## 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 2023 Offerings

Descriptions are included below under Course Descriptions.

MATH 5080 - Special Study for Teachers

Mary Ann Huntley

MATH 5200 - Differential Equations and Dynamical Systems

Yusheng Luo, TR 1:25-2:40

MATH 5220 - Applied Complex Analysis

Steven Strogatz, TR 10:10-11:25

MATH 5250 - Numerical Analysis and Differential Equations

Yunan Yang, TR 11:40-12:55

MATH 5410 - Introduction to Combinatorics I

Mario Sanchez, MWF 1:25-2:15

MATH 6110 - Real Analysis

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

MATH 6150 - Partial Differential Equations

Alex Vladimirsky, TR 10:10-11:25

MATH 6210 - Measure Theory and Lebesgue Integration

Liam Mazurowski, TR 2:55-4:10

MATH 6310 - Algebra

Dan Halpern-Leistner, MW 1:25-2:40 + dis F 1:25-2:15

MATH 6390 - Lie Groups and Lie Algebras

Birgit Speh, TR 10:10-11:25

MATH 6410 - Enumerative Combinatorics

Marcelo Aguiar, TR 11:40-12:55

MATH 6520 - Differentiable Manifolds

Reyer Sjamaar, TR 8:40-9:55 + dis F 12:20-1:10

MATH 6640 - Hyperbolic Geometry

Ben Dozier, MW 11:40-12:55

MATH 6710 - Probability Theory I

Phil Sosoe, MW 8:40-9:55

MATH 6740 - Mathematical Statistics II

Michael Nussbaum, TR 11:40-12:55

MATH 6830 - Model Theory

Slawomir Solecki, TR 1:25-2:40

MATH 7110 - Topics in Analysis: Geometric Analysis

Xiaodong Cao, MW 11:40-12:55

MATH 7130 - Functional Analysis

John Hubbard, TR 1:25-2:40

MATH 7280 - Topics in Dynamical Systems: Mathematical Biology

Steve Strogatz, WF 1:25-2:40

MATH 7510 - Berstein Seminar in Topology: 4-Manifolds

Tara Holm, MW 8:40-9:55

MATH 7570 - Topics in Topology: Category Theory

Rhiannon Griffiths, TR 10:10-11:25

MATH 7740 - Statistical Learning Theory

Marten Wegkamp, MW 10:10-11:25

MATH 7810 - Seminar in Logic

Justin Moore, MF 2:55-4:10

## Spring 2024 Offerings

Descriptions are included below under Course Descriptions.

MATH 5080 - Special Study for Teachers

Mary Ann Huntley

MATH 6120 - Complex Analysis

John Hubbard, TR 1:25-2:40 + dis W 3:35-4:25

MATH 6160 - Partial Differential Equations

Daniel Stern, TR 10:10-11:25

MATH 6220 - Applied Functional Analysis

Yunan Yang, MW 10:10-11:25

MATH 6302 - Lattices: Geometry, Cryptography and Algorithms

Noah Stephens-Davidowitz, TR 2:55-4:10

MATH 6320 - Algebra

Ravi Ramakrishna, TR 11:40-12:55 + dis W 2:30-3:20

MATH 6350 - Homological Algebra: Computational Homological Algebra

Michael Stillman, MWF 9:05-9:55

MATH 6510 - Algebraic Topology

James West, TR 8:40-9:55 + dis W 1:25-2:15

MATH 6630 - Symplectic Geometry

Morgan Weiler, MW 11:40-12:55

MATH 6720 - Probability Theory II

Philippe Sosoe, TR 8:40-9:55

MATH 6730 - Mathematical Statistics I

F. Bunea, T 11:15-1:45

MATH 7160 - Topics in Partial Differential Equations: The Allen-Cahn Equation

Liam Mazurowski, MW 10:10-11:25

MATH 7310 - Topics in Algebra: Profinite Rigidity for Groups

Martin Kassabov, TR 1:25-2:40

MATH 7390 - Topics in Lie Groups and Lie Algebras: Automorphic Forms and Representation Theory

Dan Barbasch, TR 11:40-12:55

MATH 7580 - Topics in Topology: Hyperbolic Groups

Jason Manning, MF 1:25-2:40

MATH 7670 - Topics in Algebraic Geometry: Positivity in Algebraic Geometry

Ritvik Ramkumar, TR 2:55-4:10

MATH 7710 - Topics in Probability: The Theory of Dirichlet Forms

Laurent Saloff-Coste, MW 11:40-12:55

MATH 7820 - Logic Seminar

Slawomir Solecki, MF 2:55-4:10

## Course Descriptions

### MATH 5080 - Special Study for Teachers

Fall 2023, Spring 2024. 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

Fall 2023. 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: MATH 2210-MATH 2220, MATH 2230-MATH 2240, MATH 1920-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 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 2023. 3 credits. Student option grading.

Prerequisite: MATH 2210-MATH 2220, MATH 2230-MATH 2240, MATH 1920-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 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 2023. 4 credits. Student option grading.

Prerequisite: MATH 2210, MATH 2230-MATH 2240, 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 2023. 4 credits. Student option grading.

Prerequisite: MATH 2210, MATH 2230, 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 2024. 4 credits. Student option grading.

Prerequisite: MATH 2210, MATH 2230, 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 2023. 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 measure and integration and functional analysis.

### MATH 6120 - Complex Analysis

Spring 2024. 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 2023. 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.

We will mostly follow the textbook *Partial Differential Equations* by L. C. Evans with the primary focus on Chapters 2-3 and assorted topics from Chapters 4, 8, and 10.

### MATH 6160 - Partial Differential Equations

Spring 2024. 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 2023. 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 2024. 3 credits. Student option grading.

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

Functional analysis is a branch of mathematical analysis that focuses on the study of infinite-dimensional vector spaces and the linear operators acting upon them. It builds upon results and ideas from linear algebra and real and complex analysis to develop general frameworks which 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 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 or Spring. Not offered: 2023-2024. Next 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. Not offered: 2023-2024. Next offered: 2024-2025. 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)

Spring. Not offered: 2023-2024. Next offered: 2024-2025. 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 2024. 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 2023. 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 2024. 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. Not offered: 2023-2024. Next offered: 2024-2025. 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]

Fall. Not offered: 2023-2024. Next offered: 2024-2025. 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: Computational Homological Algebra

Spring 2024. 3 credits. Student option grading.

Prerequisites: A graduate course in abstract algebra should be sufficient (modules over a commutative ring, homomorphisms of rings and modules, direct sums, tensor products, Hom will be assumed). Undergraduate abstract algebra (as long as it covered the above concepts) might be sufficient if one has a strong background in linear algebra. Offered alternate years.

This course will be an introduction to homological algebra. Rather than regarding the subject as an abstract machine for proving non-constructive existence theorems, we will present homological algebra as a vast generalization of linear algebra: matrices are replaced by appropriate sequences of linear maps. This novel perspective will emphasize "thinking in terms of complexes" and developing effective computational tools.

Together, we will explore the fundamental structures and essential constructions within homological algebra. We will examine a wide range of examples, many explicitly illustrated via the Macaulay2 software system.

Tentative range of topics include:

- Complexes: homology, homomorphisms, tensor products.
- Equivalences: mapping cones, quasi-isomorphisms, homotopy, and standard isomorphisms.
- Resolutions: free, projective, injective, and flat resolutions.
- Derived categories: construction, universal property, distinguised triangles. We attempt to make these natural and usable, without huge amounts of technical details.
- Derived functors: induced functors, right derived Hom functors,left derived tensor functor. (i.e. (derived) Ext, Tor).
- Some selected applications: the Bernstein-Gelfand-Gelfand (BGG) correspondence, homological dimension, dualizing complexes, ...

Textbook: Because there is not currently a textbook on homological algebra from a computational perspective, we will provide lecture notes.

Homework/Grading: I will provide problem sets, whose solutions will be presented by students in class, and there will be a project. Details will be provided at the beginning of the course. Students taking the course for a letter grade will be required to do these, and, because doing mathematics is not a spectator sport, they are strongly recommended for all students!

### MATH 6370 - [Algebraic Number Theory]

Spring. Not offered: 2023-2024. Next offered: 2024-2025. 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 2023. 3 credits. Student option grading.

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

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

Fall 2023. 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 2024. 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.

We will follow Allen Hatcher's Rich and Classic text which develops the subject from a decidedly geometric perspective. We will begin with some topological properties (Homotopy Extension Theorem, Fox's Theorem) and proceed through Homology and Cohomology to the Poincare' Duality Theorem. The text requires a familiarity with the contents of an introductory topology course, and an undergaduate level algebra course would be very useful.

### MATH 6520 - Differentiable Manifolds

Fall 2023. 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: 2023-2024. Next offered: 2024-2045. 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]

Fall or Spring. Not offered: 2023-2024. Next offered: 2024-2025. 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]

Spring. Not offered: 2023-2024. Next offered: 2024-2025. 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

Spring 2024. 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 2023. 3 credits. Student option grading.

Prerequisite: Strong performance in undergraduate analysis (e.g., MATH 4130 or MATH 4180) and topology/geometry (e.g., MATH 4530, MATH 4550, or MATH 4560), 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]

Fall or Spring. Not offered: 2023-2024. Next offered: 2024-2025. 3 credits. Student option grading.

Prerequisite: MATH 6310 and 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 2023. 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 4130-4140 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 2024. 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 2024. 4 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 2023. 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. Not offered: 2023-2024. Next offered: 2024-2025. 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 2023. 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 6840 - [Recursion Theory]

Fall. Not offered: 2023-2024. Next offered: 2024-2025. 3 credits. Student option grading.

Prerequisite: a course in first-order logic such as MATH 4810/PHIL 4310 or MATH 4860/CS 4860, or permission of instructor. Offered alternate years.

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]

Fall. Not offered: 2023-2024. Next offered: 2024-2025. 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 Geometric Analysis

Fall 2023. 3 credits. S/U grades only.

In the first part, I will continue what’s left in MATH 6620, to teach the basics of geometric analysis. In the second part, I will teach more advanced topics in geometric flow and 4-manifolds.

### MATH 7120 - [Topics in Analysis]

Spring. Not offered: 2023-2024. Next offered: 2024-2025. 3 credits. S/U grades only.

Selection of advanced topics from analysis. Course content varies.

### MATH 7130 - Functional Analysis

Fall 2023. 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, by 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. Not offered: 2023-2024. Next offered: 2024-2025. 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.

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: The Allen-Cahn Equation

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

In this topics course, we will discuss some analytic and geometric properties of the Allen-Cahn (Van der Waals, Cahn-Hilliard) equation. This equation is a semilinear PDE that was originally introduced as a model for phase transitions. It has deep connections with minimal surfaces and constant mean curvature surfaces in Riemannian manifolds. We will discuss the existence of solutions and their convergence to a limiting interface. Time permitting, we will also discuss the Toda system and the work of Wang-Wei and Chodosh-Mantoulidis on the multiplicity of this interface.

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

Fall or Spring. Not offered: 2023-2024. Next offered: 2024-2025. 3 credits. S/U grades only.

Selection of advanced topics from numerical analysis. Content varies.

### MATH 7280 - Topics in Dynamical Systems: Mathematical Biology

Fall 2023. 3 credits. S/U grades only.

Prerequisite: MATH 4210 or equivalent.

This course provides an introduction to mathematical biology. We will follow the classic textbook: J.D. Murray, Mathematical Biology I. An Introduction. 3rd edition. (New York: Springer, 2002). Topics will be drawn from the following areas (as listed in the book's Table of Contents): Continuous Population Models for Single Species • Discrete Population Models for a Single Species • Models for Interacting Populations • Temperature-Dependent Sex Determination (TSD): Crocodilian Survivorship • Modelling the Dynamics of Marital Interaction: Divorce Prediction and Marriage Repair • Reaction Kinetics • Biological Oscillators and Switches • BZ Oscillating Reactions • Perturbed and Coupled Oscillators and Black Holes • Dynamics of Infectious Diseases: Epidemic Models and AIDS • Reaction Diffusion, Chemotaxis, and Nonlocal Mechanisms • Oscillator-Generated Wave Phenomena and Central Pattern Generators • Biological Waves: Single-Species Models • Use and Abuse of Fractals

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

(also CS 7290)

Fall 2023, Spring 2024. 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: Profinite Rigidity for Groups

Spring. 3 credits. S/U grades only.

The profinite completion of a group is a way to encode all finite quotients. The main question is what group properties can be read from the profinite completion, i.e., P is a profinite property if whenever two discrete groups have the same profinite completion then either both have the property or neither has it. It turns out that apart from properties which are clearly profinite (like being abelain) almost no property is profinite. In the class I will focus on a related question — when a group is determined by its profinite completion. Clearly without any additional assumptions the answer is never, but the question if usually consider with the additional condition that all groups are finitely generated and residually finite. My goal is to explain recent results by Bridson-McReynolds-Reid-Spitler, that several triangle groups with geometric origin are profinitely rigid and are determined by their profinite completion.

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

Fall or Spring. Not offered: 2023-2024. Next offered: 2024-2025. 3 credits. S/U grades only.

Offered alternate years.

Selection of advanced topics from homological algebra. Content varies.

### MATH 7370 - [Topics in Number Theory]

Fall or Spring. Not offered: 2023-2024. Next offered: 2024-2025. 3 credits. S/U grades only.

Selection of advanced topics from number theory. Course content varies.

### MATH 7390 - Topics in Lie Groups and Lie Algebras: Automorphic Forms and Representation Theory

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

This course will study the theory of automorphic representations with a view towards the Langlands program. We will focus on the local and global representation theory. The main example will be the case of SL(2). Core references are the texts by D. Bump, Automorphic forms Bump, Cogdell et al, Introduction to the Langlands program There are many other texts and notes available for the representation theory of real and p-adic Lie groups. If time permits, more recent work will be discussed.

### MATH 7410 - [Topics in Combinatorics]

Spring. Not offered: 2023-2024. Next offered: 2024-2025. 3 credits. S/U grades only.

Offered alternate years.

Selection of advanced topics in combinatorics. Course content varies.

### MATH 7510 - Berstein Seminar in Topology: 4-Manifolds

Fall 2023. 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.

We will focus on the differential topology of 4-manifolds and the descriptive techniques of Kirby calculus, loosely following the textbook 4-Manifolds and Kirby Calculus (and maybe some of The Wild World of 4-Manifolds). This will be an even more audience-driven and -lectured Berstein seminar than usual. (The department chair doesn't have time to teach, but wanted to make this seminar happen!) We will begin by discussing surgery and handlebody decompositions in any dimension, then zoom in on dimensions 3 and 4, where results such as the Lickorish-Wallace theorem allow us to describe manifolds in terms of decorated link diagrams. The power of Kirby calculus lies in a set of "moves" that relate any two diagrams describing diffeomorphic manifolds. After covering this fundamental theorem, we will focus specifically on 4 dimensions, discussing fundamental examples and the role of the intersection form in classification theorems. After this, the rest of the semester will be spent on further topics chosen by the participants. Some suggested further topics are:

- The menagerie of exotic smooth structures on ℝ⁴;
- Corks and exotic smooth structures on closed manifolds;
- Non-smooth techniques, such as Casson handles;
- Branched coverings and Lefschetz fibrations;
- Spin structures and gauge-theoretic invariants;
- Whatever else piques your interest.

You may also be interested to know that there is a Kirby calculus python package developed in 2018 by a team including one of our colleagues! In light of recent applications of machine learning to knot theory (Quanta: https://www.quantamagazine.org/deepmind-machine-learning-becomes-a-mathematical-collaborator-20220215/), some of you might be interested in the possibility of applying machine learning to Kirby calculus (probably with the goal of showing ruling out certain candidates for exotic 4-spheres, by identifying sequences of "moves" that convert their Kirby diagrams into that of the standard S⁴).

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

Spring. Not offered: 2023-2024. Next offered: 2024-2025. 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: Category Theory

Fall 2023. 3 credits. S/U grades only.

An introduction to categories, functors, natural transformations, the Yoneda Lemma, adjoint functors, universal properties, limits and colimits, and monads. Time permitting, we will also discuss monoidal categories, higher categories, and more. Much of modern mathematics is done in the language of category theory; this course is intended to familiarize students with that language and with categorical methods in general. As such, we will use a wide range of examples from algebra, geometry, and topology.

### MATH 7580 - Topics in Topology: Hyperbolic Groups

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

This course is an introduction to hyperbolic groups. Hyperbolic groups have been a central topic of geometric group theory since Gromov's 1987 essay. Gromov noticed that certain ideas from the differential geometry of negatively curved manifolds were vastly simplified and generalized by framing them in terms of "large-scale" metric properties, such as uniform thinness of triangles, or the linearity of the isoperimetric inequality. We'll give examples of hyperbolic groups, and discuss some open questions about them. Time permitting we will talk about some more general classes of "negatively curved groups", such as relatively hyperbolic, acylindrically hyperbolic, and hierarchically hyperbolic groups.

### MATH 7610 - [Topics in Geometry]

Fall. Not offered: 2023-2024. Next offered: 2024-2025. 3 credits. S/U grades only.

Selection of advanced topics from modern geometry. Content varies.

### MATH 7620 - [Topics in Geometry]

Spring. Not offered: 2023-2024. Next offered: 2024-2025. 3 credits. S/U grades only.

Selection of advanced topics from modern geometry. Content varies.

### MATH 7670 - Topics in Algebraic Geometry: Positivity in Algebraic Geometry

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

This course intends to provide a follow-up to a first course in algebraic geometry and give an introduction to higher dimensional algebraic geometry. The focus of the course will be on concrete examples and applications. We will be following Rob Lazarsfeld's volumes on 'Positivity in Algebraic Geometry.' Some of the topics we will be covering include ample, nef, and big line bundles, linear series and Iitaka dimension, Castelnuovo-Mumford regularity, cone theorems, Lefschetz theorems, Fulton-Hansen theorem, Seshadri constants, vanishing theorems, generalizations to vector bundles and their positivity. Time permitting, we will also discuss multiplier ideals and Siu's theorem regarding the invariance of plurigenera. We will assume some basic algebraic geometry, at the level of Hartshorne chapters 1-3, although we will review some of the essential concepts. This includes the notion of divisors, line bundles, and sheaves on varieties and their cohomology.

### MATH 7710 - Topics in Probability Theory: The Theory of Dirichlet Forms

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

The course will offer an introduction to the theory of Dirichlet forms (and associated Markov processes), with a number of classic examples, and a discussion related geometries.

### MATH 7720 - Topics in Stochastic Processes

Fall or Spring. Not offered: 2023-2024. Next offered: 2024-2025. 3 credits. S/U grades only.

Selection of advanced topics from stochastic processes. Content varies.

### MATH 7740 - Statistical Learning Theory

Fall 2023. 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 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. Further, active high-dimensional statistical research topics such as lasso type estimators, low-rank multivariate response regression, topic models, latent factor models, and interpolation methods are presented.

### MATH 7810 - Seminar in Logic: Set Theory and Higher Derived Limits

Fall 2023. 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.

In Fall 2023, the logic seminar will explore some recent interactions between set theory and homological algebra which arose when determining the properties of strong homology on the class of locally compact second countable spaces. The calculation of strong homology groups is closely related to the calculation of higher derived limits of inverse systems of abelian groups. When these systems are indexed by a directed set of uncountable cofinality (for instance, the set of integer sequences ordered coordinatewise), the calculations of the higher derived limits can become sensitive to set theoretic assumptions. Students will lecture on a set of papers related to the interplay between set theory and the calculation of higher derived limits. The seminar will also feature talks on outside speakers, faculty and graduate students, sometimes on other topics in set theory and logic. Students are not expected to have prior knowledge of homological algebra. Students are expected to have some knowledge of set theory at the level of Kunen's text. (Contact the instructor if in doubt about your set theory background.)

### MATH 7820 - Seminar in Logic

Spring 2024. 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 7850 - [Topics in Logic]

Fall or Spring. Not offered: 2023-2024. Next offered: 2024-2025. 3 credits. S/U grades only.

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).