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Graduate Courses 2017/2018

Please note: all the other courses are described in the catalog.  To view graduate courses offered in previous years, please visit the archived pages at

FALL 2017 WINTER 2018 SPRING 2018

511 Intro to Complex Analysis I

M. Bownik (14:00)

512 Intro to Complex Analysis II

M. Bownik (14:00)

513 Intro to Analysis I

H. Lin (13:00)

514 Intro to Analysis II

H. Lin (13:00)

515 Intro to Analysis III

P. Gilkey (13:00)

521 Partial Differential Equations: Fourier Analysis I

J Isenberg (12:00-13:50 TR)

522 Partial Differential Equations: Fourier Analysis II

J Isenberg (12:00-13:50 TR)

531 Intro to Topology I

N. Proudfoot (11:00)

532 Intro to Topology II

N. Proudfoot (11:00)

533 Intro to Differential Geometry

P. Lu (14:00)

541 Linear Algebra

S. Wang (12:00)

544 Intro to Algebra I

E. Eischen (8:30-9:50 TR)

545 Intro to Algebra II

E. Eischen (8:30-9:50 TR)

546 Intro to Algebra III

E. Eischen (8:30-9:50 TR)

561 Intro Methods of Statistics I

D. Levin (9:00)

M. Warren (12:00)

562 Intro Methods of Statistics II

D. Levin (12:00)

563 Intro Methods of Statistics III

D. Levin (12:00)

607 Seminar: Algebraic Number Theory

Akhtari (10:00)

  607 Seminar: Moduli Spaces of Manifolds and Cobordism Categories

B. Botvinnik (10:00)

 607 Seminar: Homological Algebra

B. Elias (12:00)

607 Seminar: Computer Algebra

B. Young (10:00)


607 Seminar: Intro to Quantum Topology

S. Wang (14:00)


 607 Seminar: Modular Representation Theory
A. Kleshchev (14:00)

 607 Seminar: Affine Lie Groups and Quantum Groups

Ostrik (12:00)

 607 Seminar: Ricci curvature

W. He (12:00)

616 Real Analysis I

W. He (13:00)

617 Real Analysis II

W. He (13:00)

618 Real Analysis III

W. He (13:00)

634 Algebraic Topology I

D. Sinha (11:00)

635 Algebraic Topology II

D. Sinha (11:00)

636 Algebraic Topology III

D. Sinha (11:00)

637 Differential Geometry

P. Gilkey (10:00)

638 Differential Geometry

P. Gilkey (10:00)

639 Differential Geometry

P. Gilkey (10:00)

647 Abstract Algebra I

V. Ostrik (9:00)

648 Abstract Algebra II

V. Ostrik (9:00)

649 Abstract Algebra III

V. Ostrik (9:00)

681 Algebraic Geometry

N. Addington (9:00)

682 Algebraic Geometry

N. Addington (9:00)

683 Algebraic Geometry

N. Addington (9:00)

684 Advanced Analysis

C. Phillips (12:00)

685 Advanced Analysis

Q. Wang (12:00)

 686 Advanced Analysis

Q. Wang (1300)

 690 Characteristic Classes

R. Lipshitz (11:00)

 691 K-theory

D. Dugger (11:00)

Fall Seminars

 Shabnam Akhtari (10:00): 607 Algebraic Number Theory

This course is not a standard course in algebraic number theory. While a graduate course in algebraic number theory is not a prerequisite, we will recall and use some important theorems and tools from classical algebraic number theory. The goal is to discuss the following topics:

1) Heights of Algebraic Numbers
2) Heights of Vectors and Polynomials
3) Lehmer’s Problem
4) Effective Lower Bounds for Height of Algebraic Numbers
5) Effective Computations in Number Fields (ideals, fundamental system of units, …).
6) Height functions on algebraic curves.

Prerequisites: 500 Algebra and 500 Real Analysis.


Silverman’s lecture notes:

Amoroso’s lecture notes: amoroso/files-height/Tubingen.pdf

 Ben Elias(12:00) 607 Homological Algebra

We will develop the fundamentals of homological algebra, focusing on the category of modules over a ring. Topics will include: projective and injective modules, complexes, derived functors, homological dimension, Grothendieck groups, Morita equivalence, triangulated categories, and homotopy and derived categories. As time permits, we will also discuss: spectral sequences, derived Morita equivalence, stable categories, and hopfological algebra..

 Alexander Kleshchev (14:00) 607 Modular Representation Theory

I suggest to work through classical chapters of modular representation theory of finite groups:
blocks; defect groups; Brauer correspondence; vertices, sources and Green correspondence; Brauer tree algebras; if a homological algebra course to be taught simultaneously with this course treats derived categories, then in the end we could consider examples of interesting derived equivalences of blocks of finite groups related to Broue’s abelian defect group conjecture. Otherwise we can stay more classical and discuss Brauer tree algebras and cyclic blocks in more detail. There is a very good book on the subject: J. Alperin, “Local Representation Theory”.

Winter Seminars

 Ben Young (10:00) 607 Computer Algebra

Introduction to mathematical computing in sage/python, pitched particularly at graduate students in mathematics.  The target audience is people who want to use computer experiment in their own work, as well as people who wish to learn the rudiments of software development.  Topics include: crash course in python/sage, basics of analysis of algorithms, test driven development, experimental mathematics, and selected other topics in computer algebra.  There will be a project, and the class will be better if students actually have a project relevant to their research or career.

 Victor Ostrik (12:00) 607 Affine Lie Algebras and Quantum Groups

Affine Lie algebras form an interesting class of infinite dimensional Lie algebras with many similarities to simple finite dimensional Lie algebras. In this class we will discuss some aspects of representation theory of affine Lie algebras which connect them with quantum groups — some very interesting deformations of universal enveloping algebras of simple Lie algebras.

Spring Seminars

Boris Botvinnik (10:00) Moduli Spaces of Manifolds and Cobordism Categories

The course is an introduction to recent development on smooth topology.  The goal for the course will be to describe results by Madsen and Weiss as well as by Galatius and Randal-Williams on the structure of moduli spaces of manifolds. This is an active area of research with exciting applications. Prerequisite: 600-topology, bundles, characteristic classes.

Shida Wang (12:00) Introduction to Quantum Topology

Quantum topology has become a very active area since its inception in the 1980s. This course will provide a basic introduction to the subject from the aspect of knot theory.
The topics to be covered include:
1. Knots and links
2. The Jones polynomial
3. Skein modules
4. Tangles
5. Strict monoidal categories
6. The Temperley-Lieb algebra
7. Braids and the Yang-Baxter equation
8. Colored Jones polynomials
9. Quantum invariants of 3-manifolds
10. Topological quantum field theory
The prerequisites are linear algebra and some familiarity with the notion of groups.

Weiyong He (14:00) Ricci curvature

The study of structure of Riemannian manifolds with  bounded Ricci curvature (or Ricci curvature bounded below) has formed a central topic in modern theory of geometric analysis and it has been developed for decades, with many very important applications.
We will largely follow J. Cheeger’s notes “Degeneration of Riemannian metrics under Ricci curvature bounds”, try to cover the basic “Cheeger-Colding” theory on the structures of the Riemannian manifolds under Ricci curvature bounds. Along the lines of the presentation, many fundamental technique and results in geometric analysis will be covered: including Bochner’s technique, volume comparison and Laplacian comparison, strong maximum principle, Cheeger-Gromoll splitting theorem, and Cheng-Yau’s gradient estimates.
Prerequisite: Riemannian geometry; theory of elliptic PDE will help, but is not mandatory (we will develop many of these technique in the class).