# 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 2020 Offerings

Descriptions are included under Course Descriptions. Meeting days and times were updated August 12th.

MATH 6110 - Real Analysis

Phil Sosoe, TR 9:55-11:10

MATH 6210 - Measure Theory and Lebesgue Integration

Terence Harris, TR 9:55-11:10

MATH 6260 - Dynamical Systems

William Clark, MWF 8:00-8:50

MATH 6310 - Algebra

David Zywina, MW 1:25-2:40

MATH 6330 - Noncommutative Algebra

Marcelo Aguiar, MW 11:30-12:45

MATH 6340 - Commutative Algebra with Applications in Algebraic Geometry

Irena Peeva, TR 1:25-2:40

MATH 6520 - Differentiable Manifolds

Jim West, MW 8:00-9:15

MATH 6530 - K-Theory and Characteristic Classes

Inna Zakharevich, MWF 1:50-2:40

MATH 6630 - Symplectic Geometry

Tara Holm, TR 11:30-12:45

MATH 6710 - Probability Theory I

Lionel Levine, TR 11:30-12:45

MATH 6740 - Mathematical Statistics II

Michael Nussbaum, TR 11:30-12:45

MATH 6870 - Set Theory

Slawomir Solecki, MWF 1:50-2:40

MATH 7150 - Fourier Analysis

Camil Muscalu, TR 9:55-11:10

MATH 7510 - Berstein Seminar in Topology: Three-Dimensional Manifolds

Jason Manning, MWF 12:40-1:30

MATH 7610 - Topics in Geometry: Complex Geometry

John Hubbard, TR 8:00-9:15

MATH 7670 - Topics in Algebraic Geometry: Hilbert Schemes

Michael Stillman, MWF 10:20-11:10

MATH 7710 - Topics in Probability Theory: Topic TBA

Lionel Levine, TR 1:25-2:40

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

Marten Wegkamp, MW 9:55-11:10

MATH 7810 - Seminar in Logic

Justin Moore, TF 3:00-4:15

## Spring 2021 Offerings

Descriptions are included under Course Descriptions.

MATH 6120 - Complex Analysis

Justin Moore, TR 11:25-12:40

MATH 6220 - Applied Functional Analysis

Federico Fuentes, TR 2:45-4:00

MATH 6270 - Applied Dynamical Systems

Alice Nadeau, TR 9:40-10:55

MATH 6320 - Algebra

Nicolas Templier, MW 11:25-12:40

MATH 6370 - Algebraic Number Theory

Shankar Sen, MW 1:00-2:15

MATH 6390 - Lie Groups and Lie Algebras

Dan Barbasch, TR 9:40-10:55

MATH 6510 - Algebraic Topology

Inna Zakharevich, MW 9:40-10:55

MATH 6620 - Riemannian Geometry

Xin Zhou, TR 9:40-10:55

MATH 6720 - Probability Theory II

Lionel Levine, MW 1:00-2:15

MATH 6730 - Mathematical Statistics I

F. Bunea, T 11:20-1:45

MATH 6810 - Logic

Anil Nerode, TR 1:00-2:15

MATH 7120 - Topics in Analysis: Harnack Inequalities via Nash-Moser Iteration

Laurent Saloff-Coste, MWF 8:00-8:50

MATH 7280 - Topics in Dynamical Systems: Asymptotics and Perturbation Methods

Steve Strogatz, TR 1:00-2:15

MATH 7350 - Topics in Homological Algebra: Homotopical Algebra, Homotopy (Co)limits and Infinity-Categories

Yuri Berest, MW 1:00-2:15

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

Harrison Chen, MWF 11:20-12:10

MATH 7410 - Topics in Combinatorics: Matroids

Ed Swartz, MW 11:25-12:40

MATH 7520 - Berstein Seminar in Topology: Derived and Spectral Algebraic Geometry

Dan Halpern-Leistner, TR 11:25-12:40

MATH 7580 - Topics in Topology: Stable Homotopy Theory

J. D. Quigley, MW 9:40-10:55

MATH 7720 - Topics in Stochastic Processes: Critical Percolation and Planar Scaling Limits

Phil Sosoe, MW 9:40-10:55

MATH 7820 - Seminar in Logic

Slawomir Solecki, TF 2:45-4:00

## Course Descriptions

### MATH 6110 - Real Analysis

Fall 2020. 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 is an introduction to graduate real analysis. Subjects covered include abstract integration theory, Fourier series, and basic functional analysis. The theory will be illustrated with examples from partial differential equations and probability theory.

Textbook: Selected Chapters from Stein and Sakarchi (3 Books): Princeton Lectures in Analysis - Fourier Analysis, Real Analysis and Functional Analysis.

### MATH 6120 - Complex Analysis

Spring 2021. 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.

This course will be based on John Conway's text "Functions of one complex variable", chapters III-XII: power series, complex integration, classification of singularities, the Maximum Modulus Theorem, compactness and convergence in the space of analytic functions, the Riemann Mapping Theorem, Runge's theorem, analytic continuation and Riemann surfaces, harmonic functions, entire functions and their range. Time permitting we may cover additional topics outside the text such as the Prime Number Theorem, complex dynamics, Fourier analysis, and/or distribution theory.

### MATH 6150 - [Partial Differential Equations]

Fall. Not offered: 2020-2021. Next offered: 2021-2022. 4 credits. Student option grading.

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

This course emphasizes the "classical" aspects of partial differential equations (PDEs). The usual topics include fundamental solutions for the Laplace/Poisson, heat and and wave equations in *R ^{n}*, mean-value properties, maximum principles, energy methods, Duhamel's principle, and an introduction to nonlinear first-order equations, including shocks and weak solutions. Additional topics may include Hamilton-Jacobi equations, distribution theory, and the Fourier transform.

### MATH 6160 - [Partial Differential Equations]

Spring. Not offered: 2020-2021. Next offered: 2021-2022. 4 credits. Student option grading.

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

This course highlights applications of functional analysis to the theory of partial differential equations (PDEs). It covers parts of the basic theory of linear (elliptic and evolutionary) PDEs, including Sobolev spaces, existence and uniqueness of solutions, interior and boundary regularity, maximum principles, and eigenvalue problems. Additional topics may include: an introduction to variational problems, Hamilton-Jacobi equations, and other modern techniques for non-linear PDEs.

### MATH 6210 - Measure Theory and Lebesgue Integration

Fall 2020. 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.

Textbook: Bartle, Robert, The Elements of Integration and Lebesgue Measure, John Wiley & Sons, 1966 (ISBN: 0-471-04222-6) — free electronic access provided by the Cornell Math Library

### MATH 6220 - Applied Functional Analysis

Spring 2021. 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: 2020-2021. 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 2020. 4 credits. Student option grading.

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.

Textbooks: Katok and Hasselblat, Introduction to the Modern Theory of Dynamical Systems (paperback), Cambridge University Press, 1996 (ISBN: 0-521-57557-5). Arnold, V. I., Geometrical Methods in the Theory of Ordinary Differential Equations, Springer-Verlag, 1988 (Edition: 2; ISBN: 0-387-96649-8).

### MATH 6270 - Applied Dynamical Systems

(also MAE 7760)

Spring 2021. 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 STEM fields.

### MATH 6280 - [Complex Dynamical Systems]

Fall or Spring. Not offered: 2020-2021. Next offered: 2021-2022. 4 credits. Student option grading.

Prerequisite: MATH 4180.

Various topics in the dynamics of analytic mappings in one complex variable, such as: Julia and Fatou sets, the Mandelbrot set, Mañé-Sad-Sullivan's theorem on structural stability. Also covers: local theory, including repulsive cycles and the Yoccoz inequality, parabolic points and Ecalle-Voronin invariants, Siegel disks and Yoccoz's proof of the Siegel Brjuno theorem; quasi-conformal mappings and surgery: Sullivan's theorem on non-wandering domains, polynomial-like mappings and renormalization, Shishikura's construction of Hermann rings; puzzles, tableaux and local connectivity problems; and Thurston's topological characterization of rational functions, the spider algorithm, and mating of polynomials.

### MATH 6310 - Algebra

Fall 2020. 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 2021. 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 2020. 4 credits. Student option grading.

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

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.

Textbook: Two important sources are: (1) Graduate Algebra: Noncommutative View by Louis Halle Rowen, (2) Noncommutative Algebra, by Benson Farb and R. Keith Dennis. We will also rely on class notes.

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

Fall 2020. 4 credits. Student option grading.

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

### MATH 6350 - [Homological Algebra]

Spring. Not offered: 2020-2021. Next offered: 2021-2022. 4 credits. Student option grading.

Prerequisite: MATH 6310

A first course on homological algebra. Topics will include a brief introduction to categories and functors, chain and cochain complexes, operations on complexes, (co)homology, standard resolutions (injective, projective, flat), classical derived functors, Tor and Ext, Yoneda’s interpretation of Ext, homological dimension, rings of small dimensions, introduction to group cohomology.

### MATH 6370 - Algebraic Number Theory

Spring 2021. 4 credits. Student option grading.

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

This course is a basic introduction to algebraic number theory. The core of it deals with the ideal theory of Dedekind domains as applied to the rings of integers of number fields (finite extensions of Q). A major purpose of the theory is to overcome the lack of unique factorization into primes in these rings.

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

Note: As it is usually a small class, the topics may be modified a bit to suit the backgrounds of those attending. In the catalog, 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.

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

### MATH 6390 - Lie Groups and Lie Algebras

Spring 2021. 4 credits. Student option grading.

Lie groups, Lie algebras and their representations play an important role in much of mathematics, particularly in number theory, mathematical physics and topology. This is an introductory course, meant to be useful for more advanced topics and applications.

Prerequisites: The prerequisites are a basic knowledge of algebra and linear algebra at the honors undergraduate level. Some knowledge of differential and algebraic geometry are helpful. We will highlight the relation between Lie groups and Lie algebras throughout the course. There is no one textbook we will follow. Most of the references are available in the Cornell library, and in electronic form.

Topics: The starred ones are tentative.

- Basic structure and properties of Lie algebras; theorems of Lie and Engel.
- Nilpotent solvable and reductive Lie algebras.
- The relation between Lie groups and Lie algebras
- The algebraic groups version*
- Enveloping algebras and differential operators
- The structure of semisimple algebras
- Representation theory of semisimple Lie algebras; Lie algebra cohomology
- Compact semisimple groups and their representation theory.
- Chevalley groups, p-adic groups*
- Structure of real reductive groups
- Quantum groups, Kac-Moody algebras and their representations theory*

References:

- N. Bourbaki, Groupes et algebres de Lie, Hermann, Paris, 1971
- V. Chari and A. Pressley, A guide to quantum groups
- J. Dixmier, Enveloping algebras
- S. Helgason, Differential geometry, Lie groups and symmetric spaces, Academic Press, New York, 1978.
- N. Jacobson, Lie algebras
- J. Humphreys, Introduction to Lie algebras and representation theory
- V. Kac, Infinite dimensional Lie algebras
- J-P. Serre, Complex semisimple Lie algebras an advanced undergraduate level. Lie algebras 649 is also very useful.
- A. L. Onishchik, E. B. Vinberg, Lie Groups and Algebraic Groups, Springer-Verlag, Berlin, Heidelberg, 1990.
- V. S. Varadarajan, Lie Groups, Lie Algebras, and their Representations, Prentice-Hall, Engelwood Cliffs, NJ, 1974.
- F. Warner, Foundations of Differentiable manifolds and Lie groups, Scott, Foresman and Co., Glenview, IL, 1971.
- W. Rossmann, Lie groups: An Introduction Through Linear Groups, Oxford Graduate Texts in Mathematics, Number 5, Oxford University Press, 2002; ISBN 0198596839
- J.J. Duistermaat & J.A.C. Kolk, Lie groups, Universitext serie, Springer-Verlag, New York, 2000. ISBN 3-540-15293-8, cat prijs DM 79.
- T. Br ́’ocker & T. tom Dieck, Representations of compact Lie groups, Springer-Verlag, New York, 1985.

### MATH 6410 - [Enumerative Combinatorics]

Spring. Not offered: 2020-2021. Next offered: 2021-2022. 4 credits. Student option grading.

Prerequisite: MATH 4410 or permission of instructor.

An introduction to enumerative combinatorics from an algebraic, geometric and topological point of view. Topics include, but are not limited to, permutation statistics, partitions, generating functions, various types of posets and lattices (distributive, geometric, and Eulerian), Möbius inversion, face numbers, shellability, and relations to the Stanley-Reisner ring.

### MATH 6510 - Algebraic Topology

Spring 2021. 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 2020. 4 credits.

Prerequisite: strong performance in analysis (e.g., MATH 4130 and/or MATH 4140), linear algebra (e.g., MATH 4310), and point-set topology (e.g., MATH 4530), or permission of instructor.

MATH 6510-MATH 6520 are the core topology courses in the mathematics graduate program. MATH 6520 is an introduction to geometry and topology from a differentiable viewpoint, suitable for beginning graduate students. Topics include differentiable manifolds and maps, tangent and cotangent bundles, submanifolds and normal bundles, vector fields and Lie brackets, Sard's Theorem, Whitney's embedding theorem, integral curves and flows, Lie derivatives, Frobenius' Theorem on integrability of k-plane distributions, differential forms, integration, and De Rham cohomology. We will follow John Lee's Introduction to Smooth Manifolds (Springer GTM 218).

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

Fall 2020. 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]

Fall or Spring. Not offered: 2020-2021. Next offered: 2021-2022. 4 credits.

Prerequisite: MATH 6510 or permission of instructor.

This course is an introduction to some of the fundamentals of homotopy theory. Homotopy theory studies spaces up to homotopy equivalence, not just up to homeomorphism. This allows for a variety of algebraic techniques which are not available when working up to homeomorphism. This class studies the fundamentals and tools of homotopy theory past homology and cohomology. Topics may include computations of higher homotopy groups, simplicial sets, model categories, spectral sequences, and rational homotopy theory.

### MATH 6620 - Riemannian Geometry

Spring 2021. 4 credits. Student option grading.

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 2020. 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.

Textbook: Combination of: Lectures on Symplectic Geometry by Ana Cannas da Silva; Introduction to symplectic topology (3rd ed) by Dusa McDuff and Dietmar Salamon

### MATH 6640 - [Hyperbolic Geometry]

Fall. Not offered: 2020-2021. Next offered: 2021-2022. 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]

Fall or Spring. Not offered: 2020-2021. Next offered: 2021-2022. 4 credits. Student option grading.

Prerequisite: MATH 6310 or MATH 6340, or equivalent.

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

### MATH 6710 - Probability Theory I

Fall 2020. 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.

A mathematically rigorous development of probability theory from a measure-theoretic perspective. Topics include: independence, the law of large numbers, Poisson and central limit theorems, and random walks.

Textbook: Durrett, Probability: Theory & Examples

### MATH 6720 - Probability Theory II

Spring 2021. 4 credits. Student option grading.

Prerequisite: MATH 6710.

Conditional expectation, martingales, Brownian motion. Other topics such as Markov chains, ergodic theory, and stochastic calculus depending on time and interests of the instructor.

### MATH 6730 - Mathematical Statistics I

(also STSCI 6730)

Spring 2021. 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 2020. 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.

Textbook: Korostelev, A., Korosteleva, O., Mathematical Statistics. Asymptotic Minimax Theory. American Mathematical Society, 2011. Available as an electronic resource in the Cornell Library.

### MATH 6810 - Logic

Spring 2021. Offered alternate years. Next offered: 2022-2023. 4 credits. Student option grading.

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. Not offered: 2020-2021. Next offered: 2021-2022. 4 credits. Student option grading.

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

### MATH 6840 - [Recursion Theory]

Fall. Not offered: 2020-2021. Next offered: 2021-2022. 4 credits. Student option grading.

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 2020. Offered alternate years. Next offered: 2022-2023. 4 credits. S/U grades only.

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

Literature:

- A.S. Kechris, Classical Descriptive Set Theory, Springer, 1995.
- B.D. Miller, The graph-theoretic approach to descriptive set theory, Bull. Symbolic Logic
**18**(2012), 554–575.

### MATH 7110 - [Topics in Analysis]

Fall. Not offered: 2020-2021. Next offered: 2021-2022. 4 credits. S/U grades only.

Selection of advanced topics from analysis. Course content varies.

### MATH 7120 - Topics in Analysis: Harnack Inequalities via Nash-Moser Iteration

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

Prerequisite: 6110 or equivalent; Graduate introduction to PDEs is a plus but not required.

We discuss the relationships between elliptic and parabolic Harnack inequalities and Poincaré and Sobolev inequalities in the context of uniformly elliptic second-order differential operators on Euclidean domains and Riemannian manifolds. This is an important topic in the study of the related heat equation and its solutions.

We will use the book Aspects of Sobolev-Type Inequalities which is accessible online via the Cornell library.

Another useful reference is Gilbarg-Trudinger Elliptic Partial Differential Equations of Second Order, also available online via the Cornell library.

### MATH 7130 - [Functional Analysis]

Fall or Spring. Not offered: 2020-2021. Next offered: 2021-2022. 4 credits. Student option grading.

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

### MATH 7150 - Fourier Analysis

Fall 2020. Offered alternate years. Next offered: 2022-2023. 4 credits. S/U grades only.

An introduction to (mostly Euclidean) harmonic analysis. Topics usually include convergence of Fourier series, harmonic functions and their conjugates, Hilbert transform, Calderon-Zygmund theory, Littlewood-Paley theory, pseudo-differential operators, restriction theory of the Fourier transform, connections to PDE. Applications to number theory and/or probability theory may also be discussed, as well as Fourier analysis on groups.

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

Fall or Spring. Not offered: 2020-2021. Next offered: 2021-2022. 4 credits. S/U grades only.

Selection of advanced topics from numerical analysis. Content varies.

### MATH 7280 - Topics in Dynamical Systems: Asymptotics and Perturbation Methods

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

Asymptotics and perturbation methods are clever techniques for finding approximate analytical solutions to complicated problems, exploiting the presence of a large or small parameter. This course is an introduction to such methods and their applications in various branches of science and engineering. Analytical methods and concrete examples will be emphasized. Topics include asymptotic expansion of integrals via Laplace's method, stationary phase, steepest descent, and saddle points. Perturbation methods for differential equations include dominant balance, boundary layer theory, multiple scales, and WKB theory.

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

(also CS 7290)

Fall 2020, Spring 2021. 1 credits.

Talks on various methods in scientific computing, the analysis of their convergence properties and computational efficiency, and their adaptation to specific applications.

### MATH 7310 - [Topics in Algebra]

Fall or Spring. Not offered: 2020-2021. Next offered: 2021-2022. 4 credits. S/U grades only.

Selection of advanced topics from algebra. Course content varies.

### MATH 7350 - Topics in Homological Algebra: Homotopical Algebra, Homotopy (Co)limits and Infinity-Categories

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

The course will consist of two parts. In Part I, we will begin with an overview of 'classical' homotopical algebra (Quillen's theory of model categories) and discuss some of its applications. Our intention is to give a fairly thorough introduction to the theory (and practice) of homotopy limits and colimits. Besides classical homotopy theory (wherefrom these concepts originate), they play an important role in many areas of mathematics. As basic examples, we will look at applications of homotopy colimits in derived algebraic geometry (simplicial presheaves on derived affine schemes) and geometric topology (a construction of link homology theories, such as Khovanov homology and knot contact homology, in terms of homotopy (co)limits of dg-categories).

In Part II, we will try to give a gentle introduction to the theory of (\infty, 1)-categories which provides a 'mild' higher categorical extension of homotopical algebra. Again, we will try to organize our discussion around the concepts of (homotopy) limits and colimits in the framework of infinity-categories (in particular, we will look at 'realizable' homotopy colimits in the Barwick-Kan 2-category of relative categories and show how naturally they refine the classical constructions of homotopy colimits in the sense of Quillen, Grothendieck and Voevodsky).

Though the level of the course will be inevitably uneven, an effort will be made to make the discussion self-contained and accessible to the beginning graduate students.

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

Fall or Spring. Not offered: 2020-2021. Next offered: 2021-2022. 4 credits. S/U grades only.

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

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

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

Prerequisites: representation theory of Lie algebras (Vermas modules, highest weights), some algebraic topology (homology and cohomology), some algebraic geometry (sheaves).

We will cover the three categories:

- Category O of highest weight representations of a semisimple Lie algebra
- The category of perverse sheaves, i.e. certain constructible abelian sheaves, on the flag variety
- The category of D-modules on the flag variety

These are related by the Riemann-Hilbert and Beilinson-Bernstein theorems and were used to prove the Kazhdan-Lusztig conjectures. Our goal will be to gain a working knowledge of the theory and work out the example of SL2 and SL3.

### MATH 7410 - Topics in Combinatorics: Matroids

Spring 2021. Offered alternate years. Next offered: 2022-2023. 4 credits. S/U grades only.

Matroids are a combinatorial abstraction of linear independence and were introduced by Whitney in his 1935 paper, “On the abstract properties of linear dependence.” Since then the subject has grown enormously. We will start with the basic concepts and structures involved, assuming nothing more than linear algebra. Along the way we will see several applications including the topology of the complements of hyperplane arrangements (real, complex and finite), reliability and other aspects of topology, algebra and combinatorics. The latter part of the course will depend on the audience with possibilities including the explosion of variations of matroids and the applications of these variations, to the recent spectacular resolutions of some of the oldest and well known problems in geometric combinatorics.

### MATH 7510 - Berstein Seminar in Topology: Three-Dimensional Manifolds

Fall 2020. 4 credits. S/U grades only.

We will discuss a selection of topics in or inspired by three-dimensional topology. We will look at a mixture of classic results (such as the loop and sphere theorems) and recent breakthroughs. Possible areas of exploration include (but are not limited to) recent work of Markovic on the Simple Loop Conjecture, the Agol-Wise program proving the virtual Haken Conjecture, and Agol's Fibering Criterion (recently extended by Kielak to a group-theoretic context). The exact areas of current research covered will depend in part on the interests of the attendees.

This is a seminar-style course, in which the participants take turns presenting material.

### MATH 7520 - Berstein Seminar in Topology: Derived and Spectral Algebraic Geometry

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

Homotopical methods have had an impact on many subjects in algebraic geometry (deformation theory, moduli theory, mathematical physics, enumerative geometry, geometric Langlands, p-adic Hodge theory, etc.). In the other direction, ideas from algebraic geometry have been fruitfully applied in homotopy theory (topological modular forms, Morava K-theory, topological Hochschild and cyclic homology, etc.). The idea of unifying algebraic geometry and homotopy theory into a theory of derived algebraic geometry has been in the air since the 1980's, but the vision could not be fully realized due to the lack of a firm foundation for the theory of infinity-categories. That problem was solved by Lurie and Toen-Vezzosi in the second half of the 2000's. As a result, there has been a tremendous amount of activity and progress in the subject in recent years.

Participants in this seminar will take turns presenting topics in the foundations of derived algebraic geometry. We will discuss the theory of infinity-categories, and then we will work through Jacob Lurie’s thesis, culminating in the proof of the derived version of Artin’s representability criterion. The thesis works in the derived setting (i.e., with simplicial commutative rings), but we also discuss results in the spectral setting as well (i.e., working with E_\infty ring spectra). From there, we will cover more recent developments in the subject, depending on time and interest.

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

Fall 2020. 4 credits.

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

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

Spring 2021. 4 credits.

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

### MATH 7570 - [Topics in Topology]

Fall. Not offered: 2020-2021. Next offered: 2021-2022. 4 credits. S/U grades only.

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

### MATH 7580 - Topics in Topology: Stable Homotopy Theory

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

Prerequisites: graduate-level algebraic topology (e.g., MATH 6510). Familiarity with manifolds, vector bundles, and category theory would also be helpful.

Stable homotopy theory is the study of highly connected topological spaces using generalized cohomology theories, such as singular cohomology and topological K-theory. In this course, we will cover the following topics:

- The stable homotopy category, including spectra, smash products, duality, and examples.
- Classical topics, such as vector fields on spheres, the J homomorphism, and the Adams spectral sequence.
- (time permitting) Modern topics, such as algebraic K-theory, chromatic homotopy theory, or equivariant and motivic stable homotopy theory.

### MATH 7610 - Topics in Geometry

Fall 2020. 4 credits. S/U grades only.

Selection of advanced topics from modern geometry. Content varies.

### MATH 7620 - [Topics in Geometry]

Fall or Spring. Not offered: 2020-2021. Next offered: 2021-2022. 4 credits. S/U grades only.

Selection of advanced topics from modern geometry. Content varies.

### MATH 7670 - Topics in Algebraic Geometry

Fall 2020. 4 credits. S/U grades only.

This course will be an introduction to Hilbert schemes. We will focus on Hilbert schemes parametrizing subschemes of projective space. We will use deformation theory, Groebner deformations, generic initial ideals, and other methods, to understand the combinatorial structure of these schemes (e.g. that there is always a smooth point, connectedness, and the radius of the graph of irreducible components).

A tentative list of topics to be covered:

- Introduction to the basic idea of Grothendieck
- Castelnuovo-Mumford regularity of sheaves and modules
- Existence of the Hilbert scheme; representable functors
- Groebner degenerations, generic initial ideals, and Borel fixed ideals.
- Deformation theory
- Piene-Schlessinger theorem about the Hilbert scheme of the twisted cubic curve in projective 3-space.
- Smoothness of the lexicographic point
- Connectedness theorem of Hartshorne, and Reeve's theorem on radius of the graph of irreducible components
- Local equations of the Hilbert scheme
- The hilbert scheme of points in P^n
- The Hilbert scheme of curves in projective 3-space of a given degree and genus.
- Vakil's theorem: Murphy's law for Hilbert schemes

Prerequisite: We will assume some basic algebraic geometry, at the level of Hartshorne chapters 1-3, although we will review those aspects that we really need. This includes the notion of flatness, Hilbert functions and series, coherent sheaves on projective varieties and schemes, and their cohomology.

Textbook: We will not follow any one textbook, but we will often use Hartshorne "Deformation theory", and Sernesi "Deformations of algebraic schemes" (both from Springer, available online). We will also consult a number of papers on the topic.

### MATH 7710 - Topics in Probability Theory

Fall 2020. 4 credits. S/U grades only.

Selection of advanced topics from probability theory. Content varies.

### MATH 7720 - Topics in Stochastic Processes: Critical Percolation and Planar Scaling Limits

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

Introduction to the basic properties of critical percolation and related models in two dimensions, and its scaling limits. Russo-Seymour-Welsh estimates and gluing, overview of approaches to scaling limit. Introduction to Stochastic Loewner Evolution and its application to the computation of critical exponents.

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

Fall 2020. 4 credits. Student option grading.

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

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

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

Your grade will be based on a final project that can be done in a small group.

### MATH 7810 - Seminar in Logic

Fall 2020. 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 7820 - Seminar in Logic

Spring 2021. 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: 2020-2021. Next offered: 2021-2022. 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).