Graduate 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.
On this page:
Schedules | Math Course Descriptions
Schedule
| FALL 2024 | WINTER 2025 | SPRING 2025 |
| 510 Dynamical Systems and Control Murray, J 12:00 – 13:20, TR |
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| 513 Intro to Analysis I Addington, N 9:00 – 9:50 MWF |
514 Intro to Analysis II Lin, H 13:00 – 13:50 MWF |
515 Intro to Analysis III |
| 525 Statistical Methods I Pazdan-Siudeja, L 9:00 – 9:50 MWF |
525 Statistical Methods I Shahir, B 14:00 – 14:50 MWF |
525 Statistical Methods I Pazdan-Siudeja, L 10:00 – 10:50 MWF |
| 532 Intro to Topology II Zemke, I 11:00-11:50 MWF |
533 Curves and Surfaces Warren, M 11:00-11:50 MWF |
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| 544 Intro to Abstract Algebra I Vaintrob, A 14:00 – 14:50 MWF |
545 Intro to Abstract Algebra II Kleshchev, A 14:00 – 14:50 MWF |
546 Intro to Abstract Algebra III Young, B 14:00 – 14:50 MWF |
| 561 Intro Statistical Methods I Chen-Murphy, X 11:00 – 11:50 MWF |
561 Intro Statistical Methods I Pazdan-Siudeja, L 12:00 – 12:50 MWF |
562 Intro Statistical Methods II Pazdan-Siudeja, L 12:00 – 12:50 MWF |
| 562 Intro Statistical Methods II Levin, D 11:00 – 11:50 MWF |
563 Intro Statistical Methods III Levin, D 11:00 – 11:50 MWF |
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| 567 Stochastic Processes Sinclair, C 13:00 – 13:50 MWF |
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| 607 Generating Functions and Symmetric Functions Young, B 12:00 – 13:20 MW |
607 Cobordism Categories Botvinnik, B 12:00 – 13:20 TR |
607 Geometric Invariant Theory Polishchuk, A |
| 607 Gauge Theory and Special Holonomy He, Weiyong 14:00 – 15:20 TR |
607 Enumerative Combinotorics Hersh, P 12:00 – 13:20 MW |
607 Quantum Groups Berenstein, A |
| 616 Real Analysis Bownik, M 9:00 – 9:50 MWF |
617 Real Analysis Bownik, M 9:00 – 9:50 MWF |
618 Real Analysis 9:00 – 9:50 MWF |
| 634 Algebraic Topology Hersh, P 11:00 – 11:50 MWF |
635 Algebraic Topology Addington, N 11:00 – 11:50 MWF |
636 Algebraic Topology Addington, N 11:00 – 11:50 MWF |
| 647 Abstract Algebra Ostrik, V 15:00 – 15:50 MWF |
648 Abstract Algebra Berenstein, A 15:00 – 15:50 MWF |
649 Abstract Algebra Polishchuk, A 15:00 – 15:50 MWF |
| 672 Theory of Probability Levin, D 10:00-10:50 MWF |
673 Theory of Probability Levin, D 10:00-10:50 MWF |
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| 681 Representation Theory I Elias, B 14:00 – 14:50 MWF |
682 Representation Theory II Elias, B 14:00 – 14:50 MWF |
683 Representation Theory III Ostrik, V 14:00 – 14:50 MWF |
| 684 Harmonic Analysis I Murphy, J 10:00 – 10:50 MWF |
685 Harmonic Analysis II Murphy, J 10:00 – 10:50 MWF |
686 Harmonic Analysis III Bownik, M 10:00 – 10:50 MWF |
| 690 Spectral Sequences Botvinnik, B 12:00 – 12:50 MWF |
691 Characteristic Classes Sinha, D 11:00 – 11:50 MWF |
692 WETSK (What Every Topologist Should Know) Zemke, I 12:00 – 12:50 MWF |
Math Course Descriptions
Math 510 Dynamical Systems and Control
Fall 2024
Instructor: James Murray
Whether you are riding a bicycle, adjusting the thermostat, or designing a robot, feedback control is part of your daily life. This course will cover the mathematics of control theory, which provides a framework for driving dynamical systems such as bicycles and robots in optimal ways. For example, suppose we have a dynamical system described by the following differential equation: dx/dt = Ax + Bu where the vector x = x(t) may describe the configuration of a bicycle. How can the rider apply the optimal torque u = u(t) to the handlebars to guide the bike smoothly along a desired path? Beginning with relatively simple linear systems such as this one, we will develop the mathematical tools needed to extend our investigations to systems that are noisy, nonlinear, and/or partially observed. Assuming basic familiarity with calculus and linear algebra, this will lead us to introduce some more sophisticated tools including dynamical systems theory, stochastic processes, and calculus of variations. Students will come away from this class with an understanding of the core principles of control theory, including positive and negative feedback, robustness to noise, and the trade-off between stability and maneuverability. Through mathematical exercises and scientific programming exercises in Python, students will learn how to apply these principles in examples drawn from engineering, biology, neuroscience, and bicycling. Prerequisites are multi-variable calculus and undergraduate linear algebra.
Note: This is a combined course for undergraduates and graduate students, though the course is designed mostly for the former. Seats for graduate students are available but limited.
Math 607 Generating Functions and Symmetric Functions
Fall 2024
Instructor: Ben Young
This is a class in rational, algebraic and D-finite generating functions, following chapters 4-6 of Stanley’s “Enumerative Combinatorics”; we’ll at times supplement with other references such as Flajolet-Sedgewick’s “Analytic Combinatorics”, Herb Wilf’s “Generatingfunctionology,” and others.
The concept of generating function was one of the first unified pieces of theory in Algebraic Combinatorics. They were doubtless discovered and rediscovered many times by many people, but their systematic study began in the 1970s with the work of Richard Stanley and his students. Before then, to enumerate some set of objects, the method was as follows: first be astonishingly clever, and then figure out the answer. An alternative method: see if you can make a generating function appear, and then extract your answer from that generating function algebraically. Generating functions can be combined or transformed in a variety of ways; these transformations have direct parallels in the structure of the objects themselves, due to the “theory of species” of Joyal et al. Often, classical special functions such as the exponential map and even trigonometric functions have combinatorial meanings in this theory.
Moreover, there are other ways of coming across a generating function than counting (algebra, probability, experimental math, etc) and other results one can extract from a generating function (asymptotics, or equality of quantities that don’t have nice closed formulae). The three classes of generating functions described above stand out either for their simplicity and beauty (in the cases of rational and algebraic functions) or for their ubiquity and “algebraic niceness” properties (D-finiteness).
No specific prerequisites are assumed aside from some modest cleverness (which does still help, as it turns out) or willingness to hit a problem with a computer until it dies. The topics we cover are broadly applicable in many areas of mathematics, we will cover many examples from those areas. This is the chronologically “first” part in a 2-part class; the next class covers the other chapters of Stanley’s text, which are largely independent.
Math 607 Gauge Theory and Special Holonomy
Fall 2024
Instructor: Weiyong He
Topics can vary, depending on students’ interest. At the moment here’s what I have in mind:
1. The first part we briefly introduce the SW theory with an emphasize on Taube’s SW theory on symplectic four manifolds.
2. We will give an introduction to G2 geometry, and the G2 flows.
3. We will introduce a special case of G2 structure with reduction to four dimensions, the so-called hypersymplectic structure, to discuss an open problem regarding the characterization of hyperKahler structure in the setting of symplectic geometry in dimension four.
Math 607 Cobordism Categories
Winter 2025
Instructor: Boris Botvinnik
Math 607: Enumerative Combinatorics
Winter 2025
Instructor: Patricia Hersh
This course will focus on fundamental topics from enumerative combinatorics. The course will start with a discussion of classical combinatorial objects such as set partitions and a systematic analysis of classical counting problems encapsulated in the `twelve-fold’ way. This will allow us to answer questions such as how many monomials of degree d there are in n variables. Then we will turn to counting by inclusion-exclusion, learning techniques that will enable us to count things such as the number of derangements of 1,2,…,n or in other words the number of permutations of 1,2,…,n having no fixed points. Finally, we will spend the bulk of the quarter on structural properties of partially ordered sets with an emphasis on particularly important classes of partially ordered sets including geometric lattices and finite distributive lattices.
Math 607: Geometric Invariant Theory
Spring 2025
Instructor: Alexander Polishchuk
In this course we will study main constructions/results of the GIT along with some applications. The main idea of the GIT is to define a “good” open set of stable points for an action of an algebraic group on an algebraic variety, so that this open set admits a nice space of orbits. This approach is used in most constructions of moduli spaces, of which we will consider some examples.
Math 607: Quantum Groups
Spring 2025
Instructor: Arkady Berenstein
The course will be about the 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.
Here is tentative content.
* 1. Introduction to Hopf algebras
* 2. Quantum linear algebra (after Manin)
* 3. Quantum algebraic groups, quantized enveloping algebras, and their representations
Text(s):
Manin, Quantum groups and noncommutative geometry.
Majid, Foundations of Quantum Group Theory
Brown and Goodearl, Lectures on algebraic quantum groups.
Math 681: Representation Theory I
Fall 2024
Instructor: Ben Elias
This is the first term of a sequence on the representation theory of Lie groups and Lie algebras. In the first term we cover: Topological groups, Lie groups, compactness and integration on groups, the Peter-Weyl theorem. Lie algebras, the exponential map, and the Baker-Campbell-Hausdorff formula. Complexification. The representation theory of SL(2). Roots and weights in general, isogeny. The course will begin with a reminder on the representation theory of finite groups, via the McKay correspondence.
Math 682: Representation Theory II
Winter 2025
Instructor: Ben Elias
This is the second term of a sequence on the representation theory of Lie groups and Lie algebra. In the second term we cover: The abstract theory of lie algebras, nilpotent and solvable lie algebras, theorems of Engel and Lie and Cartan. The Killing form, complete reducibility, and the casimir element. Semisimple lie algebras, root systems, weight lattices. Weyl groups. Universal enveloping algebras, the PBW basis, and the Serre presentation. The classification and construction of irreducible representations. Weyl dimension formula, Weyl character formula, Steinberg tensor product formula. The BGG resolution. If time permits we will discuss web algebras for representations in type A.
Math 684/685: Harmonic Analysis I and II
Fall 2024/Winter 2025
Instructor: Jason Murphy
In the first two courses of the sequence, we will cover classical topics in harmonic analysis such as Fourier analysis with applications to partial differential equations, interpolation, maximal functions, singular integrals and Calderón-Zygmund theory, Littlewood-Paley Theory, and the analysis of oscillatory integrals. We will also cover some more modern topics such as sharp inequalities via profile decompositions and restriction theory. Additional topics (e.g. semiclassical analysis and scattering theory) will be covered as time permits.
Math 686: Harmonic Analysis III
Spring 2025
Instructor: Marcin Bownik
In the third term of the sequence we will cover other fundamental topics in harmonic analysis related to wavelets and frames. This includes Hardy H^p spaces, the construction of Stromberg, Meyer, and Daubechies wavelets, minimally supported frequency (MSF) wavelets, multiresolution analysis, frame wavelets, and characterization of function spaces by wavelet coefficients.
Math 691: Characteristic classes
Winter 2025
Instructor: Dev Sinha
Characteristic classes are cohomological measures of vector bundles (families of vector spaces parametrized by some space) and thus are used frequently in algebraic and differential geometry as well as throughout algebraic topology. The standard flavors, named after Stiefel, Whitney and Chern, are determined by a small list of axioms, which are often essentially taken on faith. We will give geometry substantial weight in a complete development, defining them through manifolds which encode linear dependence and verifying the axioms from there. We will also have a variety of applications including to vector bundles defined by group actions, which opens up a connection between representation theory and group cohomology.
Math 692: WETSK (What Every Topologist Should Know)
Spring 2025
Instructor: Ian Zemke
The course will feature student presentations on topics in topology. Topics will include 3-manifold and 4-manifold topology, gauge theory, pseudoholomorphic curves, Floer homology, Seiberg Witten theory, Instanton Theory, concordance, and homology cobordism. Students will have the opportunity to pick from a selection of papers on these topics and will present during the quarter. We will focus on one or two themes during the quarter, which will be voted on by the students.