Graduate Courses

Overview

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

Fall 2022 Offerings

Descriptions are included under Course Descriptions. Check the university class roster for locations.

MATH 5080 - Special Study for Teachers
Mary Ann Huntley

MATH 5200 - Differential Equations and Dynamical Systems
John Hubbard, TR 9:40-10:55

MATH 5220 - Applied Complex Analysis
Steven Strogatz, TR 11:25-12:40

MATH 5250 - Numerical Analysis and Differential Equations
Alexander Vladimirsky, TR 1:00-2:15

MATH 5410 - Introduction to Combinatorics I
Karola Meszaros, TR 9:40-10:55

MATH 6110 - Real Analysis
Camil Muscalu, TR 9:40-10:55

MATH 6160 - Partial Differential Equations
Timothy Healey, TR 2:45-4:00

MATH 6210 - Measure Theory and Lebesgue Integration
Philippe Sosoe, TR 9:40-10:55

MATH 6260 - Dynamical Systems
John Hubbard, TR 8:05-9:20

MATH 6302 - Lattices: Geometry, Cryptography and Algorithms
Noah Stephens-Davidowitz, TR 2:45-4:00

MATH 6310 - Algebra
Allen Knutson, MW 11:25-12:40

MATH 6330 - Noncommutative Algebra
Marcelo Aguiar, TR 1:00-2:15

MATH 6340 - Commutative Algebra with Applications in Algebraic Geometry
Irena Peeva, MF 2:45-4:00

MATH 6520 - Differentiable Manifolds
Reyer Sjamaar, TR 8:05-9:20

MATH 6530 - K-Theory and Characteristic Classes
Inna Zakharevich, MW 9:40-10:55

MATH 6710 - Probability Theory I
Lionel Levine, MW 1:00-2:15

MATH 6740 - Mathematical Statistics II
Michael Nussbaum, TR 11:25-12:40

MATH 6870 - Set Theory
Justin Moore, MWF 10:10-11:00

MATH 7350 - Topics in Homological Algebra: Nerves and Classifying Spaces
Yuri Berest, MW 1:00-2:15

MATH 7570 - Topics in Topology: Bass-Serre Theory and Complexes of Groups
Jason Manning, MW 11:25-12:40

MATH 7610 - Topics in Geometry: Variational Theory in Geometry
Xin Zhou, TR 1:00-2:15

MATH 7710 - Topics in Probability Theory: Integrable Probability
Andrew Ahn, TR 9:40-10:55

MATH 7740 - Statistical Learning Theory: Classification, Pattern Recognition, Machine Learning
Marten Wegkamp, TR 11:25-12:40

MATH 7810 - Seminar in Logic
Slawomir Solecki, TF 2:45-4:00

Spring 2023 Offerings

Descriptions are included under Course Descriptions.

MATH 5080 - Special Study for Teachers
Mary Ann Huntley

MATH 6120 - Complex Analysis
John Hubbard, TR 9:40-10:55

MATH 6230 - Differential Games and Optimal Control
Alexander Vladimirsky, TR 11:25-12:40

MATH 6270 - Applied Dynamical Systems
Steven Strogatz, TR 2:45-4:00

MATH 6320 - Algebra
Ravi Ramakrishna, TR 1:00-2:15

MATH 6370 - Algebraic Number Theory
David Zywina, TR 11:25-12:40

MATH 6390 - Lie Groups and Lie Algebras
Dan Barbasch, MW 1:00-2:15

MATH 6510 - Algebraic Topology
Jason Manning, TR 8:05-9:20

MATH 6620 - Riemannian Geometry
Kathryn Mann, MW 9:40-10:55

MATH 6670 - Algebraic Geometry
Allen Knutson, MW 9:40-10:55

MATH 6720 - Probability Theory II
Philippe Sosoe, TR 8:05-9:20

MATH 6730 - Mathematical Statistics I
Instructor TBA

MATH 6810 - Logic
Anil Nerode, TR 11:25-12:40

MATH 7150 - Fourier Analysis
Terence Harris, TR 1:00-2:15

MATH 7410 - Topics in Combinatorics: Polymatroids
Ed Swartz, MW 11:25-12:40

MATH 7520 - Berstein Seminar in Topology: Geometric Group Theory
Timothy Riley, MW 1:00-2:15

MATH 7580 - Topics in Topology: Riemann Surfaces and their Moduli
Benjamin Dozier, MW 11:25-12:40

MATH 7670 - Topics in Algebraic Geometry: Hodge Theory
Dan Halpern-Leistner, TR 2:45-4:00

MATH 7720 - Topics in Stochastic Processes: Random Walks on Groups
Ariel Yadin, TR 9:40-10:55

Course Descriptions

MATH 5080 - Special Study for Teachers

Fall 2022, Spring 2023. 1 credit. Student option grading.

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

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

MATH 5200 - Differential Equations and Dynamical Systems

Fall 2022. 4 credits. Student option grading.

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

Prerequisite: high level of performance in MATH 2210-MATH 2220, MATH 2230-MATH 2240, MATH 1920 and MATH 2940, or permission of instructor. Students will be expected to be comfortable with proofs. Enrollment limited to: graduate students. Co-meets with MATH 4200.

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

MATH 5220 - Applied Complex Analysis

Fall 2022. 4 credits. Student option grading.

Prerequisite: MATH 2210-MATH 2220, MATH 2230-MATH 2240, MATH 1920 and MATH 2940, or MATH 2130 and MATH 2310. Students will be expected to be comfortable with proofs. Enrollment limited to: graduate students. Co-meets with MATH 4220.

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

MATH 5250 - Numerical Analysis and Differential Equations

Fall 2022. 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 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 4410.

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

MATH 5420 - [Introduction to Combinatorics II]

Spring. Not offered: 2022-2023. Next offered: 2023-2024. 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 2022. 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.

Textbook: Stein and Shakarchi's Real Analysis. The grade is based on homework problems.

MATH 6120 - Complex Analysis

Spring 2023. 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. Not offered: 2022-2023. Next offered: 2023-2024. 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 Rn, mean-value properties, maximum principles, energy methods, Duhamel's principle, and an introduction to nonlinear first-order equations, including shocks and weak solutions. Additional topics may include Hamilton-Jacobi equations, distribution theory, and the Fourier transform.

MATH 6160 - Partial Differential Equations

Fall 2022. 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.

Textbook: Partial Differential Equations by L.C.. Evans, 2nd corrected edition, is recommended but not required.

MATH 6210 - Measure Theory and Lebesgue Integration

Fall 2022. 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: undergraduate analysis and linear algebra at the level of MATH 4130 and MATH 4310.

Covers measure theory, integration, and Lp spaces.

MATH 6220 - [Applied Functional Analysis]

Spring. Not offered: 2022-2023. Next offered: 2023-2024. 4 credits. Student option grading.

Prerequisite: MATH 4130 or the equivalent.

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

MATH 6230 - Differential Games and Optimal Control

Spring 2023. 4 credits. Student option grading.

A Homicidal Chauffeur tries to run over a Pedestrian. The Chauffeur's speed is much greater, but he is constrained by a minimum turn-radius. Can the Pedestrian survive this encounter? ... in the presence of obstacles? ... until the arrival of Police?

We will explore the theory of non-linear Hamilton-Jacobi PDEs using and comparing two natural perspectives: differential games/control theory and front propagation modeling. We will use the Optimality Principle to establish the properties of viscosity solutions and to build fast numerical methods by determining the direction of information flow. Throughout the course, we will highlight similarities and differences between the continuous and discrete control problems (e.g., an optimal path for a rover on the surface of Mars vs. optimal driving directions from Ithaca to New York City).

No prior knowledge of non-linear PDEs or numerical analysis will be assumed. In the numerical part of the course, the emphasis will be on the fundamental ideas rather than on implementation details. The participants' interests will determine a subset of topics to be discussed in detail:

  • Hamilton-Jacobi PDEs: general theory of viscosity solutions; interpretation of characteristics; connections to the calculus of variations; relationship to hyperbolic conservation laws; homogenization; multi-valued solutions; variational inequalities.
  • Games & Control: deterministic, piecewise-deterministic, & stochastic; discrete, continuous, & hybrid; finite, infinite time-horizon vs. exit-time problems; worst-case & on-average optimality; restricted state space and non-autonomous controls; risk-sensitive, distributionally-robust & likelihood-maximizing controls; Pareto-optimal controls; probabilistic constraints; pursuer-evader games; surveillance-evasion games; pusher-chooser games; tug-of-war games; mean field games; mean field control; optimal behavior in uncertain environments.
  • Front propagation: Legendre transform; Wulff shapes; (anisotropic) Huygens' principle; geometric optics; motion by mean curvature and degenerate ellipticity.
  • Numerical Approaches: Lagrangian, semi-Lagrangian, and Eulerian discretizations; controlled Markov chains; iterative and causal (non-iterative) methods; level set methods; model reductions and approximate dynamic programming.
  • Applications: robotics, computational geometry, path-planning, sailboats with randomly changing wind patterns, image segmentation, shape-from-shading, seismic imaging, photolithography, crystal growth, financial engineering, crowd dynamics, traffic engineering, cancer therapy scheduling, behavioral ecology, surveillance evasion, environmental crime prevention, aircraft collision avoidance.

MATH 6260 - Dynamical Systems

Fall 2022. 4 credits. Student option grading.

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

Topics include existence and uniqueness theorems for ODEs; Poincaré-Bendixon theorem and global properties of two dimensional flows; limit sets, nonwandering sets, chain recurrence, pseudo-orbits and structural stability; linearization at equilibrium points: stable manifold theorem and the Hartman-Grobman theorem; and generic properties: transversality theorem and the Kupka-Smale theorem. Examples include expanding maps and Anosov diffeomorphisms; hyperbolicity: the horseshoe and the Birkhoff-Smale theorem on transversal homoclinic orbits; rotation numbers; Herman's theorem; and characterization of structurally stable systems.

MATH 6270 - Applied Dynamical Systems

(also MAE 7760)

Spring 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 6302 - Lattices: Geometry, Cryptography, and Algorithms

Fall 2022. 3 credits. Student option grading.

Prerequisite: MATH 4310 or permission of instructor.

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

MATH 6310 - Algebra

Fall 2022. 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 2023. 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 2022. 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 include semisimple modules and rings, Wedderburn theory, the Jacobson radical and Artinian rings, local rings, group rings, path algebras of quivers, central simple algebras, the Brauer group.

I may provide notes. The following texts provide additional support:

  1. Benson Farb and R. Keith Dennis, Noncommutative Algebra
  2. Louis Halle Rowen, Graduate Algebra: Noncommutative
  3. T. Y. Lam, A First Course in Noncommutative Rings
  4. Richard S. Pierce, Associative Algebras

MATH 6340 - Commutative Algebra with Applications in Algebraic Geometry

Fall 2022. 4 credits. Student option grading.

Prerequisite: modules and ideals (e.g., strong performance in MATH 4330 and either MATH 3340 or MATH 4340), or permission of instructor. May be taken concurrently with MATH 6310.

Commutative Algebra is the theory of commutative rings and their modules. We will cover several basic topics: localization, primary decomposition, dimension theory, integral extensions, Hilbert functions and polynomials, free resolutions. The lectures will emphasize the connections between Commutative Algebra and Algebraic Geometry.

Recommended Texts:

  1. D. Eisenbud, Commutative Algebra, Springer.
  2. M. Atiyah and I. MacDonald, Introduction to Commutative Algebra, Addison Wesley.
  3. H. Matsumura, Commutative ring theory, Cambridge University Press.

MATH 6350 - [Homological Algebra]

Spring. Not offered: 2022-2023. Next offered: 2023-2024. 4 credits. Student option grading.

Prerequisite: MATH 6310. Offered alternate years.

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

MATH 6370 - Algebraic Number Theory

Spring 2023. 4 credits. Student option grading.

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

MATH 6390 - Lie Groups and Lie Algebras

Spring 2023. 4 credits. Student option grading.

Prerequisite: a basic knowledge of algebra and linear algebra at the honors undergraduate level. Some knowledge of differential and algebraic geometry are helpful.

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

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. A different viewpoint is that of algebraic groups. We will endeavor to discuss this along with the C∞ viewpoint.

Topics (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 • 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. Bröcker & T. tom Dieck, Representations of compact Lie groups, Springer-Verlag, New York, 1985.

MATH 6410 - [Enumerative Combinatorics]

Spring. Not offered: 2022-2023. Next offered: 2023-2024. 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 2023. 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 2022. 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.

An introduction to geometry and topology from a differentiable viewpoint, suitable for beginning graduate students. I will largely follow the standard syllabus. The objects of study are manifolds and differentiable maps. The collection of all tangent vectors to a manifold forms the tangent bundle, and a section of the tangent bundle is a vector field. Alternatively, vector fields can be viewed as first-order differential operators. We will study flows of vector fields and prove the Frobenius integrability theorem. We will examine vector bundles, the tensor calculus, and the exterior differential calculus and prove Stokes' theorem. If time permits, de Rham cohomology, Morse theory, or other optional topics will be covered.

This is one of the core courses in the Cornell PhD program in Mathematics and can be used to satisfy the core course requirement in the program. The course is also suitable for students in many other Cornell PhD programs, such as Physics or the Center for Applied Mathematics.

MATH 6530 - K-Theory and Characteristic Classes

Fall 2022. 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: 2022-2023. Next offered: 2023-2024. 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 2023. 4 credits. Student option grading.

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

Topics include linear connections, Riemannian metrics and parallel translation; covariant differentiation and curvature tensors; the exponential map, the Gauss Lemma and completeness of the metric; isometries and space forms, Jacobi fields and the theorem of Cartan-Hadamard; the first and second variation formulas; the index form of Morse and the theorem of Bonnet-Myers; the Rauch, Hessian, and Laplacian comparison theorems; the Morse index theorem; the conjugate and cut loci; and submanifolds and the Second Fundamental form.

MATH 6630 - [Symplectic Geometry]

Fall or Spring. Not offered: 2022-2023. Next offered: 2023-2024. 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. Not offered: 2022-2023. Next offered: 2023-2024. 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 2023. 4 credits. Student option grading.

Prerequisite: MATH 6310 and MATH 6340, or equivalent.

Schemes, sheaves, and hopefully some cohomology of sheaves, essentially [Hartshorne] chapter 2 and some of 3 (depending on clientele).

MATH 6710 - Probability Theory I

Fall 2022. 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 course in probability theory which uses measure theory but begins with the basic definitions of independence and expected value in that context. Law of large numbers, Poisson and central limit theorems, and random walks.

Textbook: Durrett, Probability: Theory and Examples, 5th edition,

MATH 6720 - Probability Theory II

Spring 2023. 4 credits. Student option grading.

Prerequisite: MATH 6710.

This course continues the study of probability theory started in MATH 6710, starting with a review of conditional expectation, and then martingales, markov chains, Brownian motion, and, time permitting, an introduction to stochastic calculus.

MATH 6730 - Mathematical Statistics I

(also STSCI 6730)

Spring 2023. 4 credits. Student option grading.

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

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

MATH 6740 - Mathematical Statistics II

(also STSCI 6740)

Fall 2022. 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, (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 and on the Canvas course website.

MATH 6810 - Logic

Spring 2023. 4 credits. Student option grading.

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

Covers basic topics in mathematical logic, including propositional and predicate calculus; formal number theory and recursive functions; completeness and incompleteness theorems, compactness and Skolem-Loewenheim theorems. Other topics as time permits.

MATH 6830 - [Model Theory]

Spring. Not offered: 2022-2023. Next offered: 2023-2024. 4 credits. Student option grading.

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

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

MATH 6840 - [Recursion Theory]

Fall. Not offered: 2022-2023. Next offered: 2023-2024. 4 credits. Student option grading.

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

Covers theory of effectively computable functions; classification of recursively enumerable sets; degrees of recursive unsolvability; applications to logic; hierarchies; recursive functions of ordinals and higher type objects; generalized recursion theory.

MATH 6870 - Set Theory

Fall 2022. 4 credits. Student option grading.

Offered alternate years.

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

This course will give a fast paced introduction to combinatorial set theory and forcing in the style of Kunen's text, although going somewhat further. It will begin with a treatment of ordinals, cardinals, transfinite recursion, the Axiom of Choice, Zorn's Lemma, and von Neumann's cumulative hierarchy. We will cover: Godel's constructible universe, stationary sets, combinatorial principles, forcing, and large cardinal axioms. Over the course of the semester, we will establish the independence of the Continuum Hypothesis and Souslin's Hypothesis from ZFC. The course will culminate in the study of Solovay's model L(R) and the Absoluteness Theorem for L(R) due to Shelah and Woodin.

Kunen's "Set Theory: an introduction to independence proofs" (either edition) is recommended as a supplemental text.

MATH 7110 - [Topics in Analysis]

Fall. Not offered: 2022-2023. Next offered: 2023-2024. 4 credits. S/U grades only.

Selection of advanced topics from analysis. Course content varies.

MATH 7120 - [Topics in Analysis]

Spring. Not offered: 2022-2023. Next offered: 2023-2024. 4 credits. S/U grades only.

Selection of advanced topics from analysis. Course content varies.

MATH 7130 - [Functional Analysis]

Fall. Not offered: 2022-2023. Next offered: 2023-2024. 4 credits. Student option grading.

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

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

MATH 7150 - Fourier Analysis

Spring. 4 credits. S/U grades only.

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

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

MATH 7160 - [Topics in Partial Differential Equations]

Fall. Not offered: 2022-2023. Next offered: 2023-2024. 4 credits. S/U grades only.

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

MATH 7270 - [Topics in Numerical Analysis]

Fall or Spring. Not offered: 2022-2023. Next offered: 2023-2024. 4 credits. S/U grades only.

Selection of advanced topics from numerical analysis. Content varies.

MATH 7280 - [Topics in Dynamical Systems]

Fall or Spring. Not offered: 2022-2023. Next offered: 2023-2024. 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 2022, Spring 2023. 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: 2022-2023. Next offered: 2023-2024. 4 credits. S/U grades only.

Selection of advanced topics from algebra. Course content varies.

MATH 7350 - Topics in Homological Algebra: Nerves and Classifying Spaces

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

Offered alternate years.

The course will consist of two parts centered around the classical notion of a classifying space. Part I will begin with a survey of basic Lie theory — the theory of compact Lie groups — from the point of view of homotopy theory. We will describe various topological constructions and properties of classifying spaces of compact Lie groups, including classical homology decomposition (approximation) theorems and the homotopy uniqueness theorem. Motivated by compact Lie groups we will then look at more general (though still closely related from topological point of view) objects called the finite loop spaces. We will describe D. L. Rector's famous homotopic classification of finite deloopings of S3 and its beautiful K-theoretic generalization to arbitrary simply connected compact Lie groups (the so-called 'fake Lie groups'). Finally, we will try to give a faithful introduction to the theory of p-compact groups (a.k.a. homotopy Lie groups) and their recent classification in terms of finite pseudo-reflection groups and p-adic root data.

In the second part of the course, we will discuss an important generalization of the notion of classifying space when a (topological) group is replaced by a (topological) category. In recent years, there have been proposed many remarkable refinements of this classical construction in the context of abstract homotopy theory and higher homotopical algebra.  We will try to survey some of these constructions and their applications. Our examples will include cyclic and cubical nerves, classifying diagrams of relative categories  (Rezk nerves), simplicial and DG homotopy coherent nerves.

MATH 7370 - [Topics in Number Theory]

Fall or Spring. Not offered: 2022-2023. Next offered: 2023-2024. 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]

Spring. Not offered: 2022-2023. Next offered: 2023-2024. 4 credits. S/U grades only.

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

MATH 7410 - Topics in Combinatorics: Polymatroids

Spring. 4 credits. S/U grades only.

Offered alternate years.

This will be a combination of an introduction to several topics which do not appear to have any obvious common ground, and the common thread they all share — the appearance of polymatroids as a player. Polymatroids were introduced by Edmonds a little over 50 years ago as a polytopal generalization of the greedy algorithm for graphs and matroids. Like matroids, polymatroids can be approached from several points of view.

The topics will probably include linear optimization, matroid-like representations for arbitrary finite groups, coding theory, the topology of subspace arrangements and their complements, the combinatorial Laplacian, and the geometry of generalized permutahedra.   Other topics have yet to be determined.

Very little background will be assumed. Each topic will get an introduction which makes as few background assumptions as is reasonable, yet (hopefully!) still gives some feeling for the subject.

MATH 7510 - [Berstein Seminar in Topology]

Fall. Not offered: 2022-2023. Next offered: 2023-2024. 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 7520 - Berstein Seminar in Topology: Geometric Group Theory

Spring 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.

We will explore three major themes of geometric group theory—growth, amenability, and quasi-isometric rigidity—by means of a tour of favorite theorems and examples.

MATH 6310 (Algebra) and MATH 6510 (Algebraic Topology) would together represent good preparation for this course.

MATH 7570 - Topics in Topology: Bass-Serre Theory and Complexes of Groups

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

This semester will start with Bass-Serre Theory, which is a tool for understanding groups via their actions on trees. Groups which act on trees can be expressed as "graphs of groups", which generalize the amalgamated free products which occur in Seifert-van Kampen. Next we will discuss actions on higher-dimensional complexes (such as buildings or cube complexes) and the corresponding concept of "complexes of groups". Complexes of groups are a common generalization of graphs of groups and orbifold fundamental groups, so we'll also spend some time talking about orbifolds. Finally we will discuss the curvature assumptions which make for a nicely behaved theory of complexes of groups, and an interpretation of complexes of groups in terms of small categories.

MATH 7580 - Topics in Topology: Riemann Surfaces and their Moduli

Spring 2023. 4 credits. S/U grades only.

We will study Riemann surfaces and their moduli, primarily from the perspective of complex analysis and hyperbolic geometry.   We will begin with various constructions of Riemann surfaces.  Further topics: Teichmuller space, character variety, the mapping class group and Nielsen-Thurston classification, modulis space and the Deligne-Mumford compactification, Kobayashi hyperbolicity, the Weil-Petersson symplectic form (and metric and measure), extremal length, quadratic differentials and the Teichmuller metric, and dynamics of the Teichmuller geodesic flow.

MATH 7610 - Topics in Geometry: Variational Theory in Geometry

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

In this course, we will lecture on two major streams of variational theory in geometry, namely the PDE approach and Geometric Measure Theory (GMT) approach. For the PDE approach, we will discuss the theory of harmonic maps and its applications in the existence theory of minimal surfaces, for instance the Sacks-Uhlenbeck theory. In the GMT approach, we will discuss the Almgren-Pitts min-max theory and the development of a Morse theory for the Area functional.

MATH 7620 - [Topics in Geometry]

Spring. Not offered: 2022-2023. Next offered: 2023-2024. 4 credits. S/U grades only.

Selection of advanced topics from modern geometry. Content varies.

MATH 7670 - Topics in Algebraic Geometry: Hodge Theory

Spring 2023. 4 credits. S/U grades only.

Hodge theory is the study of certain linear algebraic structures on the cohomology of a compact projective complex manifold X. It is a linear "shadow" of some of the the (non-linear) complex geometry of X, and has become an essential tool in complex algebraic geometry. It has grown into a huge subject, leading to the theory of motives, mixed Hodge structures and mixed Hodge modules, characteristic p and p-adic analogues, etc... This course will focus on the beginning of the subject, mostly following Voisin's books "Hodge theory and complex algebraic geometry. I and II." Students will present talks on more specialized topics that are of interest to the participants.

Prerequisites: some experience with either compact complex manifolds or complex algebraic geometry

MATH 7710 - Topics in Probability Theory: Integrable Probability

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

Prerequisite: MATH 6710 or equivalent.

In this course, we study several statistical mechanics models, including dimers, random matrices, and non-intersecting random walks. In a variety of asymptotic settings where the system size tends to infinity in these models, universal objects such as the Gaussian free field, Tracy Widom law, and Dyson Brownian motion emerge. This course will aim to introduce these universal objects and demonstrate their ubiquity by studying the asymptotics of a variety of models. The models we focus on in this course will be integrable, meaning that they exhibit some degree of algebraic structure admitting the computation of observables, with connections to the algebra of symmetric functions and representation theory. In particular, the methods studied in this course will be based on utilizing the integrability of these models to access their asymptotics.

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

Spring 2023. 4 credits. S/U grades only.

In this course we will introduce random walks on discrete groups.  These processes give us information about the structure of the underlying group, and we will explore these connections.  After introducing the basic tools — martingales and harmonic functions — we will hopefully prove one of the fundamental results in the field, namely Gromov's theorem about polynomial growth groups.

Prior knowledge required: basic measure theory / probability (including sigma-fields), basic definitions from group theory. Should be accessible to any first-year graduate student.

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

Fall 2022. 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

Fall 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 7820 - [Seminar in Logic]

Spring. Not offered: 2022-2023. Next offered: 2023-2024. 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 or Spring. Not offered: 2022-2023. Next offered: 2023-2024. 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).

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