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 RiemannHilbert 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 Ktheory 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 AtiyahSinger 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
.
Sobolev spaces associated to and to smooth
vector bundles on manifolds.
Basics of differential operators and their symbols, on
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 Ktheory and a proof (taking some algebraic topology on faith) of the AtiyahSinger 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 AtiyahSinger 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 Ktheory, 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 Ktheory 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.)