# 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 (anlaysis), MATH 6310–MATH 6320 (algebra), and MATH 6510–MATH 6520 (topology).

## Spring 2019 Offerings

The following schedule is tentative. Locations will added to the university class roster in early December. Descriptions are included below under Course Descriptions.

MATH 6120 - Complex Analysis

Robert Strichartz, MWF 1:25-2:15

MATH 6220 - Applied Functional Analysis

Yury Kudryashov, MWF 9:05-9:55

MATH 6320 - Algebra

Martin Kassabov, MWF 11:15-12:05

MATH 6370 - Algebraic Number Theory

Shankar Sen, MWF 12:20-1:10

MATH 6510 - Algebraic Topology

Jim West, TR 8:40-9:55

MATH 6620 - Riemannian Geometry

Jason Manning, TR 11:40-12:55

MATH 6670 - Algebraic Geometry

Brian Hwang, TR 11:40-12:55

MATH 6720 - Probability Theory II

Philippe Sosoe, TR 10:10-11:25

MATH 6730 - Mathematical Statistics I

MATH 6870 - Set Theory

Justin Moore, MWF 12:20-1:10

MATH 7120 - Topics in Analysis

Gennady Uraltsev, TR 1:25-2:40

MATH 7270 - Topics in Numerical Analysis: Top Ten Algorithms of the 20th Century

Alex Townsend, TR 11:40-12:55

MATH 7370 - Topics in Number Theory: Buildings

Nicolas Templier, MWF 10:10-11:00

MATH 7390 - Topics in Lie Groups and Lie Algebras: Dirac Cohomology and Infinite Dimensional Representations

Dan Barbasch, TR 2:55-4:10

MATH 7520 - Berstein Seminar in Topology: Principal G-bundles and Classifying Spaces

Brian Hwang, TR 10:10-11:25

MATH 7620 - Topics in Geometry: Ricci Flow

Xiaodong Cao, MWF 11:15-12:05

MATH 7670 - Topics in Algebraic Geometry: An Introduction to Toric Varieties

Mike Stillman, TR 10:10-11:25

MATH 7820 - Seminar in Logic

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

## Fall 2018 Offerings

For locations, check the university class roster.

MATH 6110 - Real Analysis

Gennady Uraltsev, TR 8:40-9:55

MATH 6210 - Measure Theory and Lebesgue Integration

Robert Strichartz, TR 10:10-11:25

MATH 6260 - Dynamical Systems

Nataliya Goncharuk, MWF 12:20-1:10

MATH 6310 - Algebra

David Zywina, MWF 1:25-2:15

MATH 6340 - Commutative Algebra with Applications in Algebraic Geometry

Irena Peeva, TR 1:25-2:40

MATH 6390 - Lie Groups and Lie Algebras

Martin Kassabov, MWF 9:05-9:55

MATH 6520 - Differentiable Manifolds

Reyer Sjamaar, MWF 11:15-12:05

MATH 6540 - Homotopy Theory

Inna Zakharevich, TR 11:40-12:55

MATH 6710 - Probability Theory I

Christian Noack, TR 1:25-2:40

MATH 6740 - Mathematical Statistics II

Michael Nussbaum, MWF 1:25-2:15

MATH 6810 - Logic

Richard Shore, TR 10:10-11:25

MATH 7150 - Fourier Analysis

Camil Muscalu, TR 11:40-12:55

MATH 7310 - Topics in Algebra: Generation Questions for Finite Groups

R. Keith Dennis, MWF 10:10-11:00

MATH 7350 - Topics in Homological Algebra: Simplicial Methods in Algebra and Geometry

Yuri Berest, MWF 12:20-1:10

MATH 7410 - Topics in Combinatorics: Geometric and Topological Combinatorics

Ed Swartz, MWF 10:10-11:00

MATH 7510 - Berstein Seminar in Topology: Curves on Surfaces

Jason Manning, TR 8:40-9:55

MATH 7710 - Topics in Probability Theory: Mathematical Statistical Mechanics

Phlippe Sosoe, TR 2:55-4:10

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

Marten Wegkamp, TR 2:55-4:10

MATH 7810 - Seminar in Logic

Anil Nerode, T 2:55-4:10 and W 4:00-5:15

## Course Descriptions

### MATH 6110 - Real Analysis

Fall 2018. 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. MATH 6110 aims to provide a base knowledge of real analysis and integration theory. We will introduce some notions of functional analysis and probability to the extent that they are useful for a command of the main topics.

Main topics:

- Fundamentals of Measure theory and integration
- Lebesgue measure
- Properties of
*L*^{p}spaces - Hilbert and Banach spaces
- Fourier series
- Comparison of measures and differentiation
- Operations on measures: theorems of Fubini and Tonelli

The following additional topics may be included in the course: Change of variables formulae; Notions from probability; The Fourier transform; Interpolation; Basics of theory of distributions; *L*^{p} spaces on topological groups; Dyadic martingales.

### MATH 6120 - Complex Analysis

Spring 2019. 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 (offered alternate years). Next offered 2019-2020. 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 (offered alternate years). Next offered 2019-2020. 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 2018. 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 2019. 4 credits.

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

### MATH 6230 - [Differential Games and Optimal Control]

Fall or Spring. Next offered 2019-2020. 4 credits.

This course is a self-contained introduction to the modern theory of optimal control and differential games. Dynamic programming uses Hamilton-Jacobi partial differential equations (PDEs) to encode the optimal behavior in cooperative and adversarial sequential decision making problems. The same PDEs have an alternative interpretation in the context of front propagation problems. We show how both interpretations are useful in constructing efficient numerical methods. We also consider a wide range of applications, including robotics, computational geometry, path-planning, computer vision, photolithography, economics, seismic imaging, ecology, financial engineering, crowd dynamics, and aircraft collision avoidance. Assumes no prior knowledge of non-linear PDEs or numerical analysis.

### MATH 6260 - Dynamical Systems

Fall 2018. 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)

Fall or Spring. Next offered 2019-2020. 4 credits.

Prerequisite: MAE 6750, MATH 6260 (formerly MATH 6170), 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 2019-2020. 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 2018. 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 2019. 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 6330 - [Noncommutative Algebra]

Fall or Spring. Next offered 2019-2020. 4 credits.

An introduction to basic methods of the theory of noncommutative rings and modules, with applications to some important classes of rings (semi-simple, regular, Noetherian). Topics will include Morita theory, localization, rings of quotients, dimensions (Krull, global, Gelfand-Kirillov), torsion theories, homological algebra.

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

Fall 2018. 4 credits.

Prerequisites: a good background in abstract algebra.

Commutative Algebra is the theory of commutative rings and their modules. We will cover several basic topics: localization, primary decomposition, dimension theory, integral extensions, Hilbert functions and polynomials, free resolutions. The lectures will emphasize the connections between commutative algebra and algebraic geometry.

### MATH 6350 - [Homological Algebra]

Fall or Spring. Next offered 2019-2020. 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 2019. 4 credits.

Prerequisite: an advanced course in abstract algebra at the level of MATH 4340

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), zeta-functions, distribution of primes in arithmetic progressions.

Text: None. But for those who like to see it in print: Samuel's Introduction to Algebraic Number Theory is short and elegant. Also Lang's book on number theory covers most of the material of the course (and a great deal besides).

### MATH 6390 - Lie Groups and Lie Algebras

Fall 2018. 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 (offered alternate years). Next offered 2019-2020. 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 2019. 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 2018. 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. We will study flows of vector fields and prove the Frobenius integrability theorem. In the presence of a Riemannian metric, the notions of parallel transport, curvature, and geodesics are developed. We will examine the tensor calculus and the exterior differential calculus and prove Stokes' theorem. If time permits, de Rham cohomology, Morse theory, or other optional topics will be covered.

This is one of the core courses in the Cornell Ph.D. program in Mathematics, and can be used to satisfy the core course requirement in the program. The course is also suitable for students in many other Cornell Ph.D. programs, such as Physics or the Center for Applied Mathematics.

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

Fall or Spring. Next offered 2019-2020. 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 2018. 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

Spring 2019. 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 2019-2020. 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 or Spring. Next offered 2019-2020. 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

Spring 2019. 4 credits.

Prerequisite: MATH 6310 or MATH 6340, or equivalent.

This class will be a introduction to schemes in algebraic geometry. Originating in the early 1960s, schemes are an elegant generalization of an algebraic variety, interpolating the algebraic, geometric, topological, and arithmetic aspects of the subject. While we cannot hope for an exhaustive treatment of the theory of schemes in one semester, the goal of the course is to motivate and illustrate some of the basic definitions and concepts of the theory of schemes, with an emphasis on examples and classical constructions.

The class is intended to be suitable for those with no prior knowledge of schemes, even the special case of algebraic varieties. It is meant to be accessible to those who have taken a graduate algebra course, but more important than a specific class, you should have a good working knowledge of concepts in commutative algebra such as localization, primary decomposition, and dimension theory. If you would like to take the class but believe you have less background, please do talk to me to see if we can work something out.

While we will not follow a specific textbook, good accessible references for the theory of schemes are:

- The Geometry of Schemes by Eisenbud and Harris
- Algebraic Geometry and Arithmetic Curves by Liu
- Foundations of Algebraic Geometry by Vakil
- Algebraic Geometry by Hartshorne

### MATH 6710 - Probability Theory I

Fall 2018. 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 2019. 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 2019. 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 2018. 4 credits.

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

Some familiarity with basic statistical theory is assumed, i.e. with point estimation, hypothesis testing and confidence intervals, as well as with the concepts of Bayesian and minimax decisions. The course is intended as an introduction to some modern nonparametric and Bayesian methods. The following topics will be treated: (1) a recap of Bayesian estimation, (2) empirical Bayes and shrinkage estimators; (3) unbiased estimation of risk in nonparametric estimation; (4) adaptive estimation, leading up to the study of oracle inequalities, a powerful concept which has also found applications in the related area of classification and machine learning; (5) Bayesian inference: modeling and computation, which will touch upon the use of Markov random fields for image restoration; (6) asymptotic optimality of estimators, discussing the concepts of contiguity and local asymptotic normality of statistical models.

### MATH 6810 - Logic

Fall 2018 (offered alternate years). 4 credits.

This course will be a basic introduction to mathematical logic. The content will, to some extent, depend on the background and interests of the students. As a common starting point, we will describe and develop a formal syntax for mathematical discourse along with precise semantics by defining both the notions of a formula in a given language and a structure for the language. The next step in our analysis is to give precise definitions of proofs and provability and to establish Gödel's Completeness theorem: A sentence is provable (from given axioms) iff it is true in every structure (which satisfies the axioms).

We will next develop some of the basic results of model theory such as the Compactness Theorem: A set *S* of sentences is consistent (i.e. does not prove a contradiction) iff it has a model (a structure in which every one of the sentences is true) iff every finite subset of *S* has a model. (Corollary: If a sentence of the appropriate language is true in all fields of characteristic 0, it is true in all fields of sufficiently large characteristic.) The Skolem-Löwenheim Theorem: If a countable set of sentences *S* has an infinite model then it has a countable one. We will also develop other connections between the forms of axioms and properties of models. Sample: If a sentence is true in one algebraically closed field of characteristic 0, e.g. in the complexes, then it is true in every algebraically closed field of characteristic 0.

The time devoted to these basic topics will depend on the background of the students. Other topics as time permits and student interest dictates will include some of the following:

- A brief development of the basic facts of recursion theory (computability theory) to the point that we can prove the undecidability of the halting problem as well as various mathematical theories especially by Interpreting one theory in another. Tarksi's result on the undefinability of truth. Church's result on the undecidability of of first order logic (validity).
- Decidability or undecidability of some mathematical theories, i.e. algorithms for determining of every sentence if it is a theorem or not.
- Gödel's Incompleteness Theorem: Given any reasonable, consistent theory
*T*containing arithmetic, there is a true sentence of arithmetic which is not provable in*T*(nor, of course, is its negation). - The basics of axiomatic set theory to ordinals and cardinals and their arithmetic.

The text for the course will possibly be Mathematical Logic by Ebbinghaus, Flum and Thomas plus supplementary material.

### MATH 6830 - [Model Theory]

Spring (offered alternate years). Next offered 2019-2020. 4 credits.

Introduction to model theory at the level of the books by Hodges or Chang and Keisler.

### MATH 6840 - [Recursion Theory]

Fall (offered alternate years). Next offered 2019-2020. 4 credits.

Covers theory of effectively computable functions; classification of recursively enumerable sets; degrees of recursive unsolvability; applications to logic; hierarchies; recursive functions of ordinals and higher type objects; generalized recursion theory.

### MATH 6870 - Set Theory

Spring 2019 (offered alternate years). 4 credits.

This course will cover material in combinatorial set theory and forcing, roughly following the outline of Kunen's "Set Theory" Chapters I, II, VI, VII, VIII. Over the course of the semester, we will develop the tools used to prove that the Continuum Hypothesis can neither be proved nor refuted using the axioms of ZFC. The course will start with an introduction to the axioms of ZFC and basic concepts in set theory: ordinals and cardinals, transfinite induction and recursion (Chapter I). It will then study addional axioms, tools and construction in combinatorial set theory: stationary sets, set-theoretic trees, Martin's Axiom, Jensen's diamond principle (Chapter II). We will then cover the fundamentals of the Axiom of Constructability (Chapter VI) and the method of forcing (Chapters VII and VIII). Time permitting, we will choose among the following topics: Solovay's proof that ZF is insufficient to construct a non Lebesgue measurable subset of the real line, Martin's Maximum and its consequences, Todorcevic's method of minimal walks.

### MATH 7110 - [Topics in Analysis]

Fall. Offered on demand. 4 credits.

Selection of advanced topics from analysis. Course content varies.

### MATH 7120 - Topics in Analysis

Spring 2019. 4 credits.

Selection of advanced topics from analysis. Course content varies.

### MATH 7130 - [Functional Analysis]

Fall or Spring (offered alternate years). Next offered 2019-2020. 4 credits.

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

### MATH 7150 - Fourier Analysis

Fall 2018. 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: Top Ten Algorithms of the 20th Century

Spring 2019. 4 credits.

In the January/February 2000 issue of Computing in Science and Engineering, Jack Dongarra and Francis Sullivan selected 10 algorithms with the greatest influence on science, numerical analysis, and engineering in the 20th century. In March 2016, Nick Higham (SIAM president, 2017-2018) presented a slightly revised list. In no particular order, the list is: (1) Newton and quasi-Newton methods; (2) Matrix factorizations (LU, Cholesky, QR); (3) Singular value decomposition, QR and QZ algorithms; (4) Monte-Carlo methods; (5) Fast Fourier transform; (6) Krylov subspace methods; (7) JPEG; (8) PageRank; (9) Simplex method; and (10) Kalman filter.

We will cover the motivations, ideas, history, and future impact of these ten algorithms. Honorable mentions are the fast multipole method, quicksort, and bootstrapping that may be covered if time permits.

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

(also CS 7290)

Fall 2018, Spring 2019. 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: Generation Questions for Finite Groups

Fall 2018. 4 credits.

Prerequisites: Basic group theory, e.g., MATH 4340 or MATH 6310.

About 25 years ago, Persi Diaconis asked the following question: Is it possible to develop a "K-theory of finite groups using finite generating sequences" in a fashion analogous to what is done in the classical situation?

New properties and questions about finite groups that arise from studying various approaches to answering this question will be developed. Natural connections with older work of Tarski, the Neumanns, and others will appear. Many unsolved problems about finite groups, both old and new, appear in a natural way. Diaconis' question seems to generated many interesting problems and gives an interpretation to how some of the current work in finite group theory might fit into a larger picture.

### MATH 7320 - [Topics in Algebra]

Spring. Offered on demand. 4 credits.

Selection of advanced topics from algebra. Course content varies.

### MATH 7350 - Topics in Homological Algebra - Simplicial Methods in Algebra and Geometry

Fall 2018. 4 credits.

This course will be a survey of simplicial techniques in algebra and (algebraic/differential) geometry, both old and new. In the first part, we will review some of the classical topics in simplicial homological algebra (the Dold-Kan correspondence, the Eilenberg-Zilber theorem, canonical resolutions, Barr-Beck (co)triple (co)homology). In the second part, we will turn our attention to specific algebraic categories, such as simplicial groups and simplicial (commutative) algebras. (Topics may include the Kan loop group construction and an introduction to the Andre-Quillen homology). The third (main) part of the course will focus on more recent developments: we will try to give a gentle introduction to simplicial categories and DG categories which are the main technical tools in several areas of modern mathematics (derived algebraic geometry, modern representation theory and noncommutaive geometry).

Depending on the audience preferences, topics may include a survey of \infinity-categories, simplicial presheaves on manifolds, homotopy theory of DG categories and derived Hall algebras. As for prerequisites, basic knowledge of homological algebra (at the level of MATH 6350), algebraic geometry and algebraic topology (at the level of MATH 6510) will be helpful.

### MATH 7370 - Topics in Number Theory: Buildings

Spring 2019. 4 credits.

This course will be an introduction to buildings. Buildings answer the fundamental question of how to represent elements in a group. The first applications were to the study of finite groups of Lie type. Nowadays there is a broad range of applications, from geometric representation theory and number theory, to low-dimensional topology and geometric group theory.

Key concepts include: alcove, appartment, chamber, Coxeter complex, affine and spherical building, maximal/minimal facets, special vertices, BN pairs, Iwahori and parahoric subgroups, Moy-Prasad filtration. One goal will be to prove the Iwahori-Matsumoto decomposition, which is a common generalization of the Cartan decomposition and the Bruhat decomposition. As we shall explain in the course, the theory is analogous to that of Riemannian symmetric spaces in differential geometry.

Prerequisites: Lie groups and Lie algebras (MATH 6390). Also core courses in algebra and algebraic topology (MATH 6310, 6320, 6510), or equivalent.

### MATH 7390 - Topics in Lie Groups and Lie Algebras: Dirac Cohomology and Infinite Dimensional Representations

Spring 2019. 4 credits.

Prerequisite: a one-semester course in Lie groups

The use of the Dirac operator in the representation theory of real reductive groups was initiated by Parthasarathy and Schmid in their study of the Discrete Series. The Dirac inequality plays an important role in the determination of the unitary dual. Dirac cohomology is a refinement introduced by Vogan via a conjecture, which was subsequently proved by Huang and Pandzic.

It has since seen applications to many other situations such as Affine Hecke algebra.

This course will work through the basic notions, including the basics of infinite dimensional representations of real and p-adic group, and present some applications.

### MATH 7410 - Topics in Combinatorics: Geometric and Topological Combinatorics

Fall 2018. 4 credits.

Specific topics will be chosen the first week of the course in consultation with the class.

Past topics have ranged from recent uses of Stanley-Reisner theory, including local cohomology, to a simple introduction and applications of discrete Morse theory. Many other topics are possible, including (but not limited to) matroids, hyperplane arrangements, combinatorial aspects of cell complexes and posets, and polytopes.

Prerequisites will depend on the specific topics chosen. A minimum of a semester of undergraduate topology, algebra and combinatorics will always be necessary. Frequently additional background will be required.

### MATH 7510 - Berstein Seminar in Topology: Curves on Surfaces

Fall 2018. 4 credits.

What can we say about (or using) immersions of 1-manifolds into 2-manifolds? Perhaps surprisingly, quite a lot! Specific directions will be negotiated between the instructor and the participants, but may include:

- The geometry and topology of the curve complex, including applications to understanding the mapping class group of a surface.
- Gabai's proof of the Simple Loop Conjecture for maps between surfaces, and other forms of the Simple Loop Conjecture.
- Calegari's work on stable commutator length.
- Determining minimal intersection and self-intersection of curves on surfaces.

Since this is a Berstein seminar, almost all the presentations will be given by the participants. A student who has taken the first year courses in topology and algebra will be well-prepared for this course.

### MATH 7520 - Berstein Seminar in Topology: Principal G-Bundles and Classifying Spaces

Spring 2019. 4 credits.

This Berstein seminar will be an introduction to the theory of principal G-bundles (where G is a topological group) and classifying spaces. Students will be expected to give presentations during the course.

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

Fall 2018. 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 2019. 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. Offered on demand. 4 credits.

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

### MATH 7580 - [Topics in Topology]

Spring. Offered on demand. 4 credits.

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

### MATH 7610 - [Topics in Geometry]

Fall. Offered on demand. 4 credits.

Selection of advanced topics from modern geometry. Content varies.

### MATH 7620 - Topics in Geometry: Ricci Flow

Spring 2019. 4 credits.

This course will start with a quick introduction of the Ricci flow. Including short time existence, maximal principle, Li-Yau-Hamilton type differential Harnack inequalities and their relations with Perelman's entropies, derivative estimates, Perelman's non-collapsing result and singularity analysis. We will then discuss some recent development of analytic tools and their applications to the Ricci flow.

### MATH 7670 - Topics in Algebraic Geometry: An Introduction to Toric Varieties

Spring 2019. 4 credits.

This course will be an introduction to toric varieties, assuming a first course in algebraic geometry (e.g. Shafarevich, Vol1, or Hartshorne, chapter 1, or Hasset, or Cox-Little-O'Shea), and either basic knowledge of sheaves, divisors, and schemes, or concurrent enrollment in Brian Hwang's MATH 6670.

Toric varieties are wonderful and explicit examples of affine, projective, and even more general varieties. They provide an extremely useful link of algebraic geometry with combinatorics. Toric varieties appear in many places, including in physics.

A tentative list of topics to be covered: (1) Polyhedral cones and polytopes. (2) Affine toric variety corresponding to a cone. (3) Polyhedral Fans. (4) Projective toric varieties, non-affine toric variety corresponding to a fan. (5) The torus action. (6) Singularities, smoothness, resolution of singularities (7) Divisors, properties of divisors (nef, big, ample, very ample), cones: nef cone, effective cone. (8) The homogeneous coordinate ring of a toric variety. (9) Line bundles. (10) Cohomology (topological, of line bundles, of more general vector bundles). (11) Intersection theory.

Throughout, we will see many examples, and we will use the NormalToricVarieties and Polyhedra packages in Macaulay2 to explore examples.

A note about the relationship between this course and Brian Hwang's course on algebraic geometry: We will coordinate our courses, so that after Brian introduces a concept, we will examine that concept in our course. For instance, once he defines a scheme, we will see how to glue affine toric varieties to produce more general toric varieties.

Homework: I will hand out suggested homework problems, (which will NOT be handed in), and we will discuss these in class. Some problems will be simple, making sure you understand the concepts, some will be computational, and some will be more challenging, to allow you to understand the material more deeply. Finally, some might have you explore some of the techniques that we will not have time to cover in class.

Projects: At the end of the semester, students will present short talks on a subject in toric varieties of their interest. I will have lots of possible topics to suggest.

### MATH 7710 - Topics in Probability Theory: Mathematical Statistical Mechanics

Fall 2018. 4 credits.

In this course, several mathematical models of statistical mechanics will be studied, with particular attention given to phase transitions. We will begin by studying percolation and the Ising model, describing the phase transition in both cases, as well as giving qualitative descriptions of the ordered and disordered phases. Time permitting, we will also study further models like the random cluster model.

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

Fall or Spring. Offered on demand. 4 credits.

Selection of advanced topics from stochastic processes. Content varies.

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

Fall 2018. 4 credits.

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

Learning theory has become an important topic in modern statistics. I intend to give an overview of various topics in classification, starting with Stone's (1977) stunning result that there are classifiers that are universally consistent.

Other topics include classification, plug-in methods (k-nearest neighbors), reject option, empirical risk minimization, VC theory, fast rates via Mammen and Tsybakov's margin condition, convex majorizing loss functions and support vector machines, lasso type estimators, low rank multivariate response regression, random matrix theory and current topics in high dimensional statistics.

### MATH 7810 - Seminar in Logic

Fall 2018. 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 7820 - Seminar in Logic

Spring 2019. 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 or Spring. Offered on demand. 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).