♦ Dan Freed (University of Texas at Austin)

Topics in Field Theory and Topological Phases of Matter I, II, III, IV, V 

Abstract:

Over these 5 lectures I will discuss several advances in the classification of phases of matter, including the classification of invertible field theories. I will also introduce general ideas in field theory, especially topics related to symmetry. 

♦ David Hansen (Max Plank Institute)

Geometric Eisenstein series and the Fargues-Fontaine curve I, II, III, IV

Abstract: 

The cohomology of local Shimura varieties, and of more general spaces of local shtukas, is of fundamental interest in the Langlands program. The Harris-Viehmann conjecture, roughly speaking, describes how the cohomology of local Shimura varieties interacts with parabolic induction. I will formulate a generalization of the HV conjecture, and then lay out a proof strategy, based on studying geometric Eisenstein series functors for stacks of bundles on the Fargues-Fontaine curve. Joint work in progress with Peter Scholze.

♦ Hiraku Nakajima (IPMU)

A mathematical approach towards Coulomb branches of 3d SUSY gauge theories and related topics I, II, III, IV, V   

Absratct:

I and 2. I will briefly recall a symplectic reduction, which gives Higgs branches of 3d N=4 supersymmetric gauge theories. Then I will turn to a definition of Coulomb branches. It uses some tools in geometric representation theory, such as convolution algebras on equivariant homology groups, affine Grassmannians. I will briefly review them.

3. Symplectic duality was introduced by Braden, Licata, Proudfoot and Webster, as a relation between quantization of a pair of algebraic symplectic varieties. Higgs and Coulomb branches of a common gauge theory seem to be an example of such a pair. I will explain some examples of expected relations between them and their quantizations.

4 and 5. I will explain constructions of Kac-Moody Lie algebra representations on `homology' of Higgs and Coulomb branches for the case of quiver gauge theories. Higgs case is my construction in early '90s. Coulomb case is geometric Satake for finite type quiver, and its conjectural generalization.

♦ Eric Rowell (Texas A & M University)   

The Mathematical Foundations of Topological Quantum Computation: Anyons, Braids and Categories I, II, III, IV, V  

Absratct:

I: In this lecture we will focus on motivating and modeling TQC.  We introduce the standard quantum circuit model and then discuss the algebraic/topological model for topological phases of matter. 
II: The relative computational power of TQC and the QCM will be discussed, and we will describe an algorithm for approximating Jones polynomial evaluations on a TQC and compare it to classical algorithms. 
III: How does one detect whether an anyon is abelian or non-abelian?  How do we distinguish universal anyons from non-universal anyons?  We will describe a simple rule for the first question and an equally simple, but conjectural, rules for the second. 
IV: To classify topological phases of matter into something like a periodic table we must study the related problem for modular categories.  The rank-finiteness theorem makes this a feasible problem from the classification-by-rank point of view.  We may also look at a coarser classification in terms of Witt classes (of anyon models related by certain phase transitions). 
V: The last lecture will touch on some ways in which we may extend our view to fermionic 2D systems, loop-like vortices in 3-dimensions and other situations.