Graduate Courses 2019/2020
Contents
Descriptions of Advanced Seminar Courses
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 2019 | WINTER 2020 | SPRING 2020 |
| 513 Intro to Analysis I S. Akhtari (8:30 TR) |
514 Intro to Analysis II S. Akhtari (8:30 TR) |
515 Intro to Analysis III P. Gilkey (9:00) |
| 531 Intro to Topology I N. Addington (12:00) |
532 Intro to Topology II N. Addington (12:00) |
510 De Rham Cohomology N. Proudfoot (11:00) |
| 544 Intro to Algebra I J. Brundan (14:00) |
545 Intro to Algebra II E. Eischen (14:00) |
546 Intro to Algebra III J. Brundan (14:00) |
| 607 Wonderful Geometry of Matroids N. Proudfoot (10:00) |
607 Quantum Groups A. Berenstein (14:00) |
607 Fundamentals of Number Theory S. Akhatari (10:00 TR) |
| 607 Modern Invariants of Knots R. Lipshitz (12:00) |
607 Homological Algebra N. Addington (11:00) |
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| NEW Applied Math Sequence 607 Computation & Combinatorics P. Ralph & B. Young (13:00) |
NEW Applied Math Sequence 607 Applied Stochastic Processes Y. Ahmadian & P. Ralph (13:00) |
NEW Applied Math Sequence 607 Machine Learning & Statistics L. Mazzucato (13:00) |
| 607 Topological Combinatorics P. Hersh (14:00) |
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| 616 Real Analysis I H. Lin (9:00) |
617 Real Analysis II H. Lin (9:00) |
618 Real Analysis III C. Phillips (9:00) |
| 634 Algebraic Topology I B. Botvinnik (11:00) |
635 Algebraic Topology II B. Botvinnik (11:00) |
636 Algebraic Topology III B. Botvinnik (11:00) |
| 637 Differential Geometry I P. Lu (10:00) |
638 Differential Geometry II P. Lu (10:00) |
639 Differential Geometry III P. Lu (10:00) |
| 647 Abstract Algebra I B. Elias (15:00) |
648 Abstract Algebra II B. Elias (15:00) |
649 Abstract Algebra III B. Elias (15:00) |
| 681 Algebraic Geometry I A. Polishchuk (14:00) |
682 Algebraic Geometry II A. Polishchuk (10:00) |
683 Algebraic Geometry III Y. Shen (15:00) |
| 684 Harmonic Analysis M. Bownik (10:00) |
685 Abstract Harmonic Analysis M. Bownik (10:00) |
686 Functional Analysis C. Phillips (10:00) |
| 690 Characteristic Classes B. Botvinnik (12:00) |
691 Equivariant K-theory C. Phillips (12:00) |
692 Advanced Homotopy Theory D. Dugger (12:00) |
Fall Seminar
Nicholas Proudfoot (10:00) 607 Wonderful geometry of matroids
The chromatic polynomial of a graph is the polynomial that counts the number of proper q-colorings. For example, if the graph is a triangle, the chromatic polynomial is q(q-1)(q-2) = q^3 – 3q + 2q. It was conjectured long ago that the coefficients of this polynomial form a log concave sequence, meaning that the square of each coefficient is at least as big as the product of the two neighboring coefficients. This conjecture, along with a generalization from graphs to matroids, was recently proved using Hodge theory on a variety called the “wonderful compactification” of a hyperplane arrangement. We will work through this proof and then explore some related topological and algebro-geometric constructions.
The 600 topology sequence would be a sufficient prerequisite for this course.
Winter Seminars
Robert Lipshitz (12:00) 607 Modern Invariants of Knots
This course is an introduction to two modern invariants of knots, Khovanov homology and Heegaard Floer homology, and some of their applications, particularly at the interface of 3- and 4-dimensional topology. The course will assume familiarity with material from the 600-level topology sequence, but no other background is necessary.
Outline:
- Weeks 1-2: Introduction to knot theory, Seifert genus, Jones polynomial, Alexander polynomial.
- Weeks 3-5: Khovanov homology, s-invariant, slice genus bound.
- Week 6: Concordance, facts about smooth and non-smooth 4-manifold topology, existence of exotic smooth structures on R^4.
- Weeks 7-9: Knot Floer homology via grid diagrams, tau, more concordance.
- Week 10: Overview of Heegaard Floer homology in general.
Arkady Berenstein (14:00) 607 Quantum Groups
The course will be about algebraic aspects of Quantum Groups. Quantum groups (or more precisely quantized enveloping algebras) were introduced independently by Drinfeld and Jimbo around 1985, as an algebraic framework for quantum Yang-Baxter equations. Since then numerous applications of Quantum Groups have been found in areas ranging from theoretical physics via symplectic geometry and knot theory to ordinary and modular representations of reductive algebraic groups. The course provides an introduction to the structure theory and representation theory of quantum groups.
Proposed Content:
- Introduction to Hopf algebras
- Quantum linear algebra (after Manin)
- Quantum algebraic groups, quantized enveloping algebras, and their representations
Texts:
- Manin, “Quantum groups and noncommutative geometry”
- Majid, “Foundations of Quantum Group Theory”
- Brown and Goodearl, “Lectures on algebraic quantum groups”
Peter Ralph (13:00) 607 Applied Stochastic Processes and Dynamical Systems
Goals:
proficiency with properties of common stochastic processes and their uses in modeling and computation; ability to simulate and visualize these.
Topics:
Brownian motion, Gaussian processes, point processes, diffusions and associated PDE, random matrices, linear systems theory.
Spring Seminars
Shabnam Akhtari (10:00) 607 Fundamentals of number theory
We will discuss basic results, including the three “fundamental theorems”: unique factorization of ideals, finiteness of the class number, and Dirichlet’s unit theorem. We will also discuss more modern developments and applications of these classical results. We will naturally need to discuss some topics from lattice theory and the geometry of numbers.
We will follow a very well-written book by P. Pollack, “Not Always Buried Deep: A Second Course in Elementary Number Theory”. We will see different proofs of the prime number theorem (e.g. Erdos-Selberg proof) and their extension. Other topics include Gauss’s theory of cyclotomy and its applications to rational reciprocity laws, Hilbert’s solution to Waring’s problem, Burn sieve, and Dirichlet’s theorem on primes in arithmetic progressions.
Nicolas Addington (12:00) 607 Homological Algebra
You probably encountered Ext and Tor in 600 topology as opaque technical gadgets that appear in the universal coefficient theorem and the Kunneth formula. But they have deep geometric meaning, to do with transversality. After setting up the foundations, we will explore this geometry. There will be lots of Koszul complexes.
Luca Mazzucato (13:00) 607 Machine Learning and Statistics
Goals:
Students completing this course should be able to deploy modern methods in computational inference and feedforward and recurrent neural networks, and develop familiarity with underlying theory and assumptions. The students will be able to design and train efficiently neural networks using supervised, unsupervised and reinforcement learning algorithm.
Topics:
Information theory, statistical inference, generative models, mean field theory, feedforward and recurrent neural networks, supervised/unsupervised
Patricia Hersh (14:00) 607 Topological Combinatorics
Topics include Advanced Topics in Geometry, Ring Theory, Teaching Mathematics.
Other Math Course Descriptions
684/685/686 Advanced Analysis Series
These courses introduce students to the subject of abstract harmonic analysis, which is broadly defined as Fourier analysis on groups and unitary representation theory. We will start with basic facts in Banach algebra theory and spectral theory, followed by locally compact groups, Haar measure, and unitary representations. Then we will move to analysis on Abelian groups, compact groups, and induced representations, featuring the imprimitivity theorem and its applications. In the last part of the course we explore some further aspects of the representation theory of non-compact, non-Abelian groups. The primary textbook for the course is “Abstract Harmonic Analysis” by Folland.
Chris Phillips (Winter 12:00) 691 Equivariant K-theory
This is a topics course on equivariant K-theory for actions of compact groups on compact spaces. The course will assume knowledge of topological K-theory, defined using finite dimensional vector bundles. It is intended to be accessible to both topologists and people working on Banach algebras, so Banach algebra material will be mentioned but will not be needed for most of the course.
List of topics:
- Throughout, G is a compact group, and the theory is already interesting when the group is finite
- The representation ring of G. (This is the coefficient ring for G-equivariant K-theory.)
- Basics: homotopy invariance, exact sequences, etc. These are very similar to the nonequivariant case, and will be gone through fast.
- Equivariant Bott periodicity. I don’t expect to be able to prove the most general statement, but will describe it. (The only proof I know uses elliptic pseudodifferential operators.)
- The Atiyah-Segal Completion Theorem. When G acts on X, this theorem identifies the nonequivariant (representable) K-theory of X \times_G EG with a particular completion of the G-equivariant K-theory of X. (Equivariant cohomology is sometimes defined to by the cohomology of X \times_G EG. It follows from this theorem that equivariant K-theory contains more information than that definition would suggest.)
Further topics, possibly including some discussion of the Kuenneth formula for the case when G is cyclic of prime order (what to do for more general finite groups is an open problem) and the background for the fact that equivariant K-theory is the same as the ordinary K-theory of the crossed product algebra.
Math Bio Courses
Peter Ralph (TR 10:00-11:20, Fall and Winter) BIO 610 Advanced Biological Statistic
This two-quarter graduate course aims to provide students with practical understanding of and experience with core concepts and methods in modern data analysis. The focus is on biological data, but skills will be transferable to other disciplines. Students will become familiar with major topics in univariate and multivariate statistics, analysis of large data sets, and Bayesian analysis. There is a particular emphasis on modeling and conceptual understanding of statistical noise and uncertainty. The course is advanced in that we will move through the material quickly with the goal of providing a solid foundation for subsequent learning. Students will learn to use the powerful statistical programming language R, and the flexible modeling package Stan.
*Prerequisites:* some (basic) programming.