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 2025 | WINTER 2026 | SPRING 2026 |
| 531 Intro to Topology I Addington, N 13:00-13:50 MWF |
532 Intro to Topology II Proudfoot, N 13:00-13:50 MWF |
533 Intro to Topology III Zemke, I 13:00 – 13:50, MWF |
| 544 Intro to Abstract Algebra I Ostrik, V 11:00 – 11:50 MWF |
545 Intro to Abstract Algebra II Ostrik, V 11:00 – 11:50 MWF |
546 Intro to Abstract Algebra III Ostrik, V 11:00 – 11:50 MWF |
| 561 Intro Statistical Methods I Pazdan-Siudeja, L 11:00 – 11:50 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 |
| 567 Stochastic Processes Sinclair, C 13:00 – 13:50 MWF |
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| 607 Applied Math I: Computation and Combinatorics Young, B 10:00 – 11:20 TR |
607 Applied Math II: Statistical Learning Murray, J 10:00 – 11:20 TR |
607 Applied Math III: Neural Networks Mazzucato, L 10:00-11:20 TR |
| 607 Introduction to Complex Geometry He, Weiyong 12:00 – 12:50 MWF |
607 Introduction to Homological Algebra Polishchuk, A 12:00 – 13:20 MW |
607 Modular representation theory of finite groups Kleshchev, A 14:00-14:50 MWF |
| 607 Quantum Cohomology Shen, Y 10:00 – 10:50 MWF |
607 The wonderful geometry of matroids Proudfoot, N 10:00-10:50 MWF |
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| 616 Real Analysis Levin, D 13:00 – 13:50 MWF |
617 Real Analysis Phillips, C 13:00 – 13:50 MWF |
618 Real Analysis Phillips, C 13:00 – 13:50 MWF |
| 637 Differential Geometry Warren, M 12:00 – 12:50 MWF |
638 Differential Geometry Lu, P 12:00 – 12:50 MWF |
639 Differential Geometry Lu, P 12:00 – 12:50 MWF |
| 647 Abstract Algebra Vaintrob, A 11:00 – 11:50 MWF |
648 Abstract Algebra Kleshchev, A 11:00 – 11:50 MWF |
649 Abstract Algebra Brundan, J 11:00 – 11:50 MWF |
| 681 Topics in Algebraic Geometry I Addington, N 10:00 – 10:50 MWF |
682 Topics in Algebraic Geometry II Addington, N 11:00 – 11:50 MWF |
683 Topics in Algebraic Geometry III Polishchuk, A 11:00 – 11:50 MWF |
| 684 Topics in Partial Differential Equations I Murphy, J 13:00 – 13:50 MWF |
685 Topics in Partial Differential Equations II Warren, M 13:00 – 13:50 MWF |
686 Topics in Partial Differential Equations III Warren, M 13:00 – 13:50 MWF |
| 690 Topics in Morse Theory Zemke, I 9:00 – 9:50 MWF |
691 Phillips, C 9:00 – 9:50 MWF |
692 Loop, de-loop Sinha, D 12:00 – 13:20 TR |
Math Course Descriptions
Math 607 Applied Math I: Computation and Combinatorics
Fall 2025
Instructor: Benjamin Young
This is the first in a three-class sequence in applied mathematics (broadly construed); it is suitable both for Pre-Ph.D. mathematics students intending to qualify in applied mathematics, and for Ph.D. mathematics students who want to improve their computer programming skills.
Goals: Proficiency with advanced methods for computation and algorithms, with a focus on tools from graph theory, statistical mechanics, and enumerative combinatorics. Students completing this course should be able to implement and analyze efficient algorithms in python, have a sophisticated mathematical understanding of uses and descriptive statistics of graphs, including random graphs, and have a good familiarity with some exactly solvable statistical mechanical models.
Topics: Recursive and dynamic programming, enumerative combinatorics, experimental math, algorithmic complexity, graph theory, random graphs, random spanning trees, the dimer model, the Ising model, random sampling.
Math 607 Introduction to Complex Geometry
Fall 2025
Instructor: Weiyong He
This course provides a rigorous introduction to complex geometry, focusing on the theory of complex manifolds and their rich interplay with differential geometry, complex analysis, and algebraic geometry. Students will explore the foundational concepts of holomorphic functions of several complex variables, complex structures on manifolds, and in particular Kähler geometry.
Topics include complex manifolds and holomorphic maps, Dolbeault cohomology, line bundles and divisors, Hermitian and Kähler metrics, and the Hodge theory. Additional advanced topics include the theory of positive line bundles, Kodaira embedding and Calabi-Yau theorem.
(1) Complex Manifolds: Holomorphic functions, almost complex structures, integrability (Newlander-Nirenberg theorem).
(2) Holomorphic vector bundles, line bundle and divisors, sheaf cohomology and Dobeault cohomology.
(3) Compact Kähler Manifolds: Hodge theory.
(4) Connections, curvature and Chern classes.
(5) Advanced topics: Kodaira embedding theorem, Calabi-Yau theorem, Kähler-Einstein metrics.
Math 607 Applied Math II: Statistical Learning
Winter 2026
Instructor: James Murray
This course will cover statistical and machine learning theory using classical (i.e. not neural-network based) approaches to infer structure and relationships in high-dimensional data. These include probability theory, regression, classification, kernel methods, mixture models and expectation maximization, as well as inference for sequential data using hidden Markov models and linear dynamical systems. In addition to covering the detailed mathematical derivations of these algorithms, we will implement and run them in Python using real and synthetic datasets.
Math 607: Introduction to Homological Algebra
Winter 2026
Instructor: Alexander Polishchuk
In this course we will study homological algebra mostly following Weibel’s book, but also using some material from Tennison’s sheaf theory.
Topics will include derived functors on abelian categories, in particular, those needed for the categories of modules and of sheaves; spectral sequences; and derived categories.
The only prerequisite is the 600 algebra sequence.
Math 607: Quantum Cohomology
Winter 2026
Instructor: Yefeng Shen
This course is an introduction to quantum cohomology. We plan to cover the following topics:
1) We study psi classes on moduli spaces of point curves and state the Witten conjecture/Kontsevich theorem.
2) We give an elementary introduction to moduli space of stable maps. Then we define the quantum cohomology using genus zero Gromov-Witten invariants.
3) We give many examples computing the quantum cohomology, including some toric varieties and flag varieties.
4) We study the relationship between quantum cohomology and mirror symmetry.
Math 607: Applied Math III: Neural Networks
Spring 2026
Instructor: Luca Mazzucato
This course is part of the Applied Math sequence. It is strongly interdisciplinary in nature and aimed at graduate students from Mathematics, Physics, Biology, Computer Science, Psychology, Economics, but open to everybody on campus, including faculty and postdocs. Students taking this class will develop the ability to deploy modern methods in recurrent neural networks, familiarity with underlying theory and assumptions, and proficiency in practical applications. Covered topics include: information theory, statistical inference, learning algorithms, recurrent neural networks (Hopfield networks and networks of excitatory and inhibitory neurons).
The course will be a mix of theoretical methods and practical simulations run in Python using Jupyter notebooks. Students will work out many examples in full details, with more emphasis on problem solving strategies rather than on the formal constructions and proofs. Students will learn how to deploy large language models and AI tools as coding assistants.
Course prerequisites include familiarity with computer programming in Python, equivalent to that in an introductory undergraduate programming course. Basic knowledge of Calculus, Linear Algebra, and Probability, is strongly recommended.
Math 607: Modular representation theory of finite groups
Spring 2026
Instructor: Alexander Kleshchev
We will discuss Indecomposable modules, vertices and sources, Green correspondence; blocks, defect groups, Brauer correspondence; Brauer tree algebras and blocks of cyclic defect; modular representation theory of symmetric groups as an example.
Math 607: The wonderful geometry of matroids
Spring 2026
Instructor: Nicholas Proudfoot
A matroid is an object that captures the combinatorial essence of linear dependence. The past 14 years have seen an enormous flurry of activity in the theory of matroids and hyperplane arrangements, including dramatic proofs of various conjectures from the 1970s, along with the emergence of new questions and objects of study. Much of this recent work makes use of the “wonderful variety” of a hyperplane arrangement, which is a certain compactification of the complement. The goal of this course will be to provide an introduction to matroids with an emphasis on topological and geometric interpretations of the various algebraic invariants that appear, culminating in a survey of many of these recent results.
Math 681: Topics in Algebraic Geometry I
Fall 2025
Instructor: Nicolas Addington
Basics of affine and projective varieties, their maps, dimension, degree, smoothness, etc., with an emphasis on examples. Connections to representation theory involving the Segre, Veronese, and Plücker embeddings, and determinantal varieties more generally. Text is Ellingsrud and Ottem’s “Algebraic Geometry I” notes.
Math 682: Topics in Algebraic Geometry II
Winter 2026
Instructor: Nicolas Addington
Basics of schemes, sheaves, and some cohomology, with an emphasis on examples. Connections to algebraic number theory, explaining how the ring of integers in a number field is like a curve, and Hensel’s lemma is like the implicit function theorem. Text is Ellingsrud and Ottem’s “Introduction to Schemes.”
Math 683: Topic in Algebraic Geometry III
Spring 2026
Instructor: Alexander Polishchuk
This is a continuation of Math 681, 682 sequence.
We will study more advanced topics in algebraic geometry, possibly including theorems on cohomology of coherent sheaves on algebraic varieties.
Math 684: Topics in Partial Differential Equations I
Fall 2025
Instructor: Jason Murphy
This will be the first course in a year-long sequence on partial differential equations. The focus in the first course will be primarily on evolution equations (initial-value problems), including techniques for establishing existence of solutions and analyzing their behavior over time. We will cover techniques such as the method of characteristics and Fourier-based methods. Our focus will be on hyperbolic and dispersive partial differential equations, and we will study both linear and nonlinear models.
Math 685/686: Topics in Partial Differential Equations II/III
Winter 2026/Spring 2026
Instructor: Micah Warren
The goal of this sequence is to provide a rigorous and practical foundation in the partial differential equation (PDE) techniques essential for modern geometric analysis. While many research papers gloss over the deep PDE theory with cursory references to “Schauder theory,” “elliptic regularity,” or “short-time existence,” this course aims to demystify these tools and present them with clarity and precision. Emphasis will be placed on presenting the foundational results and their correct application.
Rather than using a single textbook, I will present notes while actively referencing a range of traditional sources—including the textbooks of Gilbarg-Trudinger, Han-Lin, Evans, Lieberman, and Caffarelli-Cabré—to help students build familiarity with the standard literature.
Core Topics:
• Laplace and heat equations; maximum principles
• Nonlinear elliptic and parabolic PDEs
• Hölder and Sobolev spaces
• Fixed point theorems and the Lax-Milgram lemma
• Continuity method for nonlinear problems
• Short-time existence for geometric flows
• Schauder estimates and regularity theory
• A priori estimates and bootstrapping techniques
• De Giorgi-Nash and Moser iteration methods
• Krylov-Safonov and Krylov-Evans theorems
• Concave fully nonlinear equations
Possible Additional Topics:
• Viscosity solutions
• Applications of Hodge theory (e.g., McLean’s theorem)
• Minimal surface equations
• Fourth-order elliptic and parabolic equations
• Geometric flows of submanifolds
Math 690: Topics in Morse Theory
Fall 2025
Instructor: Ian Zemke
This course will cover the fundamentals of Morse theory. The basic idea is to study the topology of a smooth manifold by using a “height” function on the manifold. This basic idea turns out to be one of the most important and powerful techniques for understanding the topology of smooth manifolds. Morse theory gives a very concrete way of studying smooth manifolds, called handle calculus. We will pay particular attention to low dimensions, where handle calculus is encoded pictorially. We will also study Morse homology, which gives a small model for the singular homology of smooth manifolds. Time and interest permitting, we may study some more advanced topics in Morse theory, such as the h-cobordism theorem, and some descriptions of more advanced constructions in algebraic topology in terms of Morse theory. The prerequisites for the course are basic familiarity with smooth manifolds and algebraic topology, at the level of Math 636 or equivalent.
Math 691:
Winter 2026
Instructor: Chris Phillips
Math 692: Loop, de-loop
Spring 2026
Instructor: Dev Sinha
In algebraic topology, we study the category of topological spaces, which means both spaces and maps between them. For example, homotopy groups are equivalence classes of maps from spheres to a space. The story is enriched (pun intended) by the fact that the set of all maps from X to Y can be sensibly made into a space of all maps from X to Y. One main aim of this class is to study the case in which X is a sphere, in which case the space of based maps is called an iterated loop space. These spaces are generally “essentially infinite dimensional” even when X and Y are finite dimensional, though there are exceptional cases such as the fact that the space of based loops on the circle is homotopy equivalent to the integers, a discrete topological space. The components of iterated loop spaces are the homotopy groups of Y, and the components themselves have very interesting cohomology, built from the cohomology of Y along with – surprisingly at first – its cup product structure. We start of course with the first loop space, namely based maps of a circle to X, and as an application give a concrete method to approach the polynomial functions on a group.
That’s the “loop” part of the course. Next if we view taking loops as a functor of Y, we could ask whether this process can be undone. That is, can we tell if a space Z is homotopy equivalent to loops on some Y, and if so what can we say about Y? This is “de-looping”, hence the silly title of the course. This might sound like a fairly arbitrary inverse process, but in fact it plays important roles in a number of key areas in algebraic topology which we highlight, including generalized cohomology theories, bundle theory and group cohomology.
We next at these two constructions – looping and delooping – together. They form an adjoint pair. Remarkably, our algebraic topological models for them also form an adjoint pair, now between the categories of algebras and coalgebras of chain complexes. This is the first case of something called Koszul-Moore duality, a deep duality where seemingly different algebraic homotopy theories encode the same information.
Finally, we go back to iterated loops as well as general mapping spaces and give a conjectural picture, touching on topics such as nonabelian Poincare duality.