# Graduate Courses 2021/2022

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 2021 |
WINTER 2022 |
SPRING 2022 |

513 Intro to Analysis IM. Bownik (8:30-9:50 TR) |
514 Intro to Analysis IIM. Bownik (8:30-9:50 TR) |
515 Intro to Analysis IIIL. Fredrickson (8:30-9:50 TR) |

531 Intro to Topology IR. Lipshitz (12:00-12:50 MWF) |
532 Intro to Topology IIY. Shen (12:00-12:50 MWF) |
533 Intro to Topology IIITBA (11:00-11:50 MWF) |

544 Intro to Algebra IJ. Brundan (14:00-14:50 MWF) |
545 Intro to Algebra IIJ. Brundan (14:00-14:50 MWF) |
546 Intro to Algebra IIIJ. Brundan (14:00-14:50 MWF) |

607 Topological Field Theories and Tensor CategoriesV. Ostrik (10:00-11:20 TR) |
607 Number Theory IS. Akhtari (10:00-11:20 TR) |
607 Number Theory IIS. Akhtari (10:00-11:20 TR) |

607 Applied Math I: Combinatorics, Algorithms, and Stochastic ProcessesP. Ralph (14:00-15:20 TR) |
607 Applied Math II: Statistical LearningJ. Murray (13:00-13:50 MWF) |
607 Applied Math III: Machine LearningL. Mazzucato (13:00-13:50 MWF) |

607 Homological AlgebraB. Elias (12:00-13:20 TR) |
607 Computer AlgebraB. Young (12:00-12:50 MWF) |
607 Combinatorics of Coxeter GroupsP. Hersh (12:00-12:50 MWF) |

616 Real Analysis IP. Lu (9:00-9:50 MWF) |
617 Real Analysis IIP. Lu (9:00-9:50 MWF) |
618 Real Analysis IIIM. Warren (9:00-9:50 MWF) |

634 Algebraic Topology IP. Hersh (11:00-11:50 MWF) |
635 Algebraic Topology IID. Dugger (11:00-11:50 MWF) |
636 Algebraic Topology IIID. Dugger (11:00-11:50 MWF) |

637 Differential Geometry IN. Addington (10:00-10:50 MWF) |
638 Differential Geometry IIW. He (10:00-10:50 MWF) |
639 Differential Geometry IIIW. He (10:00-10:50 MWF) |

647 Abstract Algebra IA. Kleshchev (15:00-15:50 MWF) |
648 Abstract Algebra IIA. Kleshchev (15:00-15:50 MWF) |
649 Abstract Algebra IIIV. Ostrik (15:00-15:50 MWF) |

681 Algebraic Geometry IA. Polishchuk (14:00-14:50) |
682 Algebraic Geometry IIY. Shen (14:00-14:50 MWF) |
683 Algebraic Geometry IIIN. Addington (14:00-14:50 MWF) |

684 Modular FormsE. Eischen (10:00-10:50 MWF) |
685 Harmonic AnalysisM. Bownik (9:00-9:50 MWF) |
686 Symplectic GeometryW. He (9:00-9:50 MWF) |

690 Morse TheoryB. Botvinnik (12:00-12:50 MWF) |
691 Classifying SpacesD. Sinha (11:00-11:50 MWF) |
692 WETSKR. Lipshitz (11:00-11:50 MWF) |

## Math Course Descriptions

**616/617/618 Real Analysis**

We will teach the courses (Math 616 and 617) based on the book Real Analysis:

Measure Theory, Integration, and Hilbert Spaces; and Functional Analysis: Introduction to Further Topics in Analysis, by Elias M. Stein and Rami Shakarchi. We start with the concrete Lebesque measure and Lebseque integration on $R^d$.

Then we get into the differentiation. By a quick tour of the Hilbert space, then we use it to develop abstract theory of measure

and integration. This is our plan for the fall quarter.

For winter quarter, we study linear functional analysis (Banach spaces, revisiting Hilbert spaces and bounded linear operators). We begin with $L^p$-spaces. We plan to cover some harmonic analysis. If time permits we will discuss distributions and Sobolev spaces.

**647/648/649 Abstract Algebra**

647: category theory, multilinear algebra, introduction to ring theory and module theory

648: ring theory, module theory, representation theory of finite groups

**681/682/683 Advanced Algebra Series**

The 681-683 sequence is aimed at giving a thorough introduction to Algebraic Geometry.

In the first term we will follow chapter 1 of Hartshorne’s “Algebraic Geometry”.

**684/685/686 Advanced Analysis Series**

*MA 684: Introduction to Modular Forms*

This course will provide an introduction to modular forms, which play a central role in modern number theory. In addition to algebraic and analytic number theory, this topic has ties to algebraic geometry and representation theory and beyond. We will cover standard aspects of modular forms, for example definitions of modular forms, cusp forms, and Eisenstein series; dimension formulas; Hecke operators; connections with elliptic curves; and (as time allows) connections with Galois representations, congruences, L-functions, and/or automorphic forms. The recommended prerequisites for this course are the 600-level algebra and analysis sequences (although the 500-level sequences will suffice most of the time), as well as complex analysis and Fourier analysis (at the very least, being very clear on the definitions of “holomorphic” and “meromorphic,” plus being aware of the notion of a Fourier expansion).

*MA 685: Harmonic and Functional Analysis of Frames*

A frame is a generalization of the concept of a basis to sets which are overcomplete. That is, frame expansions are in general not unique and instead they satisfy a certain stability condition. Although frames were introduced in 1950’s, this area has experienced a renewed interest in recent years with the advent of wavelets. In this course we plan to explore the following topics depending on the interest of students.

1) General frames and Riesz bases in Hilbert spaces: dual frames, canonical dual frames, Naimark’s dilation theorem.

2) Frames in finite dimensional spaces: equiangular frames, fusion frames, connections with algebraic combinatorics and Littlewood-Richardson tableaux.

3) Frames in infinite dimensional spaces: Kadison’s Pythagorean Theorem, characterization of frame norms with prescribed frame operator and the Schur-Horn theorem.

4) Frames and Riesz bases in shift-invariant spaces.

5) The solution of the long standing Kadison-Singer problem and its equivalent formulation in terms of the paving conjecture, the Feichtinger conjecture, and the Bourgain-Tzafriri conjecture.

**690/691/692 Advanced Topology/Geometry Series**

*MA 691: Classifying spaces*

We will develop some relatively elementary topics { geometric cochains, Hopf invariants and

configuration spaces (GCHICS) { which can be fruitfully applied at the interface of algebraic

topology and geometric topology, algebra and combinatorics. Geometric cochains use submani-

folds to define cochains; Hopf invariants use linking numbers to distinguish homotopy as Hopf first

did; configuration spaces parametrize collections of points in a background space. Development

of these basic topics in turn rests on elementary differential topology, as treated in MA 531/2,

and this advanced course will be accessible to anyone who has done that course along with the

basic algebraic topology sequence, or is willing to fill in such material.

The course will develop three circles of thought (manifolds of thought?), where GCHICS are

applied to geometric topology, to group cohomology and related algebra, and to homotopy theory

of spaces. In comparison with first proposal(s) for this course, the current version is simultaneously

more accessible, broader, and with more interface with currently open research questions. (For

those looking forward to learning about classifying spaces, we will develop them in the second

circle of thought.) But because of this simultaneous breadth, accessibility and interface with open

mathematics, we will almost certainly only cover the first two circles of ideas, and continue the

third as a reading course in the spring.

## Math Seminars

**Math 607 Number Theory**

We will discuss a variety of topics from the Geometry of Numbers and Arithmetic Statistics. We will consider basic problems such as “how many integer points are there in a given circle?” to motivate and extend the Gauss circle problem and more general problems in enumerating lattice points. We will also discuss Minkowski Convex Body problem and some of its applications in Diophantine Geometry. Other topics include representation of integers or rationales by quadratic forms, local-global principle, Chevalley-Warning Theorem, and Diophantine equations over function fields. This sequence will be suitable for any graduate students with basic background in Algebra and Analysis. The plan is to make the two parts of the course independent. The specific topics will be decided in the beginning of each quarter based on participating students’ interests.

**Math 607 Topological field theories and tensor categories**

This will be an introductory course on topological quantum field theories from algebraic point of view. By definition such a theory is a symmetric tensor functor from the category of cobordisms to a symmetric tensor category. Thus we discuss the general theory and some examples of tensor categories (including Deligne’s categories). Some interesting examples in dimension 3 require modular tensor categories which are tensor but non-symmetric; this also will be discussed in the class.

**Math 607 Computer Algebra**

Introduction to sage/python. Experimental math; mathematical illustration; data visualization; object oriented programming; functional programming; computer algebra. Various topics and applications from graduate and undergraduate mathematics as well as “the real world”, all of which are an excuse to write code. Classes are in-class pair-programming workshops, not lectures. Suitable for students intending to use software for math research, and/or considering a career outside academia. Take this class instead of signing up for some random undergraduate programming course.

**607 Applied Math II: Statistical Learning**

This course will cover statistical and machine learning theory using foundational approaches (i.e. not neural-network based, as these will be the focus of the third course in the Applied Math sequence). These will 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. Homework assignments will be a mix of pen-and-paper calculations together with implementations and applications of machine learning algorithms to real and synthetic data using Python. The main prerequisites for this course are calculus and linear algebra. Prior familiarity with probability and statistics or with coding will be helpful but is not necessary.