## Overview

Graduate course offerings for the coming 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).

## Fall 2024 Offerings

Descriptions are included below under Course Descriptions.

MATH 5080 - Special Study for Teachers

Mary Ann Huntley

MATH 5220 - Applied Complex Analysis

Steven Strogatz, TR 10:10-11:25

MATH 5250 - Numerical Analysis and Differential Equations

Yunan Yang, TR 2:55-4:10

MATH 5410 - Introduction to Combinatorics I

Karola Meszaros, TR 2:55-4:10

MATH 6110 - Real Analysis

Camil Muscalu, MW 10:10-11:25 + dis F 10:10-11:00

MATH 6210 - Measure Theory and Lebesgue Integration

Instructor TBD, MW 10:10-11:25

MATH 6230 - Differential Games and Optimal Control

Alex Vladimirsky, TR 10:10-11:25

MATH 6260 - Dynamical Systems

John Hubbard, TR 1:25-2:40

MATH 6310 - Algebra

Allen Knutson, TR 11:40-12:55 + dis W 3:35-4:25

MATH 6390 - Lie Groups and Lie Algebras

Dan Barbasch, MW 11:40-12:55

MATH 6520 - Differentiable Manifolds

James West, TR 8:40-9:55 + dis F 11:15-12:05

MATH 6710 - Probability Theory I

Phil Sosoe, TR 8:40-9:55

MATH 6740 - Mathematical Statistics II

Michael Nussbaum, TR 11:40-12:55

MATH 6870 - Descriptive Set Theory

Slawomir Solecki, MWF 12:20-1:10

MATH 7110 - Topics in Analysis: De Girogi-Nash-Moser Theory

Xin Zhou, TR 10:10-11:25

MATH 7510 - Berstein Seminar in Topology

Moon Duchin, TR 1:25-2:40

MATH 7670 - Topics in Algebraic Geometry: Schubert Varieties and Degenerations

Allen Knutson, TR 2:55-4:10

MATH 7710 - Topics in Probability Theory: Math for AI Safety

Lionel Levine, MW 11:40-12:55

MATH 7740 - Statistical Learning Theory

Marten Wegkamp, MW 1:25-2:40

## Spring 2025 Offerings

Descriptions are included below under Course Descriptions.

MATH 5080 - Special Study for Teachers

Mary Ann Huntley

MATH 5200 - Differential Equations and Dynamical Systems

John Hubbard, TR 1:25-2:40

MATH 6120 - Complex Analysis

Yusheng Luo, MW 10:10-11:25 + dis F 10:10-11:00

MATH 6220 - Applied Functional Analysis

Yunan Yang, MW 10:10-11:25

MATH 6302 - Lattices: Geometry, Cryptography, and Algorithms

Noah Stephens-Davidowitz, TR 1:25-2:40

MATH 6320 - Algebra

Martin Kassabov, MW 8:40-9:55 + dis F 9:05-9:55

MATH 6370 - Algebraic Number Theory

David Zywina, TR 10:10-11:25

MATH 6510 - Algebraic Topology

Inna Zakharevich, TR 11:40-12:55 + dis M 12:20-1:10

MATH 6540 - Homotopy Theory

Yuri Berest, MF 1:25-2:40

MATH 6620 - Riemannian Geometry

Jason Manning, TR 1:25-2:40

MATH 6670 - Algebraic Geometry

Dan Halpern-Leistner, MW 11:40-12:55

MATH 6720 - Probability Theory II

Lionel Levine, MW 11:40-12:55

MATH 6730 - Mathematical Statistics I

Florentina Bunea, MW 2:55-4:10

MATH 6810 - Logic

Mark Poór, TR 1:25-2:40

MATH 7150 - Fourier Analysis

Camil Muscalu, TR 2:55-4:10

MATH 7160 - Topics in Partial Differential Equations: Spectral Geometry

Daniel Stern, TR 10:10-11:25

MATH 7310 - Topics in Algebra: Hilbert Scheme of Points

Ritvik Ramkumar, TR 2:55-4:10

MATH 7370 - Topics in Number Theory: The Mordell Conjecture, after Lawrence and Venkatesh

Alexander Betts, MW 10:10-11:25

MATH 7410 - Topics in Combinatorics

Ed Swartz, MWF 1:25-2:15

MATH 7620 - Topics in Geometry: Interactions of Hyperbolic Dynamics, Geometry, and Low-Dimensional Topology

Kathryn Mann, TR 8:40-9:55

MATH 7820 - Logic Seminar

Slawomir Solecki, MF 2:55-4:10

## Course Descriptions

### MATH 5080 - Special Study for Teachers

Fall 2024, Spring 2025. 1 credit. Student option grading.

Primarily for: secondary mathematics teachers and others interested in issues related to teaching and learning secondary mathematics (e.g., mathematics pre-service teachers, mathematics graduate students, and mathematicians). Not open to: undergraduate students. Co-meets with MATH 4980.

Examines principles underlying the content of the secondary school mathematics curriculum, including connections with the history of mathematics, technology, and mathematics education research. One credit is awarded for attending two Saturday workshops (see math.cornell.edu/math-5080) and writing a paper.

### MATH 5200 - Differential Equations and Dynamical Systems

Spring 2025. 3 credits. Student option grading.

Forbidden Overlap: due to an overlap in content, students will receive credit for only one course in the following group: MAE 5790, MATH 4200, MATH 4210, MATH 5200.

Prerequisite: a semester of linear algebra (MATH 2210, MATH 2230, MATH 2310, or MATH 2940) and a semester of multivariable calculus (MATH 2220, MATH 2240, or MATH 1920), or equivalent. Enrollment limited to: graduate students. Students will be expected to be comfortable writing proofs. More experience with proofs may be gained by first taking a 3000-level MATH course. Co-meets with MATH 4200.

Covers ordinary differential equations in one and higher dimensions: qualitative, analytic, and numerical methods. Emphasis is on differential equations as models and the implications of the theory for the behavior of the system being modeled and includes an introduction to bifurcations.

### MATH 5220 - Applied Complex Analysis

Fall 2024. 3 credits. Student option grading.

Prerequisite: a semester of linear algebra (MATH 2210, MATH 2230, MATH 2310, or MATH 2940) and a semester of multivariable calculus (MATH 2220, MATH 2240, or MATH 1920), or equivalent. Enrollment limited to: graduate students. Students will be expected to be comfortable writing proofs. More experience with proofs may be gained by first taking a 3000-level MATH course. Co-meets with MATH 4220.

Covers complex variables, Fourier transforms, Laplace transforms and applications to partial differential equations. Additional topics may include an introduction to generalized functions.

### MATH 5250 - Numerical Analysis and Differential Equations

Fall 2024. 4 credits. Student option grading.

Prerequisite: MATH 2210, MATH 2230-MATH 2240, MATH 2310, or MATH 2940 or equivalent and one additional mathematics course numbered 3000 or above. Enrollment limited to: graduate students. Students will be expected to be comfortable writing proofs. More experience with proofs may be gained by first taking a 3000-level MATH course. Co-meets with MATH 4250 and CS 4210.

Introduction to the fundamentals of numerical analysis: error analysis, approximation, interpolation, numerical integration. In the second half of the course, the above are used to build approximate solvers for ordinary and partial differential equations. Strong emphasis is placed on understanding the advantages, disadvantages, and limits of applicability for all the covered techniques. Computer programming is required to test the theoretical concepts throughout the course.

### MATH 5410 - Introduction to Combinatorics I

Fall 2024. 4 credits. Student option grading.

Prerequisite: MATH 2210, MATH 2230, MATH 2310, MATH 2940, or equivalent. Enrollment limited to: graduate students. Students will be expected to be comfortable writing proofs. More experience with proofs may be gained by first taking a 3000-level MATH course. Co-meets with MATH 4410.

Combinatorics is the study of discrete structures that arise in a variety of areas, particularly in other areas of mathematics, computer science, and many areas of application. Central concerns are often to count objects having a particular property (e.g., trees) or to prove that certain structures exist (e.g., matchings of all vertices in a graph). The first semester of this sequence covers basic questions in graph theory, including extremal graph theory (how large must a graph be before one is guaranteed to have a certain subgraph) and Ramsey theory (which shows that large objects are forced to have structure). Variations on matching theory are discussed, including theorems of Dilworth, Hall, König, and Birkhoff, and an introduction to network flow theory. Methods of enumeration (inclusion/exclusion, Möbius inversion, and generating functions) are introduced and applied to the problems of counting permutations, partitions, and triangulations.

### MATH 5420 - [Introduction to Combinatorics II]

Spring. Not offered: 2024-2025. Next offered: 2025-2026. 4 credits. Student option grading.

Prerequisite: MATH 2210, MATH 2230, MATH 2310, MATH 2940, or equivalent. Enrollment limited to: graduate students. Students will be expected to be comfortable writing proofs. More experience with proofs may be gained by first taking a 3000-level MATH course. Co-meets with MATH 4420.

Continuation of MATH 5410, although formally independent of the material covered there. The emphasis here is the study of certain combinatorial structures, such as Latin squares and combinatorial designs (which are of use in statistical experimental design), classical finite geometries and combinatorial geometries (also known as matroids, which arise in many areas from algebra and geometry through discrete optimization theory). There is an introduction to partially ordered sets and lattices, including general Möbius inversion and its application, as well as the Polya theory of counting in the presence of symmetries.

### MATH 6110 - Real Analysis

Fall 2024. 4 credits. Student option grading.

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 covers abstract measure and integration theory, and related topics such as the Lebesgue differentiation theorem, the Radon-Nikodym theorem, the Hardy-Littlewood maximal function, the Brunn-Minkowski inequality, rectifiable curves and the isoperimetric inequality, Hausdorff dimension and Cantor sets, and an introduction to ergodic theory.

### MATH 6120 - Complex Analysis

Spring 2025. 4 credits. Student option grading.

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. Not offered: 2024-2025. Next offered: 2025-2026. 3 credits. Student option grading.

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

This course emphasizes the "classical" aspects of partial differential equations (PDEs) — analytic methods for linear second-order PDEs and first-order nonlinear PDEs — without relying on more modern tools of functional analysis. The usual topics include fundamental solutions for the Laplace/Poisson, heat 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, Euler-Lagrange equations, similarity solutions, transform methods, asymptotics, power series methods, homogenization, distribution theory, and the Fourier transform.

### MATH 6160 - [Partial Differential Equations]

Spring. Not offered: 2024-2025. Next offered: 2025-2026. 3 credits. Student option grading.

Prerequisite: MATH 6110, MATH 6210, or the equivalent. Offered alternate years.

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 2024. 3 credits. Student option grading.

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

Prerequisite: undergraduate analysis and linear algebra at the level of MATH 4130 and MATH 4310.

Covers measure theory, integration, and Lp spaces.

### MATH 6220 - Applied Functional Analysis

Spring 2025. 3 credits. Student option grading.

Prerequisite: a first course in real analysis, including exposure to Lebesgue integration (e.g., MATH 6110 or MATH 6210).

Functional analysis is a branch of mathematical analysis that mainly focuses on the study of infinite-dimensional vector spaces and the operators acting upon them. It builds upon results and ideas from linear algebra and real and complex analysis to develop general frameworks that can be used to study analytical problems. Functional analysis plays a pivotal role in several areas of mathematics, physics, engineering, and even in some areas of computer science and economics. This course will cover the basic theory of Banach, Hilbert, and Sobolev spaces, as well as explore several notable applications, from analyzing partial differential equations (PDEs), numerical analysis, inverse problems, control theory, optimal transportation, and machine learning.

### MATH 6230 - Differential Games and Optimal Control

Fall 2024. 4 credits. Student option grading.

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 2024. 3 credits. Student option grading.

Prerequisite: MATH 4130-MATH 4140, or the equivalent. Exposure to topology (e.g., MATH 4530) will be helpful. Offered alternate years.

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. Not offered: 2024-2025. Next offered: 2026-2027. 3 credits. Student option grading.

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 6302 - Lattices: Geometry, Cryptography, and Algorithms

Spring 2025. 3 credits. Student option grading.

Prerequisite: MATH 4310 or permission of instructor.

A mathematically rigorous course on lattices. Lattices are periodic sets of vectors in high-dimensional space. They play a central role in modern cryptography, and they arise naturally in the study of high-dimensional geometry (e.g., sphere packings). We will study lattices as both geometric and computational objects. Topics include Minkowski's celebrated theorem, the famous LLL algorithm for finding relatively short lattice vectors, Fourier-analytic methods, basic cryptographic constructions, and modern algorithms for finding shortest lattice vectors. We may also see connections to algebraic number theory.

### MATH 6310 - Algebra

Fall 2024. 4 credits. Student option grading.

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 2025. 4 credits. Student option grading.

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. Not offered: 2024-2025. Next offered: 2025-2026. 3 credits. Student option grading.

Prerequisite: MATH 6310-MATH 6320, or permission of instructor. Offered alternate years.

An introduction to the theory of noncommutative rings and modules. Topics vary by semester and include semisimple modules and rings, the Jacobson radical and Artinian rings, group representations and group algebras, characters of finite groups, representations of the symmetric group, central simple algebras and the Brauer group, representation theory of finite-dimensional algebras, Morita theory.

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

Spring. Not offered: 2024-2025. Next offered: 2025-2026. 3 credits. Student option grading.

Prerequisite: modules and ideals (e.g., strong performance in MATH 4330 and either MATH 3340 or MATH 4340), or permission of instructor.

Covers Dedekind domains, primary decomposition, Hilbert basis theorem, and local rings.

### MATH 6350 - [Homological Algebra]

Spring. Not offered: 2024-2025. Next offered: 2025-2026. 3 credits. Student option grading.

Prerequisite: MATH 6310. Offered alternate years.

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 2025. 3 credits. Student option grading.

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

An introduction to number theory focusing on the algebraic theory. Topics include, but are not limited to, number fields, Dedekind domains, class groups, Dirichlet's unit theorem, local fields, ramification, decomposition and inertia groups, zeta functions, and the distribution of primes.

### MATH 6390 - Lie Groups and Lie Algebras

Fall 2024. 3 credits. Student option grading.

Prerequisite: basic knowledge of algebra and linear algebra at the honors level (e.g., MATH 4330-MATH 4340). Some knowledge of differential and algebraic geometry are helpful.

Lie groups, Lie algebras, and their representations play an important role in much of mathematics, particularly in number theory, mathematical physics, and topology. This is an introductory course, meant to be useful for more advanced topics and applications. The relationship between Lie groups and Lie algebras will be highlighted throughout the course. A different viewpoint is that of algebraic groups. We will endeavor to discuss this along with the C∞ viewpoint.

Topics: Basic structure and properties of Lie algebras; theorems of Lie and Engel. • Nilpotent solvable and reductive Lie algebras. • The relation between Lie groups and Lie algebras • The algebraic groups version (tentative) • Enveloping Algebras and Differential Operators • The structure of semisimple algebras • Representation theory of semisimple Lie algebras; Lie algebra cohomology • Compact semisimple groups and their representation theory. • Chevalley groups, p-adic groups (tentative) • Structure of real reductive groups • Quantum groups, Kac-Moody algebras and their representations theory (tentative)

### MATH 6410 - [Enumerative Combinatorics]

Fall. Not offered: 2024-2025. Next offered: 2025-2026. 3 credits. Student option grading.

Prerequisite: MATH 4410 or permission of instructor. Offered alternate years.

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 and combinatorial species, 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 2025. 4 credits. Student option grading.

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 2024. 4 credits. Student option grading.

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. This course 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. 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.

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

Fall. Not offered: 2024-2025. Next offered: 2025-2046. 3 credits. Student option grading.

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

Spring 2025. 3 credits. Student option grading.

Prerequisite: MATH 6510 or permission of instructor.

This course is an introduction to the theory of infinity-categories that provides a convenient language for homotopy theory and plays an increasingly important role in many other parts of mathematics. Along the way we will cover basics of classical homotopical algebra (model categories and simplicial sets), and as an application --- if time permits --- discuss Quillen's approach to rational homotopy theory and its modern ramifications.

### MATH 6620 - Riemannian Geometry

Spring 2025. 3 credits. Student option grading.

Prerequisite: MATH 6520 or strong performance in analysis (e.g., MATH 4130 and/or MATH 4140), linear algebra (e.g., MATH 4310), and coursework on manifolds and differential geometry at the undergraduate level, such as both MATH 3210 and MATH 4540. Offered alternate years.

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. Not offered: 2024-2025. Next offered: 2026-2027. 3 credits. Student option grading.

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. Not offered: 2024-2025. Next offered: 2025-2026. 3 credits. Student option grading.

Prerequisite: Strong performance in undergraduate analysis (e.g., MATH 4130 or MATH 4180), topology/geometry (e.g., MATH 4520, MATH 4530, or MATH 4540), and algebra (e.g., MATH 4340), or permission of instructor. Offered alternate years.

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, Klein, and Lorentzian models. We will cover both synthetic and computational approaches. We will then discuss hyperbolic structures on surfaces and 3-manifolds, and the corresponding groups of isometries (i.e., Fuchsian and Kleinian groups). Additional topics may include: Geodesic and horocycle flows and their properties, counting closed geodesics and simple closed geodesics, Mostow rigidity, infinite area surfaces.

### MATH 6670 - Algebraic Geometry

Spring 2025. 3 credits. Student option grading.

Prerequisite: MATH 6310 or MATH 6340, or equivalent.

A first course in algebraic geometry. Affine and projective varieties. The Nullstellensatz. Schemes and morphisms between schemes. Dimension theory. Potential topics include normalization, Hilbert schemes, curves and surfaces, and other choices of the instructor.

### MATH 6710 - Probability Theory I

Fall 2024. 3 credits. Student option grading.

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

Measure theory, independence, distribution of sums of iid random variables, laws of large numbers, and central limit theorem. Other topics as time permits.

### MATH 6720 - Probability Theory II

Spring 2025. 3 credits. Student option grading.

Prerequisite: MATH 6710.

The second course in a graduate probability series. Topics include conditional expectation, martingales, Markov chains, Brownian motion, and (time permitting) elements of stochastic integration.

### MATH 6730 - Mathematical Statistics I

(also STSCI 6730)

Spring 2025. 3 credits. Student option grading.

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 2024. 3 credits. Student option grading.

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

Focuses on the foundations of statistical inference, with an emphasis on asymptotic methods and the minimax optimality criterion. In the first part, the solution of the classical problem of justifying Fisher’s information bound in regular statistical models will be presented. This solution will be obtained applying the concepts of contiguity, local asymptotic normality and asymptotic minimaxity. The second part will be devoted to nonparametric estimation, taking a Gaussian regression model as a paradigmatic example. Key topics are kernel estimation and local polynomial approximation, optimal rates of convergence at a point and in global norms, and adaptive estimation. Optional topics may include irregular statistical models, estimation of functionals and nonparametric hypothesis testing.

### MATH 6810 - Logic

Spring 2025. 3 credits. Student option grading.

Prerequisite: an algebra course covering rings and fields (e.g., MATH 4310 or MATH 4330) or permission of instructor. Offered alternate years.

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]

Fall or Spring. Not offered: 2024-2025. Next offered: 2025-2026. 3 credits. Student option grading.

Prerequisite: rings and fields (e.g., MATH 4310 or MATH 4330) and a course in first-order logic at least at the level of MATH 4810/PHIL 4310, or permission of instructor. Offered alternate years.

Introduction to model theory at the level of David Marker's text.

### MATH 6870 - Descriptive Set Theory

Fall 2024. 3 credits. Student option grading.

Prerequisite: metric topology and measure theory (e.g., MATH 4130-MATH 4140 or MATH 6210) and a course in first-order logic (e.g., MATH 3840/PHIL 3300, MATH 4810/PHIL 4310, or MATH 6810), or permission of instructor. Offered alternate years.

This will be an introductory graduate course in Descriptive Set Theory, that is, a theory of definable (Borel, analytic, and co-analytic) subsets of separable, completely metrizable spaces and quotients of such spaces by definable equivalence relations. Some recently discovered aspects of the theory of quotients by Borel equivalence relations will be covered. Some connections with dynamics, classical analysis, combinatorics, and topology will be described.

### MATH 7110 - Topics in Analysis: De Girogi-Nash-Moser Theory and Applications

Fall 2024. 3 credits. S/U grades only.

We will discuss the celebrated De Girogi-Nash-Moser interaction method in elliptic PDE, and its applications in several famous geometry problems, such as the regularity theory for minimal graph equations, Allard regularity, the epsilon regularity for harmonic maps, and so on.

### MATH 7130 - [Functional Analysis]

Fall. Not offered: 2024-2025. Next offered: 2025-2026. 3 credits. Student option grading.

Prerequisite: some basic measure theory, *L ^{p}* spaces, and (basic) functional analysis (e.g., MATH 6110). Advanced undergraduates who have taken MATH 4130-MATH 4140 and linear algebra (e.g., MATH 4310 or MATH 4330), but not MATH 6110, need permission of instructor. Offered alternate years.

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

### MATH 7150 - Fourier Analysis

Spring 2025. 3 credits. S/U grades only.

Prerequisite: some basic measure theory, *L ^{p}* spaces, and (basic) functional analysis (e.g., MATH 6110). Advanced undergraduates who have taken MATH 4130-MATH 4140, but not MATH 6110, by permission of instructor. Offered alternate years.

The class is an introduction to Euclidean harmonic analysis. Topics usually include convergence of Fourier series, harmonic functions and their conjugates, Hilbert transform, Calderón-Zygmund theory, Littlewood-Paley theory, duality between the Hardy space H1 and BMO, paraproducts, Fourier restriction and applications, etc. If time permits, some applications to PDE and number theory will also be discussed.

### MATH 7160 - Topics in Partial Differential Equations: Spectral Geometry

Spring 2025. 3 credits. S/U grades only.

Prerequisite: MATH 6110 and MATH 6160 or equivalents.

This course will serve as a first introduction to spectral geometry—the study of the eigenvalues and eigenfunctions of natural elliptic operators and their relationship with geometry—a subject appearing in various guises in physics, differential geometry, number theory, dynamics, and of course PDE. To keep prerequisites to a minimum, we will focus on the Laplace operator on domains in *R ^{n}*, beginning with classic theorems like Weyl's Law and the Faber-Krahn inequality, before moving on to a selection of more modern results related to isoperimetric inequalities, nodal geometry, isospectral problems, etc.

### MATH 7280 - [Topics in Dynamical Systems]

Fall or Spring. Not offered: 2024-2025. Next offered: 2026-2027. 3 credits. S/U grades only.

Selection of advanced topics from dynamical systems. Content varies.

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

(also CS 7290)

Fall 2024, Spring 2025. 1 credits. S/U grades only.

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: Hilbert Schemes of Points

Spring 2025. 3 credits. S/U grades only.

The Hilbert scheme of points on a variety is a compact moduli space parameterizing 0-dimensional subschemes of *X*. It has a distinguished component parameterizing a reduced collection of points, which has been used extensively in fields ranging from algebraic geometry and topology to combinatorics and even computer science. In this course, we will construct the Hilbert scheme and its relatives, such as the nested Hilbert scheme and the G-Hilbert scheme. We will focus on general questions regarding their singularities, starting with basic topics like smoothness and irreducibility, and progressing to a description of the local structure around torus-fixed points and the structure of the cohomology rings.

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

Fall or Spring. Not offered: 2024-2025. Next offered: 2025-2026. 3 credits. S/U grades only.

Offered alternate years.

Selection of advanced topics from homological algebra. Content varies.

### MATH 7370 - Topics in Number Theory: The Mordell Conjecture, after Lawrence and Venkatesh

Spring 2025. 3 credits. S/U grades only.

Prerequisites: first courses in algebraic geometry (e.g. MATH 6670), Galois Theory (e.g. MATH 6320) and number fields (e.g. MATH 6370), as well as being comfortable with p-adic numbers. Useful, but not required, would be algebraic topology to the level of covering spaces (e.g. MATH 6510), and/or some basic familiarity with etale cohomology (but we'll cover what we need in class).

If *X* is a smooth projective curve of genus at least 2 defined over a number field *K*, then *X* has only finitely many points defined over *K*. This statement is known as the Mordell Conjecture, and its resolution by Gerd Faltings in 1983 is one of the crowning achievements of 20th-century arithmetic geometry. Since Faltings' proof, several other mathematicians (Vojta, Bombieri,...) have come up with different proofs, so that by now we can understand this result from many different perspectives. This course will go through the most recent proof of Mordell, due to Brian Lawrence and Akshay Venkatesh. The idea, roughly speaking, is to replace points defined over *K* by representations of its absolute Galois group, and to study them instead, eventually coming back to geometry using the theory of *p*-adic period maps. Over the course of assembling the proof, we will give user-friendly overviews of several important areas of modern arithmetic geometry, such as etale cohomology, crystalline cohomology and Fontaine's crystalline representations.

### MATH 7390 - [Topics in Lie Groups and Lie Algebras]

Fall or Spring. Not offered: 2024-2025. Next offered: 2025-2026. 3 credits. S/U grades only.

Topics will vary depending on the instructor and the level of the audience. They range from representation theory of Lie algebras and of real and p-adic Lie groups, geometric representation theory, quantum groups and their representations, invariant theory to applications of Lie theory to other parts of mathematics.

### MATH 7410 - Topics in Combinatorics

Spring 2025. 3 credits. S/U grades only.

Offered alternate years.

The course will cover topics in algebraic, topological and/or geometric combinatorics determined in consultation with graduate students during the fall 2024 semester. Neither 'algebraic', nor 'geometric' refers to algebraic geometry.

### MATH 7510 - Berstein Seminar in Topology

Fall 2024. 3 credits. S/U grades only.

This will be a project-based course on modeling and analyzing elections and redistricting that assumes no particular background. The idea is to get you to the state of the art, so that your projects are immediately applicable in the field. This course may be of interest for those wanting to work in metrics of fairness, mathematical modeling, mechanism design, and democracy.

Background topics include:

- overview of social choice theory (including computational social choice), apportionment
- domain knowledge in law, geography, policy
- rules of redistricting, including Voting Rights Act
- state of play in democracy reform

Main topics include:

- fairness axioms for electoral outcomes; fairness metrics for redistricting
- graph partitioning
- Markov chain methods
- (other) spanning-tree methods

- optimization methods for redistricting, both heuristic and exact
- statistical ranking models
- generative models of election

### MATH 7520 - [Berstein Seminar in Topology]

Spring. Not offered: 2024-2025. Next offered: 2025-2026. 3 credits. S/U grades only.

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

### MATH 7570 - [Topics in Topology]

Fall. Not offered: 2024-2025. Next offered: 2025-2026. 3 credits. S/U grades only.

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

### MATH 7580 - [Topics in Topology]

Spring. Not offered: 2024-2025. Next offered: 2025-2026. 3 credits. S/U grades only.

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

### MATH 7610 - [Topics in Geometry]

Fall. Not offered: 2024-2025. Next offered: 2025-2026. 3 credits. S/U grades only.

Selection of advanced topics from modern geometry. Content varies.

### MATH 7620 - Topics in Geometry: Interactions of Hyperbolic Dynamics, Geometry, and Low-Dimensional Topology

Spring. 3 credits. S/U grades only.

Prerequisites: MATH 6520 (Differentiable Manifolds). MATH 6260 (Dynamical systems) is recommended preparation, especially for undergraduates. Some basic familiarity with Riemannian and/or hyperbolic geometry could also be helpful but not strictly required.

This is a course on hyperbolic and partially hyperbolic diffeomorphisms and Anosov flows, from a geometric-topological viewpoint. We will cover some foundational results in smooth and hyperbolic dynamics motivated by low-dimensional examples, and then specialize to flows on three-manfiolds and relationship with the geometry of foliations. Students will do presentations on a related sub-topic of their interest at the end of the semester.

### MATH 7670 - Topics in Algebraic Geometry: Schubert Varieties and Degenerations

Fall 2024. 3 credits. S/U grades only.

Schubert varieties arise in many places in geometry and representation theory. They also serve as a combinatorially tractable source of examples of singular varieties and group actions (rather like toric varieties do). We'll compute lots of things about them, in no small part by degenerating them to unions of pieces. On the combinatorial side, we'll be studying simplicial complexes, and the Stanley-Reisner theory that associates schemes to them. On the algebra side, we'll be using Gröbner and SAGBI degenerations, and controlling them using Frobenius splitting. Once we have Schubert varieties as building blocks, we'll apply our tech to study other spaces, in particular quiver cycles and positroid varieties, maybe wonderful compactifications of groups.

### MATH 7710 - Topics in Probability Theory: Math for AI Safety

Fall 2024. 3 credits. S/U grades only.

AI holds great promise and, many believe, great peril. What can mathematicians contribute to ensuring that promise is fulfilled, and peril avoided? Topics may include: predictive coding, good regulator theorems, Markov decision processes, power-seeking theorems, signaling games, evolution of cooperation, open-source game theory, multi-agent learning, opponent shaping, logical uncertainty, usable information under computational constraints, proper scoring rules, forecast aggregation, Bayesian truth serum, coherence theorems, multi-objective optimization. This course is loosely modeled on the AI Alignment course taught by Roger Grosse at the University of Toronto.

Useful background: machine learning, game theory, and stochastic processes (at the level of MATH 4740).

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

Fall or Spring. Not offered: 2024-2025. Next offered: 2025-2026. 3 credits. S/U grades only.

Selection of advanced topics from stochastic processes. Content varies.

### MATH 7740 - Statistical Learning Theory

Fall 2024. 3 credits. Student option grading.

Prerequisite: basic mathematical statistics (STSCI/MATH 6730 or equivalent) and measure theoretic probability (MATH 6710), or permission of instructor. Enrollment limited to: graduate students.

Learning theory has become an important topic in modern statistics. This course gives 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, Vapnik-Chervonenkis theory, fast rates via Mammen and Tsybakov’s margin condition, convex majorizing loss functions, RKHS methods, support vector machines, lasso type estimators, low-rank multivariate response regression, random matrix theory, topic models, latent factor models, and interpolation methods in high dimensional statistics.

### MATH 7810 - [Seminar in Logic]

Fall. Not offered: 2024-2025. Next offered: 2025-2026. 3 credits. S/U grades only.

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 2025. 3 credits. S/U grades only.

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.