Summer Mini-Courses 2017

Abstract. We will first introduce the framework of spectral sequences, and then cover the Atiyah-Hirzebruch, Serre, Eilenberg-Moore, and Grothendieck spectral sequences.

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.

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.

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.

• §1 discusses TQFTs in general and discusses why (oo,n)-categories provide a good setting for TQFTs.
• §2 discusses the necessary technical background to make a formulation of the cobordism hypothesis, as discussed in §1, precise.
• §3 is the heart of the paper and gives the sketch of proof of the coborsism hypothesis.
• §4 discusses variants and various vistas.

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.

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:

• Day 1: Review of complex and symplectic geometry, Definition and properties of Kähler and Hyperkähler manifolds
• Day 2: Examples such as quaternionic vector spaces, Gibbons-Hawking spaces, Ooguri-Vafa space
• Day 3: Moduli spaces of quiver representations
• Day 4&5: Moduli space of Higgs bundles

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.

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.

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.

• Day 1: Introduction, four perspectives on characteristic classes
• Day 2: Stiefel-Whitney classes
• Day 3: Wu classes
• Day 4: Chern, Pontrjagin, and Euler classes
• Day 5: Genera and the Hirzebruch signature theorem

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

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.

Abstract.

Abstract.We will go through some bits of A Primer on Mapping Class Groups by Farb and Margalit.

1. Definitions and motivations
2. Hands-on practice with Dehn twists, etc.
3. Deeper into structure of MCG
4. Teichmuller Theory
5. Nielsen-Thurston

This schedule is still a bit flexible, if anyone has something they are especially interested in.

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 quasi-projective varieties, generalized Hodge structure, namely mixed Hodge structures, still exist.

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.