# Graduate Courses

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## Overview

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

## Fall 2021 Offerings

Descriptions are included under Course Descriptions.

MATH 5080 - Special Study for Teachers

Mary Ann Huntley

MATH 5250 - Numerical Analysis and Differential Equations

Alex Vladimirsky, TR 1:00-2:15

MATH 5410 - Introduction to Combinatorics I

Karola Meszaros, MWF 11:20-12:10

MATH 6110 - Real Analysis

Camil Muscalu, TR 11:25-12:40

MATH 6210 - Measure Theory and Lebesgue Integration

Ben Dozier, MW 8:05-9:20

MATH 6310 - Algebra

Marcelo Aguiar, TR 1:00-2:15

MATH 6340 - Commutative Algebra

Irena Peeva, MF 2:45-4:00

MATH 6520 - Differentiable Manifolds

James West, TR 8:05-9:20

MATH 6640 - Hyperbolic Geometry

Jason Manning, MWF 11:20-12:10

MATH 6710 - Probability Theory I

Lionel Levine, TR 9:40-10:55

MATH 6740 - Mathematical Statistics II

Michael Nussbaum, TR 11:25-12:40

MATH 6840 - Recursion Theory

Anil Nerode, TR 9:40-10:55

MATH 7110 - Topics in Analysis: Lectures on the “Helicoidal Method"

Camil Muscalu, TR 8:05-9:20

MATH 7130 - Functional Analysis

Terry Harris, TR 1:00-2:15

MATH 7160 - Topics in Partial Differential Equations: Nonlinear Elliptic PDE

Timothy Healey, MW 1:00-2:15

MATH 7370 - Topics in Algebraic Number Theory

Ravi Ramakrishna, TR 2:45-4:00

MATH 7510 - Berstein Seminar in Topology: Anosov Flows on 3-Manifolds

Kathryn Mann, MW 8:05-9:20

MATH 7610 - Topics in Geometry: Comparison Geometry and Minimal Surfaces

Xin Zhou, TR 11:25-12:40

MATH 7670 - Topics in Algebraic Geometry: Non-Archimedean Geometry

Dan Halpern-Leistner, MWF 1:30-2:20

MATH 7720 - Topics in Stochastic Processes: Random Walks on Groups

Laurent Saloff-Coste, MWF 10:10-11:00

MATH 7810 - Seminar in Logic: The Logic of Thompson's Groups and their Relatives

Justin Moore, TF 2:45-4:00

## Spring 2022 Offerings

Descriptions are included under Course Descriptions.

MATH 5080 - Special Study for Teachers

Mary Ann Huntley

MATH 5420 - Introduction to Combinatorics II

Allen Knutson, MWF 1:30-2:20

MATH 6120 - Complex Analysis

Nicolas Templier, TR 11:25-12:40

MATH 6150 - Partial Differential Equations

Laurent Saloff-Coste, TR 9:40-10:55

MATH 6220 - Applied Functional Analysis

Terrence Harris, TR 1:00-2:15

MATH 6320 - Algebra

David Zywina, TR 1:00-2:15

MATH 6350 - Homological Algebra

Yuri Berest, MW 1:00-2:15

MATH 6370 - Algebraic Number Theory

Shankar Sen, MW 11:25-12:40

MATH 6410 - Enumerative Combinatorics

Edward Swartz, MWF 10:10-11:00

MATH 6510 - Algebraic Topology

James West, MW 8:05-9:20

MATH 6540 - Homotopy Theory

Inna Zakharevich, MW 9:40-10:55

MATH 6670 - Algebraic Geometry

Michael Stillman, TR 9:40-10:55

MATH 6720 - Probability Theory II

Philippe Sosoe, TR 8:05-9:20

MATH 6730 - Mathematical Statistics I

F. Bunea, Time TBA

MATH 6830 - Model Theory

Justin Moore, MW 9:40-10:55

MATH 7120 - Topics in Analysis: Geometry and Analysis of Four-Manifolds

Xiaodong Cao, TR 11:25-12:40

MATH 7390 - Topics in Lie Groups and Lie Algebras: Symplectic Resolutions

Allen Knutson, MW 11:25-12:40

MATH 7580 - Topics in Topology: Index Theory

Yannis Loizides, TR 2:45-4:00

MATH 7620 - Topics in Geometry: Universal Rigidity and Applications

Robert Connelly, TR 2:45-4:00

MATH 7710 - Topics in Probability Theory: Limits of Discrete Random Structures

Lionel Levine, TR 11:25-12:40

MATH 7820 - Seminar in Logic

Slawomir Solecki, TF 2:45-4:00

## Course Descriptions

### MATH 5080 - Special Study for Teachers

Fall 2021, Spring 2022. 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 5250 - Numerical Analysis and Differential Equations

Fall 2021. 4 credits. Student option grading.

Prerequisite: MATH 2210 or 2940 or equivalent, one additional mathematics course numbered 3000 or above, and knowledge of programming. Students will be expected to be comfortable with proofs. Enrollment limited to: graduate students. Co-meets with MATH 4250.

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 2021. 4 credits. Student option grading.

Prerequisite: MATH 2210, MATH 2230, MATH 2310, or MATH 2940. Students will be expected to be comfortable with proofs. Enrollment limited to: graduate students. 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 2022. 4 credits. Student option grading.

Prerequisite: MATH 2210, MATH 2230, MATH 2310 , or MATH 2940. Students will be expected to be comfortable with proofs. Enrollment limited to: graduate students. 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 2021. 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 2022. 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

Spring 2022. 4 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). The usual topics include fundamental solutions for the Laplace/Poisson, heat and and wave equations in **R**^{n}, mean-value properties, maximum principles, energy methods, Duhamel's principle, and an introduction to nonlinear first-order equations, including shocks and weak solutions. Additional topics may include Hamilton-Jacobi equations, distribution theory, and the Fourier transform.

### MATH 6160 - [Partial Differential Equations]

Fall. Not offered: 2021-2022. Next offered: 2022-2023. 4 credits. Student option grading.

Prerequisite: MATH 4130, MATH 4140, or the equivalent, or permission of instructor. 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 2021. 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.

Covers measure theory, integration, and Lp spaces.

### MATH 6220 - Applied Functional Analysis

Spring 2022. 4 credits. Student option grading.

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

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

Fall. Not offered: 2021-2022. Next offered: 2022-2023. 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: 2021-2022. Next offered: 2022-2023. 4 credits. Student option grading.

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: 2021-2022. Next offered: 2022-2023. 4 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 6310 - Algebra

Fall 2021. 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 2022. 4 credits. Student option grading.

Prerequisite: MATH 6310, or permission of instructor.

MATH 6320 is the second of the two core algebra courses. It treats Galois theory, representation theory of finite groups and associative algebras, and an introduction to homological algebra. For the most part these subjects are not covered in depth, since the purpose of the course is to present a broad view with a glimpse of several topics.

Some topics to be covered:

Field theory and Galois theory — Field extensions, degree, splitting fields, algebraic closure, normal and separable extensions, fundamental theorem of Galois theory, solvability of equations by radicals, cyclotomic extensions, finite fields.

Homological algebra — Exact sequences, projective and injective modules, homological dimension, complexes, homology.

Representation theory of finite groups — Simple and semi-simple rings and modules, Wedderburn's theorem, group representations, Maschke's theorem, characters of finite groups, orthogonality relations, Frobenius reciprocity, applications to group theory.

### MATH 6330 - [Noncommutative Algebra]

Fall. Not offered: 2021-2022. Next offered: 2022-2023. 4 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 2021. 4 credits. Student option grading.

We will cover topics in graded rings, Hilbert functions, free resolutions, combinatorial commutative algebra, and computational algebra.

### MATH 6350 - Homological Algebra

Spring 2022. 4 credits. Student option grading.

Prerequisite: MATH 6310. Offered alternate years.

A first course in 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, Hochschild and cyclic homology of associative algebras, and Chevalley-Eilenberg (co)homology of Lie algebras.

### MATH 6370 - Algebraic Number Theory

Spring 2022. 4 credits. Student option grading.

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

Prerequisites: A course in modern algebra at the level of MATH 4340 either already or at least concurrently taken. A lower level algebra background than that will make much of the material hard to follow.

Topics: This course is a basic introduction to algebraic number theory. The core of it used to deal with the ideal theory of Dedekind domains as applied to the rings of integers of number fields (finite extensions of Q). A major purpose of the theory is to overcome the lack of unique factorisation into primes in these rings. A second parallel, and historically later, route runs through the theory of local fields which at first seems very different. The interweaving of the two approaches is particularly interesting. Because of lack of time we may focus more on the second, also because the first is treated extensively in many classic books (such as that by Samuels - see below). We will however 'do' the key theorems of ideal theory.

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

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

Grades: As there are no exams and no official homework (at most an occasional suggested problem) grades don't mean a whole lot. But as some grade has to be given my policy will be a semi-automatic A for graduate students and A- for undergraduates. Semi-automatic means there should be some evidence that you are attending and are interested in the material.

NOTE: Undergraduates will not be allowed to use this course to meet the algebra requirement for math majors. The reason is the lack of formal requirements (exams, hw).

### MATH 6390 - [Lie Groups and Lie Algebras]

Fall. Not offered: 2021-2022. Next offered: 2022-2023. 4 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

Spring 2022. 4 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, 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 2022. 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 2021. 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. MATH 6520 is an introduction to geometry and topology from a differentiable viewpoint, suitable for beginning graduate students. The objects of study are manifolds and differentiable maps. The collection of all tangent vectors to a manifold forms the tangent bundle and a section of the tangent bundle is a vector field. Alternatively vector fields can be viewed as first-order differential operators. This course studies flows of vector fields and prove the Frobenius integrability theorem. In the presence of a Riemannian metric, develops the notions of parallel transport, curvature and geodesics. Examines the tensor calculus and the exterior differential calculus and prove Stokes' theorem. If time permits, de Rham cohomology, Morse theory, or other optional topics are introduced.

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

Fall. Not offered: 2021-2022: Next offered: 2022-2023. 4 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 2022. 4 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: 2021-2022. Next offered: 2022-2023. 4 credits. Student option grading.

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. Not offered: 2021-2022. Next offered: 2022-2023. 4 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 2021. 4 credits. Student option grading.

Prerequisite: MATH 6510 or permission of instructor.

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

### MATH 6670 - Algebraic Geometry

Spring 2022. 4 credits. Student option grading.

This course is an introduction to basic algebraic geometry, at the level of Shafarevich's book: Basic Algebraic Geometry, vol 1. i.e. we focus on the geometry and not on the notions of scheme or sheaf.

We will cover many of the most important geometric concepts in algebraic geometry. Topics may include the following (or most of the following):

Chapter 1: Basics: affine and projective varieties, regular and rational maps, irreducible varieties, blowups, finite maps, closedness of maps, fibers of maps, products, dimension.

Chapter 2: local properties: tangent space, local rings, singularities, local parameters, Bertini's theorem, normal varieties.

Chapter 3: Divisors and differential forms, divisors on curves, Riemann-Roch for curves, structure of birational maps.

Chapter 4: Intersection theory, mainly for surfaces.

One aspect of this book that I very much like is that it doesn't blackbox the commutative algebra. It proves things from scratch. That said, we will not cover all of the algebra in class: If you have had MATH 6340, this should be fine. If not, you will want to read the algebraic proofs in the book.

Throughout, we will consider many examples (e.g. curves, surfaces, group actions, multi-projective spaces, Grassmannians, Segre and Veronese embeddings, secant loci, Calabi-Yau varieties), and we will use my computer algebra system Macaulay2 to help analyze and understand examples. We will see how to computationally deal with many of the concepts we consider. We will attempt to incorporate some important ideas from the theory of 3-folds and higher dimensional varieties as well (e.g Mori cone, nef cone, nef divisors). Many examples will come from Harris' book.

Prerequisites: MATH 6310 and MATH 6340. (Having MATH 6340 is very useful but as mentioned above, it is possible, with some extra reading and effort, to understand this material without this prerequisite.)

Homework: I will assign biweekly homework, which will not be handed in, but students will take turns presenting solutions. It is very important to actually do examples and problems, as this is the only way to really learn algebraic geometry!

### MATH 6710 - Probability Theory I

Fall 2021. 4 credits. Student option grading.

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

An introductory graduate course in Probability Theory. Core topics include measure theoretic probability, the law of large numbers, weak convergence, characteristic functions, and the central limit theory. Possible additional topics include Poisson processes, stable laws, large deviations and conditional expectation. (Conditional expectation is treated in detail in MATH 6720.)

### MATH 6720 - Probability Theory II

Spring 2022. 4 credits. Student option grading.

Prerequisite: MATH 6710.

The second part of a graduate introduction to probability, sequel to MATH 6710. Core topics include conditional expectation, martingales, Markov chains and their relation to martingales, Brownian motion. Time permitting, there will be an introduction to the basics of Ito calculus for Brownian motion.

### MATH 6730 - Mathematical Statistics I

(also STSCI 6730)

Spring 2022. credits.

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

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

### MATH 6740 - Mathematical Statistics II

(also STSCI 6740)

Fall 2021. 4 credits. Student option grading.

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

Some familiarity with basic statistical theory is assumed, i.e. with point estimation, hypothesis testing and confidence intervals, as well as with the concepts of Bayesian and minimax decisions. The course focuses on the modern theory of statistical inference, with an emphasis on nonparametric and asymptotic methods. In finding optimal decisions, a pivotal role will be played by the concept of asymptotic minimaxity. A tentative list of topics is (with chapter numbers in the textbook): (1) Fisher efficiency (recap), (2) Bayes and minimax estimators (recap), (3) asymptotic minimaxity, (4) some irregular statistical experiments, (8) estimation in nonparametric regression, (9) local polynomial approximation of the regression, (10) estimation of regression in global norms, (12) asymptotic optimality in global norms, (13) estimation of functionals, (15) adaptive estimation.

### MATH 6810 - [Logic]

Fall or Spring. Not offered: 2021-2022. Next offered: 2022-2023. 4 credits. Student option grading.

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

Spring 2022. 4 credits. Student option grading.

Offered alternate years.

This course will give a graduate-level introduction to model theory, focusing on examples in algebra and geometry. It will follow Tent and Ziegler's text. Topics covered will include the compactness and completeness theorems for first-order logic, Henkin's construction, Lowenheim-Skolem Theorems, Vaught's criteria for completeness, back and forth arguments, realizing and omitting types, spaces of types, stability, Vaughtian pairs, and indiscernibles. The course will culminate in the Baldwin-Lachlan proof of Morley's theorem.

### MATH 6840 - Recursion Theory

Fall 2021. 4 credits. Student option grading.

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: 2021-2022. Next offered: 2022-2023. 4 credits. S/U grades only.

Offered alternate years.

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

### MATH 7110 - Topics in Analysis: Lectures on the "Helicoidal Method"

Fall 2021. 4 credits. S/U grades only.

This will be an advanced course in Harmonic Analysis. The scope of it is to describe in detail what we called the "Helicoidal Method”, an iterated, and extremely robust technique, which provides in particular, new paradigms for proving (multiple) vector valued, sparse domination and mixed norm estimates for many (if not most) of the multi-quasi-linear operators of interest in Harmonic Analysis. It has been developed in the last eight years (or so), in collaboration with Cristina Benea, who was a graduate student here at Cornell, and graduated in May 2015.

The presentation will be as self contained as possible, but familiarity with the basic theories of Harmonic Analysis, should clearly be of help. I envision this class as being a class for people who like ANALYSIS, even if their particular research interests lie in distinct areas, such as PDE, Mathematical Physics, Functional Analysis, Complex Analysis, Geometric Measure Theory, Calculus of Variations, Spectral Theory, etc.

### MATH 7120 - Topics in Analysis: Geometry and Analysis of Four-Manifolds

Spring 2022. 4 credits. S/U grades only.

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

### MATH 7130 - Functional Analysis

Fall 2021. 4 credits. Student option grading.

Offered alternate years.

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

### MATH 7150 - [Fourier Analysis]

Fall. Not offered: 2021-2022. Next offered: 2022-2023. 4 credits. S/U grades only.

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: Nonlinear Elliptic PDE

Fall 2021. 4 credits. S/U grades only.

Topics from: calculus of variations (direct methods) and bifurcation theory (local and degree-theoretic methods).

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

Fall or Spring. Not offered: 2021-2022. Next offered: 2022-2023. 4 credits. S/U grades only.

Selection of advanced topics from numerical analysis. Content varies.

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

Spring. Not offered: 2021-2022. Next offered: 2022-2023. 4 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 2021, Spring 2022. 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]

Fall or Spring. Not offered: 2021-2022. Next offered: 2022-2023. 4 credits. S/U grades only.

Selection of advanced topics from algebra. Course content varies.

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

Fall or Spring. Not offered: 2021-2022. Next offered: 2022-2023. 4 credits. S/U grades only.

Offered alternate years.

Selection of advanced topics from homological algebra. Course content varies.

### MATH 7370 - Topics in Algebraic Number Theory

Fall 2021. 4 credits. S/U grades only.

The aim of this course is to be able to appreciate Ozaki's Theorem from 2011 in which he proved that for every finite p-group G, there exists a number field K whose p-Hilbert Class field Tower is G. Most of the course will be spent understanding the statements of and gaining facility with the necessary background material in class field theory.

This course presupposes some background in Number Theory, around the level of MATH 6370.

### MATH 7390 - Topics in Lie Groups and Lie Algebras: Symplectic Resolutions

Spring 2022. 4 credits. S/U grades only.

"Symplectic resolutions" are a certain family of holomorphic symplectic manifolds with many amazing properties. There are a few general constructions, each of which we will study in detail:

- hypertoric varieties, which are to hyperplane arrangements as toric varieties are to polytopes;
- Nakajima quiver varieties, which provide a "geometric representation theory" for simply-connected Kac-Moody algebras;
- slices in affine Grassmannians, which provide a "geometric representation theory" for finite-dimensional Lie groups;
- Slodowy slices to nilpotent orbit closures

(The latter three all match up, in type A, but diverge for other groups.)

Our main example will be the cotangent bundle T^* G/B to a flag variety. Two theorems of Namikawa I plan to get to:

- If the affinization of a symplectic resolution M is a complete intersection (plus a couple of other minor conditions), there is a unique reductive Lie algebra g for which M = T^* G/B. (Not for nothing has Okounkov called them "the Lie algebras of the 21st century").
- If one bounds the number and degrees of the defining equations (of the affinization), there are only finitely many symplectic resolutions up to isomorphism. (Can one hope for a classification??)

We will only scratch this burgeoning subject. The prerequisites are a solid knowledge of representation theory, and some knowledge of algebraic geometry.

### MATH 7410 - [Topics in Combinatorics]

Spring. Not offered: 2021-2022. Next offered: 2022-2023. 4 credits. S/U grades only.

Offered alternate years.

Selection of advanced topics in combinatorics. Course content varies.

### MATH 7510 - Berstein Seminar in Topology: Anasov Flows on 3-Manifolds

Fall 2021. 4 credits. S/U grades only.

We'll study the dynamical properties of geodesic flow on surfaces, and more generally Anosov flows on 3-manifolds. The techniques come from geometric/low-dimensional topology, hyperbolic geometry, and smooth dynamics. There are many interesting constructions of examples and existence and abundance results. We'll take a particularly close look at examples on hyperbolic 3-manifolds, and related "quasi-geodesic" flows, with an aim towards understanding some important open questions. As with all Berstein seminars, students will present the bulk of the material and will have some say in which subtopics we cover.

Prerequisites: At minimum, the first year differentiable manifolds course or equivalent knowledge. Ideally, you should know what a hyperbolic 3-manifold is. Last year's Berstein seminar on 3-manifolds is great preparation but not required (but if you were there, we may see some beautiful connections with the Thurston norm).

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

Spring. Not offered: 2021-2022. Next offered: 2022-2023. 4 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 7550 - Topology and Geometric Group Theory Seminar

Fall 2021. 4 credits. S/U grades only.

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

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

Spring 2022. 4 credits. S/U grades only.

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

### MATH 7580 - Topics in Topology

Spring 2022. 4 credits. S/U grades only.

This course will be about the Atiyah-Singer index theorem. This theorem describes a beautiful link between topology, differential geometry, and analysis. One instance of the theorem is the Chern-Gauss-Bonnet theorem, which says that the Euler characteristic of a compact manifold can be computed by the integral of the Euler class over the manifold; by Hodge theory the Euler characteristic is the same as the index of the de Rham-Dirac operator (these things will be explained in the course). One of the end goals will be to understand a refinement (only for Dirac operators) of the original theorem called the local index theorem. Time permitting we will study generalizations to the equivariant and families settings.

Prerequisites: I will assume familiarity with topics covered in MATH 6520 (manifolds, differential forms, de Rham cohomology, Riemannian metrics, etc.). Prior exposure to bundles, connections and curvature would be an asset, as I plan for the discussion of these topics to be brief.

### MATH 7610 - Topics in Geometry: Comparison Geometry and Minimal Surfaces

Fall 2021. 4 credits. S/U grades only.

The first part of this course will be a continuation of Math 6620: Riemannian Geometry offered in Spring 2021. We will discuss several classical comparison theorems. The second part will be a brief introduction to minimal surface theory. The topics will include the first and second variation formula, monotonicity formula, maximum principle, and classical curvature estimates. In time permits, we will discuss the existence theory and applications.

### MATH 7620 - Topics in Geometry: Universal Rigidity and Applications

Spring 2022. 4 credits. S/U grades only.

This is an introduction to the theory of rigid frameworks in general, including tensegrities, universal rigidity, and its application to engineering structures and a recent exciting application to finding the maximum likelihood threshold for maximum likelihood estimators in statistics. A framework is just a set of points in a Euclidean space, a configuration, where some pairs of points, the edges of a graph, are constrained to be at a fixed distance. Universal rigidity is where those distance constraints determine the configuration up to a global motion, in any higher dimension. There is a kind of upside-down relationship between the existence of a universally rigid configuration for a graph in a given dimension and the maximum likelihood threshold.

### MATH 7670 - Topics in Algebraic Geometry: Non-Archimedean Geometry

Fall 2021. 4 credits. S/U grades only.

Once a specialized subject, Non-Archimedean Geometry has grown into a foundational part of algebraic and arithmetic geometry. Its goal is to extend concepts and results about complex analytic spaces to spaces defined over non-archimedean fields, such as the fields of p-adic numbers or the field of formal Laurent series. For instance, Tate originally formulated the theory to extend the uniformization of complex elliptic curves, in which a curve is described as the quotient of the group of multiplicative complex numbers by a free subgroup, to an analogous construction for elliptic curves over more general non-archimedean fields. The general theory of non-archimedean spaces has since been developed by Raynaud, Berkovich, Huber, and many others, and it admits several related formulations. It has been applied in many contexts: the Langlands correspondence, the topology of algebraic varieties in higher dimensions, mirror symmetry, birational geometry, p-adic Hodge theory, modularity of Galois representations, etc..

This course will serve as an introduction to non-archimedean geometry, focusing mainly on the approaches of Raynaud and Berkovich. We will define non-archimedean spaces and study their basic properties. Then we will discuss the theories of coherent cohomology and etale cohomology on these spaces.

### MATH 7710 - Topics in Probability Theory: Limits of Discrete Random Structures

Spring 2022. 4 credits. S/U grades only.

There are many ways to take a limit of a discrete structure (such as a random graph, spanning tree, or coloring). We'll explore some of these topics:

Random Walks and Electrical Networks, Harmonic Functions, Gaussian Free Field, Scaling Limits, Infinite Volume Limits, Graph Limits (Lovasz and Benjamini-Schramm), Uniform Spanning Forest, Minimal Spanning Forest, Branching Processes, Continuum Random Tree, Concentration Inequalities, Random Graphs, Percolation, Random Cluster Model.

Background: To follow this course you'll need to know at least one semester of graduate probability at the level of MATH 6710. Familiarity with Martingales and Brownian Motion at the level of MATH 6720 is a big plus!

We will follow the book Probability on Trees and Networks, by Lyons and Peres: https://rdlyons.pages.iu.edu/prbtree/prbtree.html

I expect to cover chapters 2,4,5,6,7,9,10,11,12.

We may also cover parts of these books!:

The Random Cluster Model, by Grimmett: http://www.statslab.cam.ac.uk/~grg/books/rcm1-1.pdf

Random Graphs, by Janson, Luczak, and Rucinski: https://onlinelibrary.wiley.com/doi/book/10.1002/9781118032718

Large Networks and Graph Limits, by Lovasz: https://web.cs.elte.hu/~%20lovasz/bookxx/hombook-almost.final.pdf

### MATH 7720 - Topics in Stochastic Processes: Random Walks on Groups

Fall 2021. 4 credits. S/U grades only.

This course will provide an introduction to modern development in the study of random walks on groups. A good part of the course will be devoted to basics of random walk on groups. The second part will deal with applications of these techniques to particular classes of groups: nilpotent, polycyclic, etc. No deep knowledge of group theory is needed.

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

Fall. Not offered: 2021-2022. Next offered: 2022-2023. 4 credits. Student option grading.

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

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

### MATH 7810 - Seminar in Logic: The Logic of Thompson's Groups and their Relatives

Fall 2021. 4 credits. S/U grades only.

The logic seminar consists of lectures presented by students on a single topic, supplemented by lectures by faculty, outside speakers, and students presenting their own research. In the Fall 2021 semester, the topic component of the seminar will be "The logic of Thompson's groups and their relatives." Students enrolled in the course are expected to present 2-3 lectures on the topic.

Description of the seminar topic: While a graduate student in logic at Berkeley in the 1960s and 70s, Richard Thompson introduced three groups now known as $F$, $T$, and $V$. All three are finitely presented groups of homeomorphisms of the Cantor set. $T$ and $V$ were early examples of finitely presented infinite simple groups while $F$ is perhaps the simplest example of an nonelementary amenable group which does not contain the free group on two generators.

Since the 1980s, these groups have played an important role in group theory and topology due largely to their exotic properties. More recently, their study has touched on different parts of logic and set theory. This semester will give an introduction to these groups, as well as their relatives such as the group of piecewise linear and piecewise projective homeomorphisms of the unit interval. The focus will be on their interaction with logic, ranging from set theory to proof theory to automata theory. The seminar will also highlight a number of open problems and areas of ongoing research.

### MATH 7820 - Seminar in Logic

Spring 2022. 4 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. Not offered: 2021-2022. Next offered: 2022-2023. 4 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).