Graduate Courses 2020/2021
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 2020 | WINTER 2021 | SPRING 2021 |
| 513 Intro to Analysis I L. Fredrickson (12:30) |
514 Intro to Analysis II L. Fredrickson (12:30) |
515 Intro to Analysis III Y. Shen (12:30) |
| 531 Intro to Topology I P. Hersh (9:30) |
532 Intro to Topology II N. Proudfoot (9:30) |
510 Intro to Manifolds N. Proudfoot (9:30) |
| 544 Intro to Algebra I A. Kleshchev (14:00) |
545 Intro to Algebra II A. Kleshchev (14:00) |
546 Intro to Algebra III A. Kleshchev (14:00) |
| 607 Dimer Models B. Young (9:30) |
607 Connections and Char Classes N. Addington (9:30) |
607 Monoidal Categories J. Brundan (9:30) |
| 607 Number Theory I E. Eischen (12:15 MW) |
607 Number Theory II E. Eischen (12:15 MW) |
607 Geometric PDEs W. He (12:30) |
| 607 Total Positivity J. Matherne (12:15 TR) |
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| 616 Real Analysis I M. Bownik (8:00) |
617 Real Analysis II M. Bownik (8:00) |
618 Real Analysis III M. Bownik (8:00) |
| 619 Complex Analysis S. Akhtari (14:15 TR) |
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| 634 Algebraic Topology I N. Addington (11:00) |
635 Algebraic Topology II R. Lipshitz (11:00) |
636 Algebraic Topology III R. Lipshitz (11:00) |
| 647 Abstract Algebra I A. Polishchuk (14:00) |
648 Abstract Algebra II A. Polishchuk (14:00) |
649 Abstract Algebra III A. Polishchuk (14:00) |
| 672 Theory of Probability D. Levin (15:30) |
673 Theory of Probability P. Ralph (15:30) |
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| 681 Representation Theory J. Brundan (14:00) |
682 Representation Theory V. Ostrik (14:00) |
683 Representation Theory V. Ostrik (14:00) |
| 684 Functional Analysis H. Lin (9:30) |
685 Functional Analysis H. Lin (11:00) |
686 Functional Analysis C. Phillips (9:30) |
| 690 Characteristic Classes D. Sinha (12:30) |
691 Spectral Sequences Y. Shen (10:15 TR) |
692 Surgery Theory B. Botvinnik (11:00) |
Math Course Descriptions
619 Complex Analysis
This is an introductory course to the subject of Complex Analysis. We will discuss topics including the residue theorem, the argument principle, the maximum modulus principle, infinite product factorization, the Fourier transform and Fourier inversion. Some of these topics will be motivated by their applications in Number Theory. One particular goal is to understand the proofs of the Hermite-Lindemann-Weierstrauss Theorem and the Gelfand-Schneider Theorem in Transcendental number theory. For example, using techniques from complex analysis, we will see why the real number π cannot satisfy a polynomial equation with integer coefficients. Another goal is to study the proof of Dirichlet’s Theorem on primes in arithmetic progressions from classical analytic number theory.
681/682/683 Advanced Algebra Series
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.
690/691/692 Advanced Topology/Geometry Series
Math Seminars
Dimer Model
The dimer model is a probability distribution on the set of perfect matchings of a graph G, first studied by Kasteleyn in 1961, and by many others since. The model arises in many surprising ways in random matrix theory, statistical mechanics, algebraic geometry, cluster algebras, and statistical mechanics, with intricate combinatorics underlying all of these applications. This course will serve as an introduction to the dimer model, with the aim of reaching some rather recent developments in the closely-related “double dimer model”. If time remains, we can cover other topics according to student interest.
Monoidal Categories
This course will be a introduction to monoidal categories. The most basic of all monoidal categories is the category of finite-dimensional vector spaces over a field; the monoidal structure is the usual tensor product. A much broader source of examples comes from taking some fixed category C, then considering the new category whose objects are functors F:C — C and whose morphisms are the natural transformations between those endofunctors. Monoidal categories are also ubiquitous in representation theory, starting from the category of representations of a finite group or a semisimple Lie algebra, and more generally modules over a Hopf algebra. I will take a more diagrammatic approach to the subject, using so-called Penrose diagrams. Using them, we will discuss basic notions such as duality/adjunction, pivotal, symmetric, and braided monoidal categories. Then I will try to focus as much as possible on some of the fundamental examples starting from the Temperley-Lieb category, which is a model for the representations of the Lie algebra sl_2 and its quantum analog. This example is intimately related the definition of the Jones polynomial in knot theory. I will also discuss other important diagrammatic monoidal categories such as the HOMFLY-PT skein category, and the Heisenberg category. The latter is closely related to the representation theory of symmetric groups and the theory of symmetric functions. I will partly follow the following sources but I intend to focus on examples as much as possible! Turaev and Virelizier, Monoidal categories and topological field theory. Etingof, Gelaki, Nikshych and Ostrik, Tensor categories.
Geometric PDEs
1. The major topic is to give a detailed treatment of Yau’s proof of the Calabi conjecture—it is now known as Calabi-Yau theorem. We shall introduce briefly the Kahler manifolds and a brief history of Calabi conjecture, and then introduce necessary analytic tools (elliptic PDE theory) to solve the problem.A solid grasp of concept of differential manifolds is certainly necessary; it would be very helpful to know the basic notion of Riemannian metric and curvature, even though it is not mandatory (we shall develop all these notions in the setting of Kahler metrics).Students are supposed to have learned 600 level analysis (which would be necessary to be mathematical mature to follow the actual proof, such as L^p space, Banach space etc).
2. We shall also include some basic analytic tools which are necessary for geometric analysis, including Sobolev spaces, Holder continuity etc.
3. We shall also cover the basic idea of calculus of variations, including Hilbert’s approach to solve the classical Laplacian equation.
4. If time permits, we might introduce some relevant open problems.
Total Positivity
A matrix is totally positive (totally nonnegative) if all of its minors are positive (nonnegative) real numbers. Total positivity is related to a variety of things, including scattering amplitudes in high energy physics, probability and stochastic processes, canonical bases for quantum groups, cluster algebras, and combinatorics. After treating totally nonnegative matrices, we will move on to study the totally nonnegative Grassmannian. Describing the topology of this space is a beautiful story related to affine Bruhat order, dimer models on a disk, and a host of other combinatorial objects. If time permits and depending on the interests of the participants, we may look further than the Grassmannian: Lusztig defined the totally nonnegative partial flag variety G/P for any split real reductive group G.
The prerequisites are minimal. A sound knowledge of linear algebra is most important. The graduate algebra and topology sequences would be helpful, but topics from these courses can be developed as needed.
Algebraic Number Theory
This two-term algebraic number theory sequence will introduce standard topics essential for students considering working in number theory and also potentially useful for students working in related fields, including algebraic geometry, representation theory, and topology.These courses will focus on the structure of number fields (finite extensions of the rational numbers), including ideal class groups, unit groups, cyclotomic extensions, quadratic reciprocity, special cases of Fermat’s Last Theorem, local fields, global fields, adeles, and an introduction to class field theory. The main prerequisite is abstract algebra at the level of the 500-level algebra sequence (with an emphasis on the Galois theory covered in Math 546), as well as elementary knowledge of modules. Students who have seen some commutative algebra (at the level covered in the 600-level algebrasequence) will be better prepared. Students who have not seen commutative algebra will need to do a little extra work or accept a few facts as black boxes. Topics and pace may be adjusted according to the background and interests of the students enrolled in the course (e.g. since some of these topics have been covered in the student number theory seminar in 2017-2018).
Surgery theory and applications
1. Elementary surgery theory, i.e., Morse functions on manifolds and handle decomposition.
2. h-cobordism and s-cobordism theorems.
3. Basics on Whitehead torsion.
4. Normal maps and surgery problem.
5. Surgery of simply-connected manifolds.
6. Surgery applications to differential geometry.
Connections and characteristic classes
I will begin with connections on vector bundles and principal bundles, and several equivalent definitions of curvature, which is (roughly) a matrix-valued 2-form. By taking the trace, determinant, Pfaffian, etc. we get forms representing characteristic classes. The goal of the course will be the Chern-Gauss-Bonnet theorem, which says that for an oriented real vector bundle, the Pfaffian gives the Euler class, which is Poincare dual to the vanishing locus of a generic section. We will also discuss Chern classes of complex vector bundles, which are Poincare dual to the dependency locus of n generic sections.The 690 course on characteristic classes is not a prerequisite. This material is complementary to that material, and has quite a differentflavor: geometry rather than homotopy.