This is the schedule for the summer 2017 mathematics graduate studentrun minicourses at UT Austin. You can sort by subject, speaker, and date.
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Topic  Speaker(s)  Dates  Time and Location  Abstract  Notes 

Intro to Spectral Sequences  Adrian Clough, Arun Debray, Ernie Fontes, Richard Hughes, Richard Wong 
May 812  RLM 10.176 1112pm, 12pm 

Abstract. We will first introduce the framework of spectral sequences, and then cover the AtiyahHirzebruch, Serre, EilenbergMoore, and Grothendieck spectral sequences. 

Spectral Sequences in Equivariant/Stable Homotopy Theory  Ernie Fontes, Richard Wong  May 1519  RLM 10.176 1112pm, 12pm 

Abstract. We will focus on spectral sequences used in homotopy theory, namely the homotopy fixed point spectral sequence, the Adams spectral sequence, and the May spectral sequence. 

Review: Equivariant Stable Homotopy Theory  Adrian Clough, Arun Debray, Richard Wong 
May 2226  RLM 10.176 1112pm, 12pm 

Abstract. This is a review of Andrew Blumberg's course on equivariant stable homotopy theory. We will focus on calculating examples of Bredon cohomology, Mackey functors, and Tambara functors. 

Classification of TQFTs (Cobordism Hypothesis)  Adrian Clough, Yuri Sulyma  May 29  June 2  RLM 10.176 1112pm, 12pm 

Abstract: we will discuss Lurie's expository article on the classification of topological quantum field theories, which famously sketches a proof of the cobordism hypothesis. The article is divided into four sections.
Yuri will discuss §1&2 on Monday. From Tuesday to Thursday we will discuss §3, with Adrian speaking on Tuesday and the other two days to be determined. On Friday we will either discuss §4 or have a general discussion. 

Hyperkähler geometry/Higgs bundles  Omar Kidwai, Sebastian Schulz  June 1216  RLM 11.176 1112pm, 12pm 

Abstract. This course will largely be a recap of Andy Neitzke's course on Higgs bundles. Depending on the audience, the topics might vary but a suggestion is as follows:
All of these are not only interesting geometric structures but carry a lot of exciting physics that we could try to talk about if time permits. 

Entropy Theory in Symbolic Dynamics  Frank Lin  June 1216  RLM 12.176 1112pm 

Abstract. Entropy is a way of measuring the complexity or randomness of a dynamical system. We give an introduction to the classical theories of topological and measure theoretic entropy, discussing important examples and theorems, including the variational principle which connects the two theories. Then we give a survey of entropy theory when the acting group is more general than the integers  including amenable and nonamenable (residually finite and sofic) groups. Most of the time we will focus on symbolic dynamical systems. 

Characteristic Classes  Arun Debray  June 1923  RLM 11.176 1112pm, 12pm 

Abstract. In this course, we'll introduce the use of characteristic classes in algebraic topology, with the goal of learning how to use them to solve problems.


Review: Riemannian Geometry  Max Stolarski  June 2630  RLM 11.176 1112pm 

Abstract. This will be a review of Dan Freed’s Spring ’17 Riemannian geometry course. The course website can be found here. 

Kähler Geometry  Max Stolarski  July 1014  RLM 11.176 1112pm 

Abstract. Kahler geometry lies at the intersection of Riemannian and complex geometry. We will introduce the basic concepts of Kahler geometry in the course of proving the Kodaira embedding theorem, which states that a compact complex manifold embeds into projective space if and only if it has a positively curved (or very ample) holomorphic line bundle. 

Topics in Parabolic PDE  TBD  July 1721  TBD  
Abstract. 

Mapping Class Groups  Max Riestenberg  July 1721  RLM 11.176 1112pm, 12pm 

Abstract.We will go through some bits of A Primer on Mapping Class Groups by Farb and Margalit.
This schedule is still a bit flexible, if anyone has something they are especially interested in. 

Period Mappings and Period Domains  Jonathan Lai, Yan Zhou  Aug 711  RLM 11.176 12pm 

Abstract. The goal of this week will be to use concrete examples to illustrate special structures on the cohomology groups of compact Kahler manifolds, namely Hodge structures, and discuss their basic properties and applications . In the special cases of complex curves and K3 surfaces, we will see that global Torelli theorem exists. If time permits, we will see that in the cases of singular varieties and quasiprojective varieties, generalized Hodge structure, namely mixed Hodge structures, still exist. 

Primer on Algebraic Number Theory  Tom Gannon  Aug. 1418  RLM 13.156 12pm 

Abstract. We'll go over some things that will be assumed as prerequisite knowledge for the fall's Algebraic Number Theory. Two topics I had in mind to review were the proof that any field extension L/F can be written as a separable extension and a purely inseparable extension, and review modules over a principal ideal domain, and in particular prove that any torsion free finitely generated module over a PID is free. If there's something you want to go over, email Tom Gannon. 