Graduate Courses

Overview

Graduate course offerings for the academic year 2025-2026 are included below along with course descriptions that are in some 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 2025 Offerings

Descriptions are included below under Course Descriptions.

MATH 5080 - Special Study for Teachers
Mary Ann Huntley

MATH 5220 - Applied Complex Analysis
Adrian Chu, MWF 9:05-9:55

MATH 5250 - Numerical Analysis and Differential Equations
Alex Townsend, TR 2:55-4:10

MATH 5410 - Introduction to Combinatorics I
Karola Meszaros, TR 1:25-2:40

MATH 6110 - Real Analysis
Camil Muscalu, MW 8:40-9:55 + dis F 12:20-1:10

MATH 6150 - Partial Differential Equations
Daniel Stern, TR 10:10-11:25

MATH 6210 - Measure Theory and Lebesgue Integration
Laurent Saloff-Coste, MW 8:40-9:55

MATH 6260 - Dynamical Systems
Yusheng Luo, MF 1:25-2:40

MATH 6310 - Algebra
Alex Betts, TR 11:40-12:55 + dis W 2:30-3:20

MATH 6330 - Noncommutative Algebra
Martin Kassabov, MW 10:10-11:25

MATH 6390 - Lie Groups and Lie Algebras
Yuri Berest, TR 1:25-2:40

MATH 6410 - Enumerative Combinatorics
Marcelo Aguiar, TR 1:25-2:40

MATH 6520 - Differentiable Manifolds
Allen Knutson, TR 2:55-4:10 + dis W 3:35-4:25

MATH 6530 - K-Theory and Characteristics Classes
Inna Zakharevich, TR 10:10-11:25

MATH 6640 - Hyperbolic Geometry
Inhyeok Choi, TR 8:40-9:55

MATH 6710 - Probability Theory I
Lionel Levine, MW 10:10-11:25

MATH 6840 - Recursion Theory
Slawomir Solecki, MWF 1:25-2:15

MATH 7130 - Functional Analysis
Camil Muscalu, TR 2:55-4:10

MATH 7370 - Topics in Number Theory: Monodromy and Galois Groups
David Zywina, TR 11:40-12:55

MATH 7670 - Topics in Algebraic Geometry: Moduli Theory
Dan Halpern-Leistner, TR 8:40-9:55

MATH 7740 - Statistical Learning Theory
Marten Wegkamp, MW 11:40-12:55

MATH 7810 - Seminar in Logic: Large Cardinals and Saturation Properties of Ideals
Justin Moore, MF 2:55-4:10

Spring 2026 Offerings

Descriptions are included below under Course Descriptions. Meeting times are to be determined at a later date.

MATH 5080 - Special Study for Teachers
Mary Ann Huntley

MATH 5200 - Differential Equations and Dynamical Systems
Yusheng Luo

MATH 5420 - Introduction to Combinatorics II
Edward Swartz

MATH 6120 - Complex Analysis
Benjamin Dozier

MATH 6160 - Partial Differential Equations
Daniel Stern

MATH 6220 - Applied Functional Analysis
Yunan Yang

MATH 6302 - Lattices: Geometry, Cryptography, and Algorithms (tentative)

MATH 6320 - Algebra
Nicolas Templier

MATH 6340 - Commutative Algebra with Applications in Algebraic Geometry
Michael Stillman

MATH 6350 - Homological Algebra
Yuri Berest

MATH 6370 - Algebraic Number Theory
Andreea Iorga

MATH 6510 - Algebraic Topology
Inna Zakharevich

MATH 6670 - Algebraic Geometry
Rachel Webb

MATH 6720 - Probability Theory II
Philippe Sosoe

MATH 6730 - Mathematical Statistics I (tentative)

MATH 6810 - Logic
Mark Poór

MATH 7110 - Topics in Analysis: Topic TBA
Vlassis Mastrantonis

MATH 7120 - Topics in Analysis: Topic TBA
Xin Zhou

MATH 7520 - Berstein Seminar in Topology: High-Dimensional Hyperbolic Manifolds
Jason Manning

MATH 7580 - Topics in Topology
Anubhav Nanavaty

MATH 7620 - Topics in Geometry: Rigidity Theory
Robert Connelly

MATH 7670 - Topics in Algebraic Geometry
Dan Halpern-Leistner

MATH 7820 - Logic Seminar
Slawomir Solecki, MF 2:55-4:10

Course Descriptions

MATH 5080 - Special Study for Teachers

Fall 2025, Spring 2026. 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 2026. 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 2025. 3 credits. Student option grading.

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 2025. 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 2025. 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 2026. 4 credits. Student option grading.

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 2025. 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 2026. 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 2025. 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 Rⁿ, 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 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 2025. 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 2026. 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

Not offered 2025-2026. 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 2025. 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)

Not offered 2025-2026. 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

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 2025. 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 2026. 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 2025. 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 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 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 2026. 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 2025. 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.

The course is intended to be an introduction to classical Lie theory: that is, the theory of compact Lie groups and finite-dimensional Lie algebras. Time permitting, we will cover some additional topics, such as classifying spaces of Lie groups and their topological generalizations: the finite loop spaces (aka 'fake Lie groups') and the p-compact groups (aka 'homotopy Lie groups').

MATH 6410 - Enumerative Combinatorics

Fall 2025. 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 for Fall 2025 include: Stanley's Twelvefold way q-analogs. Inversions of permutations and finite vector spaces. The finite Grassmannian Posets. Möbius inversion. Lattices. Generating functions. Exponential formula. Lagrange inversion.

We will draw from various textbooks and our own sources, including the following: Enumerative Combinatorics I, 2nd edition, by Richard Stanley, published by Cambridge U Press. Chapters 1, 2, 3, and a little of 4. A course in enumeration, by Martin Aigner, published by Springer. Chapters 1, 2, 3, 5, 7. A course in Combinatorics, 2nd edition, by J. H. van Lint and R. M. Wilson, published by Cambridge U Press. Chapters 10, 13, 14, 15, 24, 25, and a little of 2 and 6. Other references may be indicated in class.

MATH 6510 - Algebraic Topology

Spring 2026. 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 2025. 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 2025. 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

Not offered 2025-2026. 3 credits. Student option grading.

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

Not offered 2025-2026. 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

Not offered 2025-2026. 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 2025. 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 2026. 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 2025. 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 2026. 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 2026. 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)

Not offered 2025-2026. 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 2026. 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

Not 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 - Set Theory

Not offered 2025-2026. 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.

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

MATH 7110 - Topics in Analysis

Spring 2026. 3 credits. S/U grades only.

Selection of advanced topics from analysis. Course content varies.

MATH 7120 - Topics in Analysis

Spring 2026. 3 credits. S/U grades only.

Selection of advanced topics from analysis. Course content varies.

MATH 7130 - Functional Analysis

Fall 2025. 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

Not 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, but not MATH 6110, by permission of instructor. 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 7160 - Topics in Partial Differential Equations

Not offered 2025-2026. 3 credits. S/U grades only.

Selection of advanced topics from partial differential equations. Content varies.

MATH 7280 - Topics in Dynamical Systems

Not offered 2025-2026. 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 2025, Spring 2026. 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

Not offered 2025-2026. 3 credits. S/U grades only.

Selection of advanced topics from algebra. Course content varies.

MATH 7350 - Topics in Homological Algebra

Not offered 2025-2026. 3 credits. S/U grades only.

Selection of advanced topics from homological algebra. Content varies.

MATH 7370 - Topics in Number Theory: Monodromy and Galois Groups

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

Prerequisites: familiarity with basic algebraic geometry (e.g., MATH 6670) and algebraic number theory (e.g., MATH 6370).

We will study Galois groups and their geometric generalization etale fundamental groups. We are especially interested in various actions of these groups that arise in number theory. Applications include some cases of the Inverse Galois Problem as well as some theorems of Serre on the images of Galois representations arising from abelian varieties.

MATH 7390 - Topics in Lie Groups and Lie Algebras

Not 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

Not offered 2025-2026. 3 credits. S/U grades only.

Offered alternate years.

Selection of advanced topics in combinatorics. Course content varies.

MATH 7510 - Berstein Seminar in Topology

Not 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 7520 - Berstein Seminar in Topology

Spring 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

Not 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 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

Not offered 2025-2026. 3 credits. S/U grades only.

Selection of advanced topics from modern geometry. Content varies.

MATH 7620 - Topics in Geometry

Spring 2026. 3 credits. S/U grades only.

Selection of advanced topics from modern geometry. Content varies.

MATH 7670 - Topics in Algebraic Geometry: Moduli Theory

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

A recurring theme in algebraic geometry is that for many kinds of geometric objects, one can construct a space, called a moduli space, parameterizing all isomorphism classes of these objects. Projective space, Grassmannians, and Hilbert schemes are all examples of moduli spaces, and there are many others. The geometry of a moduli space often sheds light on the objects that it parameterizes. In addition, auxiliary constructions of moduli spaces are used to define sophisticated invariants of algebraic varieties, such as Gromov-Witten invariants or Donaldson invariants. In modern language, a moduli problem, properly formulated, is represented by an object called an algebraic stack. Over the past decade there has been a lot of progress in determining when an algebraic stack admits a moduli space, and what other structures an algebraic stack might exhibit when it does not have a moduli space. This course will be an introduction to the theory of algebraic stacks and the tools that people use to study moduli problems in algebraic geometry: moduli spaces, semistability, and Theta-stratifications. The course will follow my web book: https://book.themoduli.space

MATH 7710 - Topics in Probability Theory

Not offered 2025-2026. 3 credits. S/U grades only.

Selection of advanced topics from probability theory. Content varies.

MATH 7720 - Topics in Stochastic Processes

Not offered 2025-2026. 3 credits. S/U grades only.

Selection of advanced topics from stochastic processes. Content varies.

MATH 7740 - Statistical Learning Theory

Fall 2025. 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. 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 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, 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 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.

This semester the seminar will examine how strong large cardinal axioms can be used to establish the consistent existence of ideals on successor cardinals with strong saturation properties. We will start with Kunen's construction of a saturated ideal on the first uncountable cardinal starting from a huge cardinal. We will then cover the work of Foreman in which he extends and generalizes this result. The culmination of the seminar will be a recent paper of Eskew and Hayut in which they construct a model in which, for each $n \geq 1$, there is an $\omega_1$-dense ideal on $\omega_n$ starting from a model with a huge cardinal. While the course will begin with a fast review of large cardinals and forcing, students are expected to have a solid foundation in set theory.

MATH 7820 - Seminar in Logic

Spring 2026. 3 credits. S/U grades only.