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 D. Sinha 11:00 – 11:50 MWF 636 Algebraic Topology D. Sinha 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 14:00 – 14:50 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

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.)