This is the schedule for the Summer 2020 mathematics graduate student-run minicourses at UT Austin. Note that all times are in Central Daylight Time (UTC -5). Most courses will be run through Zoom.

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Topic Speaker(s) Dates Time (CDT) Abstract
Introduction to Representation Theory Tom Gannon May 18-29 MWF 1:30-3:30pm

Abstract. In this course, we will learn the fundamentals of the representation theory of (complex) Lie groups and Lie algebras. We'll first define Lie groups, and then discuss why the representation theory of Lie groups (mostly) reduces to studying representations of their Lie algebras, that is, the tangent space of the group at the identity. We'll then discuss a very important class of Lie algebras, called semisimple Lie algebras. We'll then examine the representation theory of two of the most basic Lie algebras, the Lie algebras of sl_2 and sl_3. Using these examples, we will build the vocabulary to classify representations of all semisimple Lie algebras!

This course will run on Monday, Wednesday, and Friday from May 18 to May 29.

Malliavin calculus Joe Jackson May 25-29 11am-12pm, 2-3pm

Abstract: The goal of this course is to learn the basics of Malliavin calculus, and then see some applications to stochastic differential equations and partial differential equations. We'll start by working through a few chapters of Nualart's book, and then review some basic SDE theory. Finally, we'll describe the connection between Hörmander's Theorem and SDEs, and explain how Malliavin calculus can be used to prove Hörmander's Theorem.

Boundaries of groups and spaces Theodore Weisman June 1-5 10-11am, 1-2pm

Abstract: This course will be an investigation of various kinds of boundaries of spaces that sometimes show up when doing geometric group theory. We'll be focusing on CAT(0) spaces, hyperbolic groups, and relatively hyperbolic groups. We'll define each of these kinds of spaces/groups, and give (at least) one construction for a natural notion of "boundary at infinity" for each of them. The main goal of the course will be to explore the various kinds of structures and properties these boundaries can have, and how they interact with the properties of the space they bound. Specifically, we'll look at things like

  • different topologies and metrics on the boundary
  • quasi-symmetry (and maybe quasi-conformality)
  • quasi-isometric invariance (and lack thereof)
  • dynamics of group actions on the boundary
  • limit sets and quasiconvex subgroups
This is a lot, so I emphatically do NOT expect to get through everything on this list.

The Yang-Mills equations Jeffrey Jiang June 1-5 11am-12pm

Abstract: In this course we will mostly follow the paper "The Yang-Mills Equations over Riemann Surfaces" by Atiyah and Bott. We will first review principal bundles and connections, and then define the Yang-Mills equations. We will then restrict to the case of Riemann surfaces, where we relate the space Yang-Mills connections on principal U(n)-bundles with moduli spaces of holomorphic vector bundles.

What have Higgs bundles ever done for me? Sebastian Schulz June 8-12 11am-12:30pm

Abstract: This is an introduction to Higgs bundles and the various places they appear in geometry. The lectures will roughly be split up as follows:

  1. Motivation and review of hyperkähler geometry
  2. The moduli space of (Higgs) bundles
  3. Non-abelian Hodge theory
  4. SYZ mirror symmetry for moduli spaces of Higgs bundles
Lie groups and symmetric spaces Max Riestenberg June 15-19 1-3pm

Abstract: This minicourse is an introduction to symmetric spaces and related topics. Here’s a rough outline:

  1. Riemmannian symmetric spaces
  2. More general homogeneous spaces
  3. Symmetric spaces of noncompact type
  4. Boundaries of noncompact symmetric spaces
  5. Discrete subgroups of Lie groups
As time permits, I might talk about classifying real semisimple Lie group, Mostow rigidity, proper actions on homogenous spaces, or Anosov representations.

Abelian varieties and the Mordell-Weil Theorem Zachary Gardner July 6-10 10am-12pm

Abstract: The goal of this course will be to prove the Mordell-Weil Theorem for abelian varieties over global fields. Here's a rough outline"

  1. Introduction, group schemes, abelian schemes and varieties
  2. Families of line bundles on abelian varieties, dual abelian varieties
  3. Isogeny, torsion, start proof of Weak Mordell-Weil Theorem
  4. Finish proof of Weak Mordell-Weil Theorem, start height theory
  5. Finish height theory, Neron-Tate pairing, proof of Mordell-Weil Theorem
Characteristic classes Arun Debray July 6-10 1-2pm, 2-3pm

Abstract: In this minicourse, I will give an overview of characteristic classes, focusing on applications. I'll cover the basics of Stiefel-Whitney classes, Wu classes, Chern classes, Pontrjagin classes, and the Euler class. I will also spend some time on Chern-Weil theory.

Introduction to entropy theory in dynamical systems Yuqing (Frank) Lin July 6-10 11am-12pm, 3-4pm

Abstract: The plan is to go through the following topics:

  • Shannon entropy
  • Kolmogorov-Sinai entropy
  • Topological entropy in symbolic dynamics
  • Shannon-McMillan-Breiman theorem
  • Ornstein theory
Min-max theory and minimal surfaces Daniel Restrepo July 6-10 11am-12pm

Abstract: This course is designed for people with some background in variational methods in analysis and familiar, in some sense, with the basic notions in geometric analysis (perimeter, BV functions, Hausdorff measures, coarea formulas and related concepts). Nonetheless, in the first part, we will start reviewing basic concepts of Sobolev spaces, BV spaces, Calculus in Banach spaces and the direct method. The second part is devoted to the study of the main tool in min-max theory: the deformation lemma. Time provided, we will discuss the link between min-max theory, Schauder degree and Morse theory. In the third session, we will start exploring some applications in semilinear PDE including the Allen-Cahn equation, which will serve as a link between minimal surfaces and this theory. Fourth and fifth sessions will be focused in the applications to the Almgren-Pitts theory recently developed by Fernando C. Marques and André Neves (and their students) as a limiting case of the classical results in Min-max theory.

oo-categories tutorial Adrian Clough July 20-24 10-11:30am

Tentative Description: A tutorial for the use of quasi-categories where I will work though a bunch of simple examples, exhibiting the mechansms through which one obtains both rigorous statements and proofs.

Intersection theory and enumerative geometry Rok Gregoric July 20-24 12-1:30pm, TTH 2-3pm

Tentative Description: There are about 5819539783680 or so wonderful computations in enumerative geometry, where you set up a geometric question (how many BLAH BLAH BLAH), and algebraic geometry politely responds: finitely many, and here's a bizzare number! The goal of this minicourse would be to learn enough intersection theory to then be able to do some of these kind of computations, and have great fun.

Heegaard Splittings and Dehn Surgery Jonathan Johnson and Feride Ceren Kose July 20-24 10-11am, 2-4pm

Abstract: This mini-course covers some standard topics in 3-manifold topology including Heegaard splittings, the mapping class groups of closed surfaces, Seifert fibered spaces and Dehn surgery. This course will closely follow lectures 1 and 2 of Saveliev's Lectures on the Topology of 3-Manifolds, 2nd Edition. This course will be followed by the Kirby Calculus and 4-Manifolds mini-course covering lectures 3, 5, and 6. The glossary at the beginning of Saveliev's book is a good list of prerequisite topics. However, this course should be accessible to anyone comfortable with algebraic topology and differential topology preliminary material.

Kirby Calculus and 4-manifolds Jonathan Johnson and Feride Ceren Kose July 27-31 10-11am, 2-4pm

Abstract: This mini-course covers some standard topics in 4-manifold topology including Kirby calculus, intersection forms, Seifert homology spheres and the Rohlin invariant. This course will closely follow lectures 3, 5 and 6 of Saveliev's Lectures on the Topology of 3-Manifolds, 2nd Edition. This course will be preceded by the Heegaard Splittings and Dehn Surgery mini-course covering lectures 1 and 2. However, you are welcome to attend this course even if you did not attend the previous mini-course. The glossary at the beginning of Saveliev's book is a good list of prerequisite topics. However, this course should be accessible to anyone familiar with the basic concepts of Dehn surgery and comfortable with algebraic topology and differential topology preliminary material.

The Geometric Satake Equivalence Alberto San Miguel Malaney and Saad Slaoui July 27-31 3:30-5pm

Tentative Description: The Geometric Satake Theorem is a statement which says that the category of representations of some reductive algebraic group G can be viewed as sheaves on a special infinite dimensional space called the affine Grassmannian of the Langlands dual group of G. This course would discuss the motivation, precise statement, and the proof of this theorem. Ideally, we could also discuss how this theorem allows us to define "Hecke eigensheaves" in the Geometric Langlands program.

A Primer on Derived Algebraic Geometry Rok Gregoric Aug. 3-7 11am-12:30pm

Abstract.Derived algebraic geometry is a relatively young field, incorporating ideas from homotopy theory into algebraic geometry. The goal of this minicourse is to provide a glimpse into the theory, with no delusions of completeness. We wish to give somebody who is perhaps decently-enough familiar with usual (post-Grothendieck) algebraic geometry some idea about: 1.) what DAG is, and 2.) some of its applications to other areas.

Spectral Sequences Training Montage Arun Debray and Richard Wong Aug. 10-14 1-3pm

Abstract: We plan to focus on examples rather than theory, taking for granted the construction/existence of these spectral sequences and using them to compute interesting things. As both of us are algebraic topologists, our examples will primary be motivated from algebraic topology. Familiarity with a first course in algebraic topology, including homology/cohomology, is suggested for this minicourse.

There will also be an interactive problem session every day, and participation is strongly encouraged.

  • Monday:
    • What is a spectral sequence?
    • the Serre spectral sequence for a fibration
    • multiplicative structures
  • Tuesday:
    • Serre spectral sequence examples coming from group cohomology
    • the homotopy fixed-points spectral sequence
  • Wednesday:
    • the Atiyah-Hirzebruch spectral sequence
    • Steenrod squares and differentials in the Atiyah-Hirzebruch spectral sequence
  • Thursday:
    • comparison maps between spectral sequences and examples
    • comparing between homotopy fixed-points and Tate spectral sequences; the Greenlees spectral sequence
    • other useful and less-known tricks
  • Friday:
    • the Adams spectral sequence over 𝓐(1)
    • Margolis' theorem
    • algebraic structure on E2-page and use in extension questions
Introduction to the Geometric Langlands Correspondence Tom Gannon Aug. 17-21 2-3:30pm

Abstract: In this mini course, we'll provide context and give motivation related to the Geometric Langlands conjecture. We will start off by discussing the Langlands correspondence for the group GL_1, which is also known as class field theory. We will then discuss what the Langlands correspondence says for general reductive groups G for function fields. After that, we will talk about the generalization of this correspondence to the Geometric Langlands conjecture, which will lead us to the notion of a Hecke eigensheaf. We will then discuss why the naive Geometric Langlands conjecture is false, and how it can be corrected with the notion of indcoherent sheaves with nilpotent singular support. Time and interest permitting, we will also discuss the notion of an oper and the construction of a Hecke eigensheaf associated to an oper or discuss the local geometric Langlands program.