# Graduate Courses

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## Overview

Graduate course offerings for the current year are included below along with course descriptions that are in many cases more detailed than those included in the university catalog, especially for topics courses. The core courses in the mathematics graduate program are MATH 6110–MATH 6120 (analysis), MATH 6310–MATH 6320 (algebra), and MATH 6510–MATH 6520 (topology).

## Spring 2020 Offerings

Descriptions are included under Course Descriptions.

MATH 6120 - Complex Analysis

John Hubbard

MATH 6160 - Partial Differential Equations

Tim Healey

MATH 6220 - Applied Functional Analysis

Alex Townsend

MATH 6270 - Applied Dynamical Systems

Steve Strogatz

MATH 6320 - Algebra

Nicolas Templier

MATH 6350 - Homological Algebra

Martin Kassabov

MATH 6370 - Algebraic Number Theory

Shankar Sen

MATH 6410 - Enumerative Combinatorics

Karola Meszaros

MATH 6510 - Algebraic Topology

Jim West

MATH 6720 - Probability Theory II

Lionel Levine

MATH 6730 - Mathematical Statistics I

MATH 6830 - Model Theory

Justin Moore

MATH 7280 - Topics in Dynamical Systems: Asymptotics and Perturbation Methods

Steve Strogatz

MATH 7370 - Topics in Number Theory: Elliptic Curves and Arithmetic Geometry

David Zywina

MATH 7390 - Topics in Lie Groups and Lie Algebras: Representation Theory of p-adic Reductive Groups

Dan Barbasch

MATH 7520 - Berstein Seminar in Topology: Mostow Rigidity

Kathryn Mann

MATH 7580 - Topics in Topology: Orderable Groups and Dynamics

James Hyde

MATH 7620 - Topics in Geometry: Some Topics in Geometry of Four-Manifolds

Xiaodong Cao

MATH 7670 - Topics in Algebraic Geometry: New Methods in Moduli Theory

Daniel Halpern-Leistner

MATH 7710 - Topics in Probability: Random nonlinear partial differential equations

Phil Sosoe

MATH 7820 - Seminar in Logic

Slawomir Solecki, T 2:55-4:10 and W 4:00-5:15

## Fall 2019 Offerings

Descriptions are included under Course Descriptions. For locations, check the university class roster.

MATH 6110 - Real Analysis

Camil Muscalu, TR 10:10-11:25

MATH 6150 - Partial Differential Equtions

Gennady Uraltsev, TR 8:40-9:55

MATH 6210 - Measure Theory and Lebesgue Integration

Robert Strichartz, TR 1:25-2:40

MATH 6230 - Differential Games, Optimal Control, Front Propagation, and Dynamic Programming

Alex Vladimirsky, TR 11:40-12:55

MATH 6310 - Algebra

Marcelo Aguiar, TR 1:25-2:40

MATH 6340 - Commutative Algebra with Applications in Algebraic Geometry

Mike Stillman, MWF 10:10-11:00

MATH 6390 - Lie Groups and Lie Algebras

Birgit Speh, TR 10:10-11:25

MATH 6520 - Differentiable Manifolds

Allen Knutson, TR 11:40-12:55

MATH 6640 - Hyperbolic Geometry

Martin Kassabov, MWF 9:05-9:55

MATH 6670 - Algebraic Geometry

Harrison Chen, MWF 11:15-12:05

MATH 6710 - Probability Theory I

Lionel Levine, MW 8:40-9:55

MATH 6840 - Recursion Theory

Richard Shore, TR 10:10-11:25

MATH 7130 - Functional Analysis

Tim Healey, TR 2:55-4:10

MATH 7310 - Topics in Algebra: Quiver Varieties

Allen Knutson, MWF 12:20-1:10

MATH 7510 - Berstein Seminar in Topology: Algebraic K-Theory

Inna Zakharevich, MWF 10:10-11:00

MATH 7610 - Topics in Geometry: Discrete Geometry, Rigidity of Frameworks, Packing of Disks and Balls

Robert Connelly, MWF 1:25-2:15 (course website)

MATH 7740 - Statistical Learning Theory: Classification, Pattern Recognition, Machine Learning

Marten Wegkamp, TR 2:55-4:10

MATH 7810 - Seminar in Logic: Woodin's Pmax Extension

Justin Moore, T 2:55-4:10 and W 4:00-5:15

MATH 7850 - Topics in Logic

Slawomir Solecki, MWF 12:20-1:10

## Course Descriptions

### MATH 6110 - Real Analysis

Fall 2019. 4 credits.

Forbidden Overlap: Due to an overlap in content, students will not receive credit for both MATH 6110 and MATH 6210.

Prerequisite: Strong performance in an undergraduate analysis course at the level of MATH 4140, or permission of instructor.

MATH 6110-6120 are the core analysis courses in the mathematics graduate program. The plan for MATH 6110 is to use a combination of the three books by Stein and Shakarchi, devoted to Real, Fourier and Functional Analysis. The main topics to be covered usually vary, but traditionally they include:

- Abstract measure and integration theory.
- Differentiation of integrals. Functions of bounded variation. Absolutely continuous functions.
- Fourier series, Fourier transform.
- Hilbert spaces, Banach spaces, aspects of spectral theory.
- Introduction to distribution theory.
- Basic ergodic theory.

### MATH 6120 - Complex Analysis

Spring 2020. 4 credits.

Prerequisite: Strong performance in an undergraduate analysis course at the level of MATH 4140, or permission of instructor.

MATH 6110-6120 are the core analysis courses in the mathematics graduate program. MATH 6120 covers complex analysis, Fourier analysis, and distribution theory.

### MATH 6150 - Partial Differential Equations

Fall 2019 (offered alternate years). 4 credits.

Prerequisite: MATH 4130, MATH 4140, or the equivalent, or permission of instructor.

This course emphasizes the "classical" aspects of partial differential equations (PDEs). The usual topics include fundamental solutions for the Laplace/Poisson, heat and and wave equations in *R ^{n}*, mean-value properties, maximum principles, energy methods, Duhamel's principle, and an introduction to nonlinear first-order equations, including shocks and weak solutions. Additional topics may include Hamilton-Jacobi equations, distribution theory, and the Fourier transform.

### MATH 6160 - Partial Differential Equations

Spring 2020 (offered alternate years). 4 credits.

Prerequisite: MATH 4130, MATH 4140, or the equivalent, or permission of instructor.

This course highlights applications of functional analysis to the theory of partial differential equations (PDEs). It covers parts of the basic theory of linear (elliptic and evolutionary) PDEs, including Sobolev spaces, existence and uniqueness of solutions, interior and boundary regularity, maximum principles, and eigenvalue problems. Additional topics may include: an introduction to variational problems, Hamilton-Jacobi equations, and other modern techniques for non-linear PDEs.

### MATH 6210 - Measure Theory and Lebesgue Integration

Fall 2019. 4 credits.

Forbidden Overlap: Due to an overlap in content, students will not receive credit for both MATH 6110 and MATH 6210.

Covers measure theory, integration, and Lp spaces.

### MATH 6220 - Applied Functional Analysis

Spring 2020. 4 credits.

Covers basic theory of Hilbert and Banach spaces and operations on them. Applications.

### MATH 6230 - Differential Games and Optimal Control

Fall 2019. 4 credits.

A Homicidal Chauffeur tries to run over a Pedestrian. The Chauffeur's speed is much greater, but he is constrained by a minimum turn-radius. Can the Pedestrian survive this encounter? ... in the presence of obstacles? ... until the arrival of Police?

We will explore the theory of non-linear Hamilton-Jacobi PDEs using and comparing two natural perspectives: differential games/control theory and front propagation modeling. We will use the Optimality Principle to establish the properties of viscosity solutions and to build fast numerical methods by determining the direction of information flow. Throughout the course, we will highlight similarities and differences between the continuous and discrete control problems (e.g., an optimal path for a rover on the surface of Mars vs. optimal driving directions from Ithaca to New York City). No prior knowledge of non-linear PDEs or numerical analysis will be assumed. In the numerical part of the course, the emphasis will be on the fundamental ideas rather than on implementation details. The participants' interests will determine a subset of topics to be discussed in detail:

**Hamilton-Jacobi PDEs**: general theory of viscosity solutions; interpretation of characteristics; connections to the calculus of variations; relationship to hyperbolic conservation laws; homogenization; multi-valued solutions; variational inequalities.**Games & Control**: deterministic & stochastic; discrete, continuous, & hybrid; finite & infinite time-horizon vs. exit-time problems; worst-case & on-average optimality; restricted state space and non-autonomous controls; robust, risk-sensitive & likelihood-maximizing controls; Pareto-optimal controls; pursuer-evader games; surveillance-evasion games; pusher-chooser games; tug-of-war games; mean field games; optimal behavior in uncertain environments.**Front propagation**: Legendre transform; Wulff shapes; (anisotropic) Huygens' principle; geometric optics; motion by mean curvature and degenerate ellipticity.**Numerical Approaches**: Lagrangian, semi-Lagrangian, and Eulerian discretizations; controlled Markov chains; iterative and causal (non-iterative) methods; level set methods; model reductions and approximate dynamic programming.**Applications**: robotics, computational geometry, path-planning, image segmentation, shape-from-shading, seismic imaging, photolithography, crystal growth, financial engineering, crowd dynamics, aircraft collision avoidance.

### MATH 6260 - [Dynamical Systems]

Fall. Next offered: 2020-2021. 4 credits.

Topics include existence and uniqueness theorems for ODEs; Poincaré-Bendixon theorem and global properties of two dimensional flows; limit sets, nonwandering sets, chain recurrence, pseudo-orbits and structural stability; linearization at equilibrium points: stable manifold theorem and the Hartman-Grobman theorem; and generic properties: transversality theorem and the Kupka-Smale theorem. Examples include expanding maps and Anosov diffeomorphisms; hyperbolicity: the horseshoe and the Birkhoff-Smale theorem on transversal homoclinic orbits; rotation numbers; Herman's theorem; and characterization of structurally stable systems.

### MATH 6270 - Applied Dynamical Systems

(also MAE 7760)

Spring 2020. 4 credits.

Prerequisite: MAE 6750, MATH 6260, or equivalent.

Topics include review of planar (single-degree-of-freedom) systems; local and global analysis; structural stability and bifurcations in planar systems; center manifolds and normal forms; the averaging theorem and perturbation methods; Melnikov’s method; discrete dynamical systems, maps and difference equations, homoclinic and heteroclinic motions, the Smale Horseshoe and other complex invariant sets; global bifurcations, strange attractors, and chaos in free and forced oscillator equations; and applications to problems in solid and fluid mechanics.

### MATH 6280 - [Complex Dynamical Systems]

Fall or Spring. Next offered 2020-2021. 4 credits.

Prerequisite: MATH 4180.

Various topics in the dynamics of analytic mappings in one complex variable, such as: Julia and Fatou sets, the Mandelbrot set, Mañé-Sad-Sullivan's theorem on structural stability. Also covers: local theory, including repulsive cycles and the Yoccoz inequality, parabolic points and Ecalle-Voronin invariants, Siegel disks and Yoccoz's proof of the Siegel Brjuno theorem; quasi-conformal mappings and surgery: Sullivan's theorem on non-wandering domains, polynomial-like mappings and renormalization, Shishikura's construction of Hermann rings; puzzles, tableaux and local connectivity problems; and Thurston's topological characterization of rational functions, the spider algorithm, and mating of polynomials.

### MATH 6310 - Algebra

Fall 2019. 4 credits.

Prerequisite: strong performance in an undergraduate abstract algebra course at the level of MATH 4340, or permission of instructor.

MATH 6310-6320 are the core algebra courses in the mathematics graduate program. MATH 6310 covers group theory, especially finite groups; rings and modules; ideal theory in commutative rings; arithmetic and factorization in principal ideal domains and unique factorization domains; introduction to field theory; tensor products and multilinear algebra. (Optional topic: introduction to affine algebraic geometry.)

### MATH 6320 - Algebra

Spring 2020. 4 credits.

Prerequisite: MATH 6310, or permission of instructor.

MATH 6310-6320 are the core algebra courses in the mathematics graduate program. MATH 6320 covers Galois theory, representation theory of finite groups, introduction to homological algebra.

### MATH 6340 - Commutative Algebra with Applications in Algebraic Geometry

Fall 2019. 4 credits.

We will cover the basic concepts of commutative algebra that are useful in algebraic geometry.

Prerequisites: We assume knowledge of algebra from math 6310, including: modules, Noetherian rings, field extensions, UFD's, Hilbert nullstellensatz, and basic notions in homological algebra.

Algebraic geometry (and consequently commutative algebra) is a field with a rich set of examples. We will use Macaulay2 to explore such examples.

A tentative list of topics:

- Quick resume of Groebner bases and Macaulay2
- Localization and local rings
- Primary Decomposition
- Tensor products and Hom (including flatness, Tor, Ext))
- Graded rings (normal cones, blowups, Artin-Rees lemma)
- Finite and integral extensions, normal rings
- Dimension Theory (including Noether normalization, Hilbert polynomials, Krull's principal ideal theorem)
- Syzygies and free resolutions (including depth, Koszul complex, Cohen-Macaulay rings)

### MATH 6350 - Homological Algebra

Spring 2020. 4 credits.

Prerequisite: MATH 6310

A first course on homological algebra. Topics will include a brief introduction to categories and functors, chain and cochain complexes, operations on complexes, (co)homology, standard resolutions (injective, projective, flat), classical derived functors, Tor and Ext, Yoneda’s interpretation of Ext, homological dimension, rings of small dimensions, introduction to group cohomology.

### MATH 6370 - Algebraic Number Theory

Spring 2020. 4 credits.

Prerequisites: Math 434 or equivalent.

Note: The default description is given below, but as it is usually a small class the topics may be modified a bit to suit the backgrounds of those attending. In the catalogue 'distribution of primes' is mentioned. But as that is primarily an analytic topic it won't be covered except perhaps for brief discussions of Dirichlet's theorem on primes in arithmetic progressions and Cebotarev's density theorem.

Topics: This course is a basic introduction to algebraic number theory. The core of it deals with the ideal theory of Dedekind domains as applied to the rings of integers of number fields (finite extensions of Q). A major purpose of the theory is to overcome the lack of unique factorisation into primes in these rings.

The course will also cover the fundamental finiteness theorems: the finiteness of the ideal class group (via Minkowski's geometric theory of numbers), and the structure (finite generation, determination of the rank etc.) of the unit group. Additional topics which will be discussed if time permits: law of quadratic reciprocity, elementary Diophantine equations, completions (p-adic numbers).

### MATH 6390 - Lie Groups and Lie Algebras

Fall 2019. 4 credits.

Prerequisite: an advanced course in linear algebra at the level of MATH 4310 and a course in differentiable manifolds.

The course is an introduction to Lie groups and Lie algebras and covers the basics of Lie groups and Lie algebras. Topics include real and complex Lie groups, relations between Lie groups and Lie algebras, exponential map, homogeneous manifolds and the classification of simple Lie algebras.

### MATH 6410 - Enumerative Combinatorics

Spring 2020 (offered alternate years). 4 credits.

Prerequisite: MATH 4410 or permission of instructor.

An introduction to enumerative combinatorics from an algebraic, geometric and topological point of view. Topics include, but are not limited to, permutation statistics, partitions, generating functions, various types of posets and lattices (distributive, geometric, and Eulerian), Möbius inversion, face numbers, shellability, and relations to the Stanley-Reisner ring.

### MATH 6510 - Algebraic Topology

Spring 2020. 4 credits.

Prerequisite: strong performance in an undergraduate abstract algebra course at the level of MATH 4340 and point-set topology at the level of MATH 4530, or permission of instructor.

MATH 6510–MATH 6520 are the core topology courses in the mathematics graduate program. MATH 6510 is an introductory study of certain geometric processes for associating algebraic objects such as groups to topological spaces. The most important of these are homology groups and homotopy groups, especially the first homotopy group or fundamental group, with the related notions of covering spaces and group actions. The development of homology theory focuses on verification of the Eilenberg-Steenrod axioms and on effective methods of calculation such as simplicial and cellular homology and Mayer-Vietoris sequences. If time permits, the cohomology ring of a space may be introduced.

### MATH 6520 - Differentiable Manifolds

Fall 2019. 4 credits.

Prerequisite: strong performance in analysis (e.g., MATH 4130 and/or MATH 4140), linear algebra (e.g., MATH 4310), and point-set topology (e.g., MATH 4530), or permission of instructor.

MATH 6510-MATH 6520 are the core topology courses in the mathematics graduate program. MATH 6520 is an introduction to geometry and topology from a differentiable viewpoint, suitable for beginning graduate students. The objects of study are manifolds and differentiable maps. The collection of all tangent vectors to a manifold forms the tangent bundle and a section of the tangent bundle is a vector field. Alternatively vector fields can be viewed as first-order differential operators. This course studies flows of vector fields and prove the Frobenius integrability theorem. In the presence of a Riemannian metric, develops the notions of parallel transport, curvature and geodesics. Examines the tensor calculus and the exterior differential calculus and prove Stokes' theorem. If time permits, de Rham cohomology, Morse theory, or other optional topics are introduced.

### MATH 6530 - [K-Theory and Characteristic Classes]

Fall or Spring. Next offered 2020-2021. 4 credits.

Prerequisite: MATH 6510, or permission of instructor.

An introduction to topological K-theory and characteristic classes. Topological K-theory is a generalized cohomology theory which is surprisingly simple and useful for computation while still containing enough structure for proving interesting results. The class will begin with the definition of K-theory, Chern classes, and the Chern character. Additional topics may include the Hopf invariant 1 problem, the J-homomorphism, Stiefel-Whitney classes and Pontrjagin classes, cobordism groups and the construction of exotic spheres, and the Atiyah-Singer Index Theorem.

### MATH 6540 - [Homotopy Theory]

Fall or Spring. Next offered: 2021-2022. 4 credits.

Prerequisite: MATH 6510 or permission of instructor.

This course is an introduction to some of the fundamentals of homotopy theory. Homotopy theory studies spaces up to homotopy equivalence, not just up to homeomorphism. This allows for a variety of algebraic techniques which are not available when working up to homeomorphism. This class studies the fundamentals and tools of homotopy theory past homology and cohomology. Topics may include computations of higher homotopy groups, simplicial sets, model categories, spectral sequences, and rational homotopy theory.

### MATH 6620 - [Riemannian Geometry]

Fall or Spring. Next offered: 2020-2021. 4 credits.

Topics include linear connections, Riemannian metrics and parallel translation; covariant differentiation and curvature tensors; the exponential map, the Gauss Lemma and completeness of the metric; isometries and space forms, Jacobi fields and the theorem of Cartan-Hadamard; the first and second variation formulas; the index form of Morse and the theorem of Bonnet-Myers; the Rauch, Hessian, and Laplacian comparison theorems; the Morse index theorem; the conjugate and cut loci; and submanifolds and the Second Fundamental form.

### MATH 6630 - [Symplectic Geometry]

Fall or Spring. Next offered 2020-2021. 4 credits.

Prerequisite: MATH 6510 and MATH 6520, or permission of instructor.

Symplectic geometry is a branch of differential geometry which studies manifolds endowed with a nondegenerate closed 2-form. The field originated as the mathematics of classical (Hamiltonian) mechanics and it has connections to (at least!) complex geometry, algebraic geometry, representation theory, and mathematical physics. In this introduction to symplectic geometry, the class will begin with linear symplectic geometry, discuss canonical local forms (Darboux-type theorems), and examine related geometric structures including almost complex structures and Kähler metrics. Further topics may include symplectic and Hamiltonian group actions, the orbit method, the topology and geometry of momentum maps, toric symplectic manifolds, Hamiltonian dynamics, symplectomorphism groups, and symplectic embedding problems.

### MATH 6640 - Hyperbolic Geometry

Fall 2019. 4 credits.

Prerequisite: MATH 6510 or permission of instructor.

An introduction to the topology and geometry of hyperbolic manifolds. The class will begin with the geometry of hyperbolic $n$-space, including the upper half-space, Poincaré disc, and Lorentzian models. Particular attention will be paid to the cases $n=2$ and $n=3$. Hyperbolic structures on surfaces will be parametrized using Teichmüller space, and discrete groups of isometries of hyperbolic space will be discussed. Other possible topics include the topology of hyperbolic manifolds and orbifolds; Mostow rigidity; hyperbolic Dehn filling; deformation theory of Kleinian groups; complex and quaternionic hyperbolic geometry; and convex projective structures on manifolds.

### MATH 6670 - Algebraic Geometry

Fall 2019. 4 credits.

Prerequisite: MATH 6310 or MATH 6340, or equivalent.

A continuation of the previous semester of algebraic geometry. Vector bundles, cohomology, derived categories, Cech resolutions. Smoothness, flatness, base change, projection formulas. Potential further topics include application to curves and surfaces, Grothendieck-Riemann-Roch, intersection theory, equivariant sheaves and Borel-Weil-Bott.

### MATH 6710 - Probability Theory I

Fall 2019. 4 credits.

Prerequisite: knowledge of Lebesgue integration theory, at least on the real line. Students can learn this material by taking parts of MATH 4130-4140 or MATH 6210.

A mathematically rigorous development of probability theory from a measure-theoretic perspective. Topics include: independence, the law of large numbers, Poisson and central limit theorems, and random walks.

### MATH 6720 - Probability Theory II

Spring 2020. 4 credits.

Prerequisite: MATH 6710.

Conditional expectation, martingales, Brownian motion. Other topics such as Markov chains, ergodic theory, and stochastic calculus depending on time and interests of the instructor.

### MATH 6730 - Mathematical Statistics I

(also STSCI 6730)

Spring 2020. credits.

Prerequisite: STSCI 4090/BTRY 4090, MATH 6710, or permission of instructor.

This course will focus on the finite sample theory of statistical inference, emphasizing estimation, hypothesis testing, and confidence intervals. Specific topics include: uniformly minimum variance unbiased estimators, minimum risk equivariant estimators, Bayes estimators, minimax estimators, the Neyman-Pearson theory of hypothesis testing, and the construction of optimal invariant tests.

### MATH 6740 - [Mathematical Statistics II]

(also STSCI 6740)

Fall. Next offered: 2020-2021. 4 credits.

Prerequisite: MATH 6710 (measure theoretic probability) and STSCI 6730/MATH 6730, or permission of instructor.

Focuses on the modern theory of statistical inference, with an emphasis on asymptotic methods. Topics include M-estimation, related exponential inequalities, U-statistic theory as well as nonparametric density and regression estimation. Asymptotic optimality is discussed in relation to the concepts of contiguity and local approximation by normal models. Optional topics may include resampling methods, Bayesian inference, and classification.

### MATH 6810 - [Logic]

Fall or Spring (offered alternate years). Next offered: 2020-2021. 4 credits.

Covers basic topics in mathematical logic, including propositional and predicate calculus; formal number theory and recursive functions; completeness and incompleteness theorems, compactness and Skolem-Loewenheim theorems. Other topics as time permits.

### MATH 6830 - Model Theory

Spring 2020 (offered alternate years). 4 credits.

This course will give an introduction to model theory from an algebraic and geometric perspective. It will be based on Chapters 2-5 of David Marker's text. Topic covered will include: Henkin's construction, Löwenheim-Skolem theorems, Vaught's criteria for the completeness of a theory, back and forth arguments, quantifier elimination and its relationship to algebraic geometry, types, saturated and homogeneous models, and indiscernibles. The course will culminate with the Baldwin-Lachlan proof of Morley's Categoricity Theorem. Students are expected to have some familiarity with first order logic and also with undergraduate algebra (groups, rings, and fields).

### MATH 6840 - Recursion Theory

Fall 2019 (offered alternate years). 4 credits.

MATH 6840 will be a first course in the theory of computability. We will assume some background in logic. MATH 6810 or CS 6820 should be more than sufficient. The pace and content of the course will depend on the background of the students. Plausible outlines are as follows:

We will begin with a brief discussion of the basic properties of a reasonable model of computability: universal machines, the enumeration, s-m-n and recursion theorems, r.e. (effectively or computably enumerable) sets and the halting problem. Next will come the notions of relative computability, the Turing jump operator and the arithmetical hierarchy. Then there will be some development of construction procedures for non-r.e. sets, in particular, the Kleene-Post finite extension method (really Cohen forcing in arithmetic). An example or two of other forcing type constructions such as with trees (perfect set forcing) to construct a minimal degree may also be presented later.

At this point there are two likely scenarios.

One will concentrate on the recursively (computably) enumerable sets and degrees. The primary text will then be some version of Recursively Enumerable Sets and Degrees by R. I. Soare. The heart of the course will be the development of various kinds of priority arguments for the construction of r.e. sets including finite and infinite injury as well as tree arguments. We will use these methods to analyze the structure of the (Turing) degrees of r.e. sets and something of their set theoretic structure as well.

The second scenario will instead study the structure of the Turing degrees of all sets and functions as well as important substructures such as the degrees below 0' (the Halting problem) and the degrees of the arithmetic sets (those definable in first order arithmetic or equivalently computable form some finite iteration of the Turing jump). The primary techniques will be forcing arguments in the setting of arithmetic rather than set theory. We begin with the development of construction procedures such as the Kleene-Post finite extension method (now seen as Cohen forcing in arithmetic) and minimal degree constrictions by forcing with trees (perfect set forcing), forcing with Pi-0-1 classes (closed sets) and others.

Relations with rates of growth and the jump hierarchy will be explored. We will prove the basic results about the complexity of theories of these structures such as that the theories of the degrees and the degrees below 0' are of the same complexity as second order arithmetic and first order arithmetic, respectively. We will also study the restrictions on possible automorphisms of the structures and definability results: which apparently external (but natural) relations on the structures can be defined internally. In particular, we may reach the proof that the Turing jump which captures quantification in arithmetic is definable in terms of relative computability alone.

In either case, connections between degree theoretic and other properties such as types of approximations, rates of growth and complexity of definition will be considered.

I will also make available notes covering some of the material.

### MATH 6870 - [Set Theory]

Spring (offered alternate years). Next offered: 2020-2021. 4 credits.

First course in axiomatic set theory at the level of the book by Kunen.

### MATH 7110 - [Topics in Analysis]

Fall. Next offered: 2020-2021. 4 credits.

In this course, we will cover some important topics in geometric analysis.

### MATH 7120 - [Topics in Analysis]

Spring. Next offered: 2020-2021. 4 credits.

Selection of advanced topics from analysis. Course content varies.

### MATH 7130 - Functional Analysis

Fall 2019 (offered alternate years). 4 credits.

Covers topological vector spaces, Banach and Hilbert spaces, and Banach algebras. Additional topics selected by instructor.

### MATH 7150 - Fourier Analysis

Fall (offered alternate years). Next offered: 2020-2021. 4 credits.

Offered alternate years.

An introduction to (mostly Euclidean) harmonic analysis. Topics usually include convergence of Fourier series, harmonic functions and their conjugates, Hilbert transform, Calderon-Zygmund theory, Littlewood-Paley theory, pseudo-differential operators, restriction theory of the Fourier transform, connections to PDE. Applications to number theory and/or probability theory may also be discussed, as well as Fourier analysis on groups.

### MATH 7270 - [Topics in Numerical Analysis]

Fall or Spring. Next offered: 2020-2021. 4 credits.

Selection of advanced topics from numerical analysis. Content varies.

### MATH 7280 - Topics in Dynamical Systems: Asymptotics and Perturbation Methods

Spring 2020. 4 credits.

An introduction to asymptotics and perturbation methods, using the textbook by Bender and Orzsag. The course will be taught in an applied style, emphasizing examples, methods, and intuition as opposed to theorems and proofs. We will learn a lot of clever, useful techniques for finding approximate formulas for integrals and solutions to ordinary differential equations, by exploiting the presence of a small or large parameter in the problem. A few applications to PDEs will be given at the end, but the main applications are to integrals and ODEs.

Topics: Asymptotic expansion of integrals. Perturbation methods for algebraic equations. Perturbation methods for differential equations: dominant balance; boundary layer theory; WKB theory; and multiple scales.

There won't be any assigned homework or exams, and the audience is graduate students.

Prerequisites: Familiarity with complex analysis and ordinary differential equations.

### MATH 7290 - Seminar on Scientific Computing and Numerics

(also CS 7290)

Fall 2019, Spring 2020. 1 credits.

Talks on various methods in scientific computing, the analysis of their convergence properties and computational efficiency, and their adaptation to specific applications.

### MATH 7310 - Topics in Algebra: Quiver Varieties

Fall 2019. 4 credits.

"Geometric representation theory" is misnamed; it is a collection of examples of groups acting on the homology of various varieties. One of the two best examples of such actions is Nakajima's 1994 action of a Lie group on the top homology groups of "quiver varieties", where the Lie group and the varieties are both defined from a (rather generalized) Dynkin diagram. This prompts the question of what should act on the total homology or K-theory of quiver varieties, and it turns out to be the corresponding "quantized loop algebra".

Such representations had already appeared in statistical mechanics, where they provide tools for "completely integrating" some interesting stat mech problems. The connection of these "R-matrices" to the quiver variety geometry was spelled out in 2012 by Maulik and Okounkov. In the last couple of years Zinn-Justin and I have been using these tools to solve old problems in the cohomology rings of Grassmannians and other flag manifolds, a circle of problems called "Schubert calculus".

All of this will be built from the ground up. I will assume basic knowledge of homology and cohomology (though equivariant cohomology and K-theory would be nice too), and some exposure to Lie groups and their representations. Some results in algebraic geometry will be taken on faith.

### MATH 7350 - [Topics in Homological Algebra]

Fall or Spring. Next offered: 2020-2021. 4 credits.

Selection of advanced topics from algebra, algebraic number theory, and algebraic geometry. Content varies.

### MATH 7370 - Topics in Number Theory: Elliptic Curves and Arithmetic Geometry

Spring 2020. 4 credits.

An introduction to the arithmetic of elliptic curves. Elliptic curves are smooth projective curves of genus 1 with a fixed point. Despite their simple definition, they hold a distinguished role in modern number theory. After covering the basic theory and working over different fields (complex numbers, local fields, finite fields, number fields), we will study some open conjectures and advanced topics. The advanced topics, which will depend on the interests of the class, may include: modular curves, the theory of complex multiplication, Sato-Tate, Serre's open image theorem.

The course will assume some familiarity with the basic notions of algebraic geometry, Galois theory, and number theory.

### MATH 7390 - Topics in Lie Groups and Lie Algebras: Representation Theory of p-adic Reductive Groups

Spring 2020. 4 credits.

Prerequisite: a one-semester course in Lie groups

We will cover the basics of the theory of infinite dimensional representations of p-adic groups as detailed in the notes of Casselman and Bernstein. Other topics will include the connections to the representation theory of Iwahori-Hecke algebras, unitarity and character theory.

### MATH 7410 - [Topics in Combinatorics]

Fall (offered alternate years). Next offered: 2020-2021. 4 credits.

Selection of advanced topics in combinatorics. Course content varies.

### MATH 7510 - Berstein Seminar in Topology

Fall 2019. 4 credits.

This seminar is an introduction to algebraic K-theory from various perspectives, including geometry, topology, number theory, and category theory. The seminar will be student-led, with students taking turns presenting on a selection of papers investigating the origins, applications, and technical aspects of the field.

### MATH 7520 - Berstein Seminar in Topology: Mostow Rigidity

Spring 2020. 4 credits.

A seminar on an advanced topic in topology or a related subject. Content varies. The format is usually that the participants take turns to present.

This course will showcase several proofs of, and consequences of, what I consider one of the most influential and beautiful theorems of the past century - Mostow's strong rigidity. One formulation of this theorem is "Suppose you have two compact manifolds of dimension at least 3, each with a metric of constant curvature –1. If their fundamental groups are isomorphic, then the manifolds are isometric." In other words, the fundamental group completely determines the geometry of such a manifold: in principle, one should be able to read off all geometric invariants of M (diameter, volume, length of longest closed geodesic, etc.) from a presentation for its fundamental group (!)

Mostow rigidity also has a purely algebraic statement, in terms of lattices in Lie groups, and many more general formulations than the one given above. It was a precursor and inspiration for many ideas in geometric group theory, homogeneous dynamics, and several areas of geometric topology. We will start with little assumed background (some knowledge of hyperbolic geometry is useful, but not strictly required if you’re willing to catch up quick) and aim to understand a few different proofs of Mostow’s theorem and their consequences.

### MATH 7550 - Topology and Geometric Group Theory Seminar

Fall 2019. 4 credits.

A weekly seminar in which visiting or local speakers present results in topology, geometric group theory, or related subjects.

### MATH 7560 - Topology and Geometric Group Theory Seminar

Spring 2020. 4 credits.

A weekly seminar in which visiting or local speakers present results in topology, geometric group theory, or related subjects.

### MATH 7570 - [Topics in Topology]

Fall. Next offered: 2021-2022. 4 credits.

Selection of advanced topics from modern algebraic, differential, and geometric topology. Content varies.

### MATH 7580 - Topics in Topology: Orderable Groups and Dynamics

Spring 2020. 4 credits.

Prerequisites: A solid foundation in undergraduate algebra and analysis.

We will go through the monograph "Groups Orders and Dynamics" by B. Deroin, A. Navas and C. Rivas. After developing the basics of the theory of left-orderable groups we will go through some examples including braid groups and Thompson's group F and then move on to topics such as amenability and random walks on left-orderable groups. Depending on time and the interests of the participants we may cover some other areas of geometric group theory or topology.

### MATH 7610 - Topics in Geometry: Discrete Geometry, Rigidity of Frameworks, Packing of Disks and Balls

Fall 2019. 4 credits. (course website)

The rigidity of a framework consisting of a finite set of points, where some pairs of the points are constrained to be at a fixed distance, is pervasive in all corners of geometry, combinatorics and algebra. We will cover the basics starting with configurations in the plane, infinitesimal rigidity which applies linear algebra to the rigidity problem, and a tiny bit of algebraic geometry that can be applied to global rigidity, and quadratic forms which can be applied to the rigidity of frameworks in all higher dimensions. A very pleasant application of this theory is to the study of packings of disks in the plane. For example, a very recent result on "sticky disks" in the plane is very elementary and seems to bypass even some of the beginning results in combinatorial rigidity. (See arXiv:1809.02006v2 on the Math ArXiv) This is motivated by a question in the theory of granular materials about circle or sphere packings with random radii. (The three-dimensional version of this question is still unknown.) It is quite possible to get to the forefront of this very exciting theory with only a good knowledge of linear algebra and basic mathematics.

### MATH 7620 - Topics in Geometry: Some Topics in Geometry of Four-Manifolds

Spring 2020. 4 credits.

In this course, we will update some recent progress in the direction of geometry of four-manifolds. In particular, Einstein 4-manifolds, Ricci flow on 4-manifolds. We will also discuss some classical techniques such as Bochner technique, conformal geometry and their application to four-manifolds.

### MATH 7670 - Topics in Algebraic Geometry: New Methods in Moduli Theory

Spring 2020. 4 credits.

In algebraic geometry, the phrase "moduli problem" refers to the problem of studying how geometric objects vary in families. We will discuss the modern framework for studying moduli problems based on algebraic stacks. First we will introduce the basic definitions and examples of algebraic stacks, and discuss the main results of the theory (Artin's criteria, etc.). Then we will discuss some recent results on the structure of algebraic stacks: local quotient presentations, and the construction of moduli spaces and canonical stratifications. The main example of interest will be the moduli of principal G-bundles on a smooth projective curve. Our goal is to develop enough structure on this stack to prove the Teleman-Woodward index formula (a generalization of the Verlinde formula) for K-theory classes on the stack of G-bundles.

### MATH 7710 - Topics in Probability Theory: Random nonlinear partial differential equations

Spring 2020. 4 credits.

We present several instances of the use of probabilistic techniques in the study of nonlinear partial differential equations.

Invariant measures for discrete and continuum nonlinear Schroedinger, and improved well-posedness results for wave equations following work of Burq, Tzvetkov and others. If time permits, we will discuss singular PDEs with stochastic forcing, like the KPZ equation.

We will assume the audience has taken a graduate PDE course like MATH 6510. Familiarity with probability at the level of MATH 6710 and 6720 is helpful but not strictly required.

### MATH 7720 - [Topics in Stochastic Processes]

Fall or Spring. Next offered: 2020-2021. 4 credits.

Selection of advanced topics from stochastic processes. Content varies.

### MATH 7740 - Statistical Learning Theory: Classification, Pattern Recognition, Machine Learning

Fall 2019. 4 credits.

Prerequisite: basic mathematical statistics (MATH 6730 or equivalent) and measure theoretic probability (MATH 6710).

The course aims to present the developing interface between machine learning theory and statistics. Topics include an introduction to classification and pattern recognition; the connection to nonparametric regression is emphasized throughout. Some classical statistical methodology is reviewed, like discriminant analysis and logistic regression, as well as the notion of perception which played a key role in the development of machine learning theory. The empirical risk minimization principle is introduced, as well as its justification by Vapnik-Chervonenkis bounds. In addition, convex majoring loss functions and margin conditions that ensure fast rates and computable algorithms are discussed. Today's active high-dimensional statistical research topics such as oracle inequalities in the context of model selection and aggregation, lasso-type estimators, low rank regression and other types of estimation problems of sparse objects in high-dimensional spaces are presented.

### MATH 7810 - Seminar in Logic: Woodin's Pmax Extension

Fall 2019. 4 credits.

In the 1980s, Shelah and Woodin proved that if there is a supercompact cardinal, then every set of reals which can be defined using real and ordinal parameters is Lebesgue measurable. Moreover, the inner model L(R) first considered by Solovay satisfies ZF together with Dependent Choice, all sets of reals are Lebesgue measurable, have the property of Baire, and the perfect set property. Moreover they proved, again from the assumption of a supercompact cardinal, that the theory of L(R) can not be changed by forcing.

In subsequent work, Woodin isolated a forcing extension - the Pmax extension - of L(R) which satisfies ZFC as well as many strong combinatorial statements about the first uncountable cardinal. In this seminar we will present Woodin's Pmax forcing construction and analyze the many striking properties of the generic extension. Students who enroll in the course will be expected to present lectures on the material in the seminar. Students should be familiar with forcing as it is presented in, e.g., Kunen's text and have some knowledge of large cardinals. The spring 2019 offering of MATH 6870 will be adequate preparation for the course. The seminar will also feature several talks on other topics in logic, including talks by outside speakers.

### MATH 7820 - Seminar in Logic

Spring 2020. 4 credits.

A twice weekly seminar in logic. Typically, a topic is selected for each semester, and at least half of the meetings of the course are devoted to this topic with presentations primarily by students. Opportunities are also provided for students and others to present their own work and other topics of interest.

### MATH 7850 - Topics in Logic

Fall 2019. 4 credits.

Covers topics in mathematical logic which vary from year to year, such as descriptive set theory or proof theory. May also be used to further develop material from model theory (MATH 6830), recursion theory (MATH 6840), or set theory (MATH 6870).