Department of Mathematics

MATH 30200-30300. Computability Theory I-II.

The courses in this sequence are offered in alternate years.

MATH 30500. Computability and Complexity Theory. 100 Units.

Part one of this course consists of models for defining computable functions: primitive recursive functions, (general) recursive functions, and Turing machines; the Church-Turing Thesis; unsolvable problems; diagonalization; and properties of computably enumerable sets. Part two of this course deals with Kolmogorov (resource bounded) complexity: the quantity of information in individual objects. Part three of this course covers functions computable with time and space bounds of the Turing machine: polynomial time computability, the classes P and NP, NP-complete problems, polynomial time hierarchy, and P-space complete problems.

Instructor(s): A. Razborov     Terms Offered: Winter
Prerequisite(s): Consent of department counselor and instructor
Note(s): Not offered in 2016-17.
Equivalent Course(s): CMSC 38500

MATH 30900-31000. Model Theory I-II.

MATH 30900 covers completeness and compactness; elimination of quantifiers; omission of types; elementary chains and homogeneous models; two cardinal theorems by Vaught, Chang, and Keisler; categories and functors; inverse systems of compact Hausdorf spaces; and applications of model theory to algebra. In MATH 31000, we study saturated models; categoricity in power; the Cantor-Bendixson and Morley derivatives; the Morley theorem and the Baldwin-Lachlan theorem on categoricity; rank in model theory; uniqueness of prime models and existence of saturated models; indiscernibles; ultraproducts; and differential fields of characteristic zero.

MATH 31200-31300-31400. Analysis I-II-III.

Analysis I-II-III

MATH 31700-31800-31900. Topology and Geometry I-II-III.

Topology and Geometry I-II-III

MATH 32500-32600-32700. Algebra I-II-III.

Algebra I-II-III

MATH 34100. Geometric Literacy-1. 100 Units.

This ongoing course might be subtitled:  "what every good geometer should know". The topics will intersperse more elementary background with topics close to current research, and should be understandable to second year students. The individual modules (2-5 weeks each) might be logically interrelated, but we will try to maintain a "modular structure" so that people who are willing to assume certain results as "black boxes" will be able to follow more advanced modules before formally learning all the prerequisites. This years topics might include: basics of symplectic geometry, harmonic maps in geometry, pseudo-Anosov homeomorphisms and Thurston's compactification of Teichmuller space, algebraic geometry for non-algebraic geometers. Prereq: First year graduate sequence.

Instructor(s): Benson Farb     Terms Offered: Autumn
Prerequisite(s): First year graduate sequence.

MATH 34300. Geometric Literacy-3. 100 Units.

This ongoing course might be subtitled:  "what every good geometer should know". The topics will intersperse more elementary background with topics close to current research, and should be understandable to second year students. The individual modules (2-5 weeks each) might be logically interrelated, but we will try to maintain a "modular structure" so that people who are willing to assume certain results as "black boxes" will be able to follow more advanced modules before formally learning all the prerequisites. This years topics might include: basics of symplectic geometry, harmonic maps in geometry, pseudo-Anosov homeomorphisms and Thurston's compactification of Teichmuller space, algebraic geometry for non-algebraic geometers. Prereq: First year graduate sequence.

Instructor(s): Benson Farb     Terms Offered: Spring
Prerequisite(s): First year graduate sequence.

MATH 35506. Tpoics in Harmonic Analysis. 100 Units.

We will discuss some developments in harmonic analysis, but also cover more basic material.

Instructor(s): Wilhelm Schlag     Terms Offered: Spring

MATH 35600. Topics in Dynamical Systems. 100 Units.

An independent study in topics in dynamical systems.

Instructor(s): Anne Wilkinson     Terms Offered: Autumn

MATH 36000. Topology Proseminar. 100 Units.

This informal "proseminar" is devoted to topics in algebraic topology and neighboring fields. Talks are given by graduate students, postdocs, and senior faculty. They range from basic background through current research.

Instructor(s): Staff

MATH 36100. Topology Proseminar. 100 Units.

This informal "proseminar" is devoted to topics in algebraic topology and neighboring fields. Talks are given by graduate students, postdocs, and senior faculty. They range from basic background through current research.

Instructor(s): J. Peter May     Terms Offered: Winter

MATH 36200. Topology Proseminar. 100 Units.

The Spring proseminar is a more formal version of the Fall and Winter topology proseminar. It will be taught primarily or completely by May, on topics of interest to the participants.

Best,

Instructor(s): J. Peter May     Terms Offered: Spring

MATH 37102. Introduction to Minimal Surfaces and applications. 100 Units.

Minimal surfaces have long been a central tool in geometry. I will present the general theory of existence and regularity for minimal surfaces. I will then focus on applications of the existence of minimal surfaces, namely its connection with positive scalar curvature, the Positive Mass Theorem, and classification of manifolds with positive isotropic curvature.

Instructor(s): Andre Neves     Terms Offered: Winter
Prerequisite(s): Elliptic theory of PDE's and Riemannian Geometry

MATH 37106. Topics in Geometric Measure Theory-2. 100 Units.

A measure is a way to assign a size to collections of points. Lebesgue measure is the most important example but, depending upon the application, the 'size' of a set may be measured in many different, very interesting ways. The interplay between measure and geometry can be extremely subtle and has given rise to powerful ideas that are used in energy minimisation problems, the theory of partial differential equations and the study of fractal geometry. This is an advanced course on geometric measure theory and its applications. 

Instructor(s): Marianna Csornyei     Terms Offered: Spring

MATH 37203. Koszul Duality. 100 Units.

Koszul algebras are a class of graded algebras with especially nice homological properties. Koszul duality is an equivalence of derived categories of graded modules over a pair of Koszul dual algebras. A basic example of such a pair is the Symmetric algebra of a vector space and the exterior algebra of the dual vector space and the exterior algebra of the dual vector space. We will discuss applications of Koszul duality to geometric representation theory and to the topology of hyperplane arrangements.

Instructor(s): Victor Ginzburg     Terms Offered: Winter

MATH 37204. Geometric Satake. 100 Units.

Geometric Satake is a basic ingredient in the formulation of the Geometric Langlands Program. Our main goal is to introduce the Satake category of perverse sheaves on the affine Grassmannian and to prove the Satake equivalence theorem that states that the Satake category is equivalent, as a monoidal category, to the category of finite dimensional representations of the Langlands dual group. Time permitting, we will then discuss a derived category refinement of the Satake equivalence.

Instructor(s): Victor Ginzburg     Terms Offered: Spring

MATH 37500. Algorithms in Finite Groups. 100 Units.

We consider the asymptotic complexity of some of the basic problems of computational group theory. The course demonstrates the relevance of a mix of mathematical techniques, ranging from combinatorial ideas, the elements of probability theory, and elementary group theory, to the theories of rapidly mixing Markov chains, applications of simply stated consequences of the Classification of Finite Simple Groups (CFSG), and, occasionally, detailed information about finite simple groups. No programming problems are assigned.

Instructor(s): L. Babai     Terms Offered: Spring
Prerequisite(s): Consent of department counselor. Linear algebra, finite fields, and a first course in group theory (Jordan-Holder and Sylow theorems) required; prior knowledge of algorithms not required
Note(s): This course is offered in alternate years.
Equivalent Course(s): CMSC 36500

MATH 37609. Topics In PDE. 100 Units.

We will study elliptic partial differential equations including regularity results such as De Giorgi/Nash theory, Krylov-Safonov theory, and nonlinear equations.

Instructor(s): Luis Silvestre     Terms Offered: Autumn
Prerequisite(s): Math 31200, 31300, 31400

MATH 37760. Modern Signal Processing. 100 Units.

This course covers contemporary developments from time-frequency transforms and wavelets (1980s) to compressed sensing (2000s), a period during which signal processing significantly evolved and broadened to become the "mathematics of information". Topics: Review of classical sampling theory: Shannon-Nyquist, aliasing, filtering. Time-frequency transforms. Frame theory. Wavelet bases and filterbanks. Sparsity and nonlinear approximation. Algorithms: basis pursuit and matching pursuit. Compressed sensing. Matrix completion. Special topics: curvelets, phase retrieval, superresolution. Students who already have an interest in medical imaging (MRI, CT), or geophysical data processing (seismic, e-m), for instance, are welcome. The course assumes some affinity with undergraduate mathematics. The evaluation will consist of homework problems, and a project of the student's choice. The project can either consist in reproducing results from the literature, or can be research-oriented.

Terms Offered: Autumn
Prerequisite(s): Linear algebra and multivariate calculus
Note(s): Not offered in 2017-18
Equivalent Course(s): STAT 37760

MATH 38100. Geometry, Complexity, and Algorithms. 100 Units.

This course will try to explore these three topics and their interactions. Among the topics likely to be discussed are metric measure geometry (e.g. concentration of measure) and its use designing algorithms, machine learning, manifold learning, the complexity of the construction of isotopies and nullcobordisms, the Blum-Cucker-Smale theory of real computation and estimates for the complexity of root finding and related problems, persistence homology and applications, and other topics that seem like a good idea as the course develops.

Instructor(s): Shmuel Weinberger     Terms Offered: Winter
Prerequisite(s): Undergraduate mathematics, the idea of a Turing machine and what an algorthm is, ideally a quarter of each of algebra, algebraic topology, differential topology and complex variables (even at the undergraduate level) and the willingness to work.

MATH 38300. Numerical Solutions to Partial Differential Equations. 100 Units.

This course covers the basic mathematical theory behind numerical solution of partial differential equations. We investigate the convergence properties of finite element, finite difference and other discretization methods for solving partial differential equations, introducing Sobolev spaces and polynomial approximation theory. We emphasize error estimators, adaptivity, and optimal-order solvers for linear systems arising from PDEs. Special topics include PDEs of fluid mechanics, max-norm error estimates, and Banach-space operator-interpolation techniques.

Instructor(s): L. R. Scott     Terms Offered: Spring. This course is offered in alternate years.
Prerequisite(s): Consent of department counselor and instructor
Equivalent Course(s): CMSC 38300

MATH 38302. Height functions in Number Theory. 100 Units.

I explain height functions on algebraic varieties over number fields and related subjects.

Instructor(s): Kazuya Kato     Terms Offered: Winter

MATH 38305. A Second Course In Number Theory. 100 Units.

The goal of this course will be to introduce some new methods and techniques beyond those covered in Math 32700, and also to give many worked examples on how to use these ideas in practice. Specific topics may include L-functions, applications of class field theory, and Diophantine equations.

Instructor(s): Frank Calegari     Terms Offered: Autumn
Prerequisite(s): Math 32700

MATH 38405. Arithmetic Combinatorics. 100 Units.

This course covers a variety of topics in arithmetic combinatorics such as inverse problems, incidence geometry, uniformity, regularity and pseudo-randomness. A special attention will be paid to connections to classical mathematics and theoretical computer science. 

Instructor(s): Alexander Razborov     Terms Offered: Spring

MATH 38509. Brownian Motion and Stochastic Calculus. 100 Units.

This is a rigorous introduction to the mathematical theory of Brownian motion and the corresponding integration theory (stochastic integration). This is material that all analysis graduate students should learn at some point whether or not they are immediately planning to use probabilistic techniques. It is also a natural course for more advanced math students who want to broaden their mathematical education and to increase their marketability for nonacademic positions. In particular, it is one of the most fundamental mathematical tools used in financial mathematics (although we will not discuss finance in this course). This course differs from the more applied STAT 39000 in that concepts are developed precisely and rigorously.

Terms Offered: Autumn
Prerequisite(s): STAT 38300 or MATH 31200, or permission of the instructor.
Equivalent Course(s): STAT 38500

MATH 38511. Brownian Motion and Stochastic Caluculus. 100 Units.

This is a rigorous introduction to the mathematical theory of Brownian motion and the corresponding integration theory (stochastic integration). This is material that all analysis graduate students should learn at some point whether or not they are immediately planning to use probabilistic techniques. It is also a natural course for more advanced math students who want to broaden their mathematical education and to increase their marketability for nonacademic positions. In particular, it is one of the most fundamental mathematical tools used in financial mathematics (although we will not discuss finance in this course). This course differs from the more applied STAT 39000 in that concepts are developed precisely and rigorously.

Instructor(s): G. Lawler     Terms Offered: Autumn
Prerequisite(s): The usual prerequisites are either the first-year graduate mathematical analysis sequence (mainly the material in MATH 31200) or STAT 38100-38300, the first two quarters of the statistics measure-theoretic probability sequence.
Equivalent Course(s): STAT 38510

MATH 38704. Quantitative Unique Continuation For Elliptic Equations. 100 Units.

In this course we will discuss classical unique continuation results for second order eigenvalue problems. Such results are of interest in themselves, but also in nonlinear pde problems, in mathematical physics and in nodal geometry. In this connection, we will also discuss the recent breakthrough works of Logunov and Malinikova on the Yau and Nadirashvili conjectures in nodal geometry.

Instructor(s): Carlos Kenig     Terms Offered: Autumn

MATH 38815. Geometric Complexity. 100 Units.

This course provides a basic introduction to geometric complexity theory, an approach to the P vs. NP and related problems through algebraic geometry and representation theory. No background in algebraic geometry or representation theory will be assumed.

Instructor(s): K. Mulmuley     Terms Offered: Autumn. This course is offered in alternate years.
Prerequisite(s): Consent of department counselor and instructor
Note(s): Background in algebraic geometry or representation theory not required
Equivalent Course(s): CMSC 38815

MATH 38900. Geometry, Complexity, and Algorithms. 100 Units.

This course will try to explore these three topics and their interactions. Among the topics likely to be discussed are metric measure geometry (e.g. concentration of measure) and its use designing algorithms, machine learning, manifold learning, the complexity of the construction of isotopies and nullcobordisms, the Blum-Cucker-Smale theory of real computation and estimates for the complexity of root finding and related problems, persistence homology and applications, and other topics that seem like a good idea as the course develops.

Instructor(s): Shmuel Weinberger     Terms Offered: Winter
Prerequisite(s): Undergraduate mathematics, the idea of a Turing machine and what an algorthm is, ideally a quarter of each of algebra, algebraic topology, differential topology and complex variables (even at the undergraduate level) and the willingness to work.

MATH 39701. Low-Dimensional Topology. 100 Units.

This course will be an introduction to geometric and topological methods in the study of low-dimensional manifolds, especially in dimension 4.

Instructor(s): Danny Calegari     Terms Offered: Winter

MATH 41005. Sheaf Theory And Homological Algebra-1. 100 Units.

An introduction to Grothendieck's six functor formalism, aimed at second year students.

Instructor(s): Madhav Nori     Terms Offered: Autumn

MATH 42900. Mathematical Modeling of Large-Scale Brain Activity 1. 100 Units.

An independant study in mathematical modeling.

Instructor(s): Jack Cowan     Terms Offered: Autumn

MATH 42901. Mathematical Modeling of Large-Scale Brain Activity 2. 100 Units.

Independent study in Mathematical Modeling of Large-Scale Brain Activity 2.

Instructor(s): Jack Cowan     Terms Offered: Spring
Equivalent Course(s): CPNS 42901

MATH 47000. Geometric Langlands Seminar. 100 Units.

This seminar is devoted not only to the Geometric Langlands theory but also to related subjects (including topics in algebraic geometry, algebra and representation theory). We will try to learn some modern homological algebra (Kontsevich's A- infinity categories) and some "forgotten" parts of D- module theory (e.g. the microlocal approach).

Instructor(s): Alexander Beilinson, Vladimir Drinfeld     Terms Offered: Autumn

MATH 47100. Geometric Langlands Seminar. 100 Units.

This seminar is devoted not only to the Geometric Langlands theory but also to related subjects (including topics in algebraic geometry, algebra and representation theory). We will try to learn some modern homological algebra (Kontsevich's A- infinity categories) and some "forgotten" parts of D- module theory (e.g. the microlocal approach).

Instructor(s): Alexander Beilinson, Vladimir Drinfeld     Terms Offered: Winter

MATH 47200. Geometric Langlands Seminar. 100 Units.

This seminar is devoted not only to the Geometric Langlands theory but also to related subjects (including topics in algebraic geometry, algebra and representation theory). We will try to learn some modern homological algebra (Kontsevich's A- infinity categories) and some "forgotten" parts of D- module theory (e.g. the microlocal approach).

Instructor(s): Alexander Beilinson, Vladimir Drinfeld     Terms Offered: Spring