit | Grading: Letter or Credit/No Credit
Instructors: ; Venkatesh, A. (PI)

## MATH 249C:Topics in Number Theory

Terms: Spr | Units: 3 | Repeatable for credit | Grading: Letter or Credit/No Credit

## MATH 252:Algebraic Groups

Smooth affine groups over general fields, quotients, tori, solvable groups, reductive groups, root systems, Existence and Isomorphism theorem, structure theory. If time permits, classification theory over interesting fields. Prerequisites: 210A, 210B, and familiarity with algebraic varieties over general fields.
Terms: not given this year | Units: 3 | Repeatable for credit | Grading: Letter or Credit/No Credit

## MATH 254:Geometric Methods in the Theory of Ordinary Differential Equations

Topics may include: structural stability and perturbation theory of dynamical systems; hyperbolic theory; first order PDE; normal forms, bifurcation theory; Hamiltonian systems, their geometry and applications. May be repeated for credit.
Terms: not given this year | Units: 3 | Repeatable for credit | Grading: Letter or Credit/No Credit

## MATH 256A:Partial Differential Equations

The theory of linear and nonlinear partial differential equations, beginning with linear theory involving use of Fourier transform and Sobolev spaces. Topics: Schauder and L2 estimates for elliptic and parabolic equations; De Giorgi-Nash-Moser theory for elliptic equations; nonlinear equations such as the minimal surface equation, geometric flow problems, and nonlinear hyperbolic equations.
Terms: Spr | Units: 3 | Grading: Letter or Credit/No Credit
Instructors: ; Vasy, A. (PI)

## MATH 256B:Partial Differential Equations

Continuation of 256A.
Terms: Win | Units: 3 | Repeatable for credit | Grading: Letter or Credit/No Credit
Instructors: ; Vasy, A. (PI)

## MATH 257A:Symplectic Geometry and Topology

Linear symplectic geometry and linear Hamiltonian systems. Symplectic manifolds and their Lagrangian submanifolds, local properties. Symplectic geometry and mechanics. Contact geometry and contact manifolds. Relations between symplectic and contact manifolds. Hamiltonian systems with symmetries. Momentum map and its properties. May be repeated for credit.
Terms: not given this year | Units: 3 | Repeatable for credit | Grading: Letter or Credit/No Credit

## MATH 257B:Symplectic Geometry and Topology

Continuation of 257A. May be repeated for credit.
Terms: not given this year | Units: 3 | Repeatable for credit | Grading: Letter or Credit/No Credit

## MATH 258:Topics in Geometric Analysis

May be repeated for credit.
Terms: Spr | Units: 3 | Repeatable for credit | Grading: Letter or Credit/No Credit
Instructors: ; Wang, Y. (PI)

## MATH 259:mirror symmetry

Terms: not given this year | Units: 3 | Repeatable for credit | Grading: Letter or Credit/No Credit

## MATH 261A:Functional Analysis

Geometry of linear topological spaces. Linear operators and functionals. Spectral theory. Calculus for vector-valued functions. Operational calculus. Banach algebras. Special topics in functional analysis. May be repeated for credit.
Terms: not given this year | Units: 3 | Repeatable for credit | Grading: Letter or Credit/No Credit

## MATH 262:Applied Fourier Analysis and Elements of Modern Signal Processing (CME 372)

Introduction to the mathematics of the Fourier transform and how it arises in a number of imaging problems. Mathematical topics include the Fourier transform, the Plancherel theorem, Fourier series, the Shannon sampling theorem, the discrete Fourier transform, and the spectral representation of stationary stochastic processes. Computational topics include fast Fourier transforms (FFT) and nonuniform FFTs. Applications include Fourier imaging (the theory of diffraction, computed tomography, and magnetic resonance imaging) and the theory of compressive sensing.
Terms: Win | Units: 3 | Grading: Letter or Credit/No Credit
Instructors: ; Candes, E. (PI)

## MATH 264:Infinite Dimensional Lie Algebra

Terms: not given this year | Units: 3 | Repeatable for credit | Grading: Letter or Credit/No Credit

## MATH 266:Computational Signal Processing and Wavelets

Theoretical and computational aspects of signal processing. Topics: time-frequency transforms; wavelet bases and wavelet packets; linear and nonlinear multiresolution approximations; estimation and restoration of signals; signal compression. May be repeated for credit.
Terms: not given this year | Units: 3 | Grading: Letter or Credit/No Credit

## MATH 269:Topics in symplectic geometry

May be repeated for credit.
Terms: Spr | Units: 3 | Repeatable for credit | Grading: Letter or Credit/No Credit
Instructors: ; Eliashberg, Y. (PI)

## MATH 270:Geometry and Topology of Complex Manifolds

Complex manifolds, Kahler manifolds, curvature, Hodge theory, Lefschetz theorem, Kahler-Einstein equation, Hermitian-Einstein equations, deformation of complex structures. May be repeated for credit.
Terms: not given this year | Units: 3 | Repeatable for credit | Grading: Letter or Credit/No Credit

## MATH 271:The H-Principle

The language of jets. Thom transversality theorem. Holonomic approximation theorem. Applications: immersion theory and its generaliazations. Differential relations and Gromov's h-principle for open manifolds. Applications to symplectic geometry. Microflexibility. Mappings with simple singularities and their applications. Method of convex integration. Nash-Kuiper C^1-isometric embedding theorem.
Terms: not given this year | Units: 3 | Grading: Letter or Credit/No Credit

## MATH 272:Topics in Partial Differential Equations

Terms: not given this year | Units: 3 | Repeatable for credit | Grading: Letter or Credit/No Credit

## MATH 273A:Quantum Mechanics I

Terms: not given this year | Units: 3 | Grading: Letter or Credit/No Credit

## MATH 273B:QUANTUM MECHANICS II

Terms: not given this year | Units: 3 | Grading: Letter or Credit/No Credit

## MATH 280:Evolution Equations in Differential Geometry

Terms: Win | Units: 3 | Repeatable for credit | Grading: Letter or Credit/No Credit
Instructors: ; Bamler, R. (PI)

## MATH 282A:Low Dimensional Topology

The theory of surfaces and 3-manifolds. Curves on surfaces, the classification of diffeomorphisms of surfaces, and Teichmuller space. The mapping class group and the braid group. Knot theory, including knot invariants. Decomposition of 3-manifolds: triangulations, Heegaard splittings, Dehn surgery. Loop theorem, sphere theorem, incompressible surfaces. Geometric structures, particularly hyperbolic structures on surfaces and 3-manifolds. May be repeated for credit up to 6 total units.
Terms: Spr | Units: 3 | Repeatable for credit | Grading: Letter or Credit/No Credit
Instructors: ; Mirzakhani, M. (PI)

## MATH 282B:Homotopy Theory

Homotopy groups, fibrations, spectral sequences, simplicial methods, Dold-Thom theorem, models for loop spaces, homotopy limits and colimits, stable homotopy theory. May be repeated for credit up to 6 total units.
Terms: Win | Units: 3 | Repeatable for credit | Grading: Letter or Credit/No Credit
Instructors: ; Galatius, S. (PI)

## MATH 282C:Fiber Bundles and Cobordism

Possible topics: principal bundles, vector bundles, classifying spaces. Connections on bundles, curvature. Topology of gauge groups and gauge equivalence classes of connections. Characteristic classes and K-theory, including Bott periodicity, algebraic K-theory, and indices of elliptic operators. Spectral sequences of Atiyah-Hirzebruch, Serre, and Adams. Cobordism theory, Pontryagin-Thom theorem, calculation of unoriented and complex cobordism. May be repeated for credit up to 6 total units.
Terms: Spr | Units: 3 | Repeatable for credit | Grading: Letter or Credit/No Credit
Instructors: ; Carlsson, G. (PI)

## MATH 283:Topics in Algebraic and Geometric Topology

May be repeated for credit.
Terms: not given this year | Units: 3 | Repeatable for credit | Grading: Letter or Credit/No Credit

## MATH 283A:Topics in Topology

Terms: not given this year | Units: 3 | Grading: Letter or Credit/No Credit

## MATH 284:Topics in Geometric Topology

Incompressible surfaces, irreducible manifolds, prime decomposition, Morse theory, Heegaard diagrams, Heegaard splittings, the Thurston norm, sutured manifold theory, Heegaard Floer homology, sutured Floer homology.
Terms: Win | Units: 3 | Repeatable for credit | Grading: Letter or Credit/No Credit
Instructors: ; Kerckhoff, S. (PI)

## MATH 284A:Geometry and Topology in Dimension 3

The Poincare conjecture and the uniformization of 3-manifolds. May be repeated for credit.
Terms: not given this year | Units: 3 | Repeatable for credit | Grading: Letter or Credit/No Credit

## MATH 284B:Geometry and Topology in Dimension 3

The Poincare conjecture and the uniformization of 3-manifolds. May be repeated for credit.
Terms: not given this year | Units: 3 | Repeatable for credit | Grading: Letter or Credit/No Credit

## MATH 286:Topics in Differential Geometry

May be repeated for credit.
Terms: Win, Spr | Units: 3 | Repeatable for credit | Grading: Letter or Credit/No Credit

## MATH 287:Introduction to optimal transportation

This will be an introductory course on Optimal Transportation theory. We will study Monge's problem, Kantorovich's problem, c-concave functions (also in the Riemannian setting), Wasserstein distance and geodesics (including a PDE formulation), applications to inequalities in convex analysis, as well as other topics, time permitting.
Terms: not given this year | Units: 3 | Grading: Letter or Credit/No Credit

## MATH 290B:Model Theory B (PHIL 350B)

Decidable theories. Model-theoretic background. Dense linear orders, arithmetic of addition, real closed and algebraically closed fields, o-minimal theories.
Terms: Aut | Units: 1-3 | Repeatable for credit | Grading: Letter or Credit/No Credit
Instructors: ; Mints, G. (PI)

## MATH 292A:Set Theory (PHIL 352A)

The basics of axiomatic set theory; the systems of Zermelo-Fraenkel and Bernays-Gödel. Topics: cardinal and ordinal numbers, the cumulative hierarchy and the role of the axiom of choice. Models of set theory, including the constructible sets and models constructed by the method of forcing. Consistency and independence results for the axiom of choice, the continuum hypothesis, and other unsettled mathematical and set-theoretical problems. Prerequisites: PHIL151 and MATH 161, or equivalents.
Terms: not given this year | Units: 3 | Grading: Letter or Credit/No Credit

## MATH 293A:Proof Theory (PHIL 353A)

Gentzen's natural deduction and sequential calculi for first-order propositional and predicate logics. Normalization and cut-elimination procedures. Relationships with computational lambda calculi and automated deduction. Prerequisites: 151, 152, and 161, or equivalents.
Terms: not given this year | Units: 3 | Grading: Letter or Credit/No Credit

## MATH 295:Computation and Algorithms in Mathematics

Use of computer and algorithmic techniques in various areas of mathematics. Computational experiments. Topics may include polynomial manipulation, Groebner bases, computational geometry, and randomness. May be repeated for credit.
Terms: not given this year | Units: 3 | Repeatable for credit | Grading: Letter or Credit/No Credit

## MATH 301:Advanced Topics in Convex Optimization

Modern developments in convex optimization: semidefinite programming; novel and efficient first-order algorithms for smooth and nonsmooth convex optimization. Emphasis on numerical methods suitable for large scale problems arising in science and engineering. Prerequisites: convex optimization (EE 364), linear algebra (Math 104), numerical linear algebra (CME 302); background in probability, statistics, real analysis and numerical optimization.
Terms: not given this year | Units: 3 | Repeatable for credit | Grading: Letter or Credit/No Credit

## MATH 310:Top Ten Algorithms of the 20th Century (CME 329)

A high-level survey course covering one algorithm per week: metropolis, simplex method, conjugate gradient, QR, quicksort, fast fourier transform, maxcut, fast multipole method, integer relation detection, and convex/semi-definite programming.
Terms: Aut | Units: 3 | Grading: Letter or Credit/No Credit

Required of and limited to first-year Mathematics graduate students.
Terms: Spr | Units: 1 | Grading: Satisfactory/No Credit

Terms: Aut, Win, Spr, Sum | Units: 1-10 | Repeatable for credit | Grading: Letter or Credit/No Credit

## MATH 361:Research Seminar Participation

Participation in a faculty-led seminar which has no specific course number.
Terms: Aut, Win, Spr, Sum | Units: 1-3 | Repeatable for credit | Grading: Letter or Credit/No Credit

## MATH 381:Seminar in Analysis

Terms: by arrangement | Units: 1-3 | Repeatable for credit | Grading: Letter or Credit/No Credit

## MATH 382:Qualifying Examination Seminar

Terms: Sum | Units: 1-3 | Repeatable for credit | Grading: Satisfactory/No Credit
Instructors: ; Ionel, E. (PI)

## MATH 384:Seminar in Geometry

Terms: by arrangement | Units: 1 | Repeatable for credit | Grading: Satisfactory/No Credit

## MATH 385:Seminar in Topology

Terms: by arrangement | Units: 1-3 | Repeatable for credit | Grading: Satisfactory/No Credit

## MATH 388:Seminar in Probability and Stochastic Processes

Terms: by arrangement | Units: 1-3 | Repeatable for credit | Grading: Letter or Credit/No Credit

## MATH 389:Seminar in Mathematical Biology

Terms: by arrangement | Units: 1-3 | Repeatable for credit | Grading: Letter or Credit/No Credit

## MATH 394:Classics in Analysis

Original papers in analysis.
Terms: not given this year | Units: 3 | Repeatable for credit | Grading: Letter or Credit/No Credit

## MATH 395:Classics in Geometry and Topology

Original papers in geometry and in algebraic and geometric topology. May be repeated for credit.
Terms: not given this year | Units: 3 | Repeatable for credit | Grading: Letter or Credit/No Credit