Graduate Courses 2022-2023

The class schedule is subject to change at any time. Such changes are not always reflected immediately on this page. Please check classes.uoregon.edu for the accurate, live schedule for the term.

All the other courses are described in the University of Oregon Catalog. To view graduate courses offered in previous years, please visit the Graduate Course History page.

FALL 2022 WINTER 2023 SPRING 2023
513 Intro to Analysis I
M. Bownik
9:00 – 9:50 MWF		 514 Intro to Analysis II
M. Bownik
9:00 – 9:50 MWF		 515 Intro to Analysis III
M. Warren
9:00 – 9:50 MWF
531 Intro to Topology I
Y. Shen
12:00 – 12:50 MWF		 532 Intro to Topology II
L. Fredrickson
12:00 – 12:50 MWF		 534 Intro to Topology III
L. Fredrickson
12:00 – 12:50 MWF
544 Intro to Abstract Algebra I
P. Hersh
14:00 – 14:50 MWF		 545 Intro to Abstract Algebra II
P. Hersh
14:00 – 14:50 MWF		 546 Intro to Abstract Algebra III
A. Berenstein
14:00 – 14:50 MWF
607 Analysis Methods in Geometry and Topology
P. Lu
10:00 – 11:20 TR		 607 Cluster Algebras
A. Berenstein
14:00 – 14:50 MWF		 607 Riemann-Hilbert Correspondence
Y. Shen
14:00 – 14:50 MWF
607 Intro to Homological Algebra
A. Polishchuk
12:00 – 13:20 TR		 607 Derived Categories in Algebraic Geometry
A. Polishchuk
12:00 – 13:20 TR		 607 Machine Learning
D. Levin
12:00 – 12:50 MWF
616 Real Analysis
H. Lin
9:00 – 9:50 MWF		 617 Real Analysis
H. Lin
9:00 – 9:50 MWF		 618 Real Analysis
M. Bownik
9:00 – 9:50 MWF
634 Algebraic Topology
B. Botvinnik
11:00 – 11:50 MWF		 635 Algebraic Topology
B. Botvinnik
11:00 – 11:50 MWF		 636 Algebraic Topology
B. Botvinnik
11:00 – 11:50 MWF
647 Abstract Algebra
J. Brundan
15:00 – 15:50 MWF		 648 Abstract Algebra
Y. Shen
15:00 – 15:50 MWF		 649 Abstract Algebra
V. Ostrik
15:00 – 15:50 MWF
672 Theory of Probability
C. Sinclair
10:00 – 10:50 MWF		 673 Theory of Probability
C. Sinclair
10:00 – 10:50 MWF
681 Representation Theory
B. Elias
14:00 – 14:50 MWF		 682 Representation Theory II
J. Brundan
14:00 – 14:50 MWF		 683 Representation Theory III
B. Elias
10:00 – 10:50 MWF
684 Advanced Analysis
C. Phillips
10:00 – 10:50 MWF		 685 Advanced Analysis
C. Phillips
12:00 – 12:50 MWF		 686 Hodge Theory
W. He
12:00 – 12:50 MWF
690 Characteristic Classes
B. Botvinnik
12:00 – 12:50 MWF		 691 K-theory
C. Phillips
11:00 – 11:50 MWF		 692 Infinity Categories
A. Cepek
11:00 – 11:50 MWF

 

Math Course Descriptions

684/685 Advanced Analysis
The goal of this course is to develop the theory of elliptic pseudodifferential operators on compact smooth manifolds, to a sufficient extent to cover all the analysis needed for the proof of the Atiyah-Singer Index Theorem for families. This includes:

Basics of compact operators on Hilbert space.

Basics of vector bundles. (Not analysis, but an essential ingredient for the analysis.)

Basics of Fredholm operators and families of Fredholm operators on Hilbert space, and the Fredholm index.

Brief reminder of the basics of the Fourier transform on
{\mathbb{R}}^n.

Sobolev spaces associated to {\mathbb{R}}^n and to smooth
vector bundles on manifolds.

Basics of differential operators and their symbols, on
{\mathbb{R}}^n and on smooth vector bundles on manifolds.

Pseudodifferential operators and the calculus of symbols.

Maps on Sobolev spaces induced by pseudodifferential operators, including boundedness theorems and compactness theorems (Rellich’s Lemma).

Elliptic pseudodifferential operators on compact smooth manifolds are Fredholm.

Time permitting, I will then give a survey of K-theory and a proof (taking some algebraic topology on faith) of the Atiyah-Singer Index Theorem for families.

Material on compact operators and Fredholm theory is put first, since it is important for many functional analysts whose interests are unrelated to the Atiyah-Singer Index Theorem, or even to C*-algebras. As an inducement to C*-algebraists: the only proof I know of the general Bott periodicity theorem in equivariant K-theory, when the group is not abelian, depends on the material above, although backwards: instead of computing the index of a family of elliptic operators using algebraic topological data, it constructs a class in equivariant K-theory by constructing an equivariant family of elliptic operators whose index is the desired class.

Comments about C*-algebras will be made in passing, as appropriate, but little time will be spent on them. (For example, the index of a family of Fredholm operators is a special case of the index of a Fredholm operator between Hilbert modules over a C*-algebra, but almost nothing will be said about this theory beyond several definitions and a pointer to further reading.)