KIAS Summer 2021

Lecture Series at the Korea Institute for Advanced Study (Online Seminar)

Talk 1: Operators and higher categories in quantum field theory 

I. A complete mathematical definition of quantum field theory does not yet exist. Following the example of quantum mechanics, I will indicate what a good definition in terms could look like. In this good definition, QFTs are defined in terms of their operator content (including extended operators), and the collection of all operators is required to satisfy some natural properties.

II. After reviewing some classic examples, I will describe the construction of Noether currents and the corresponding extended symmetry operators.

III. One way to build topological extended operators is by "condensing" lower-dimensional operators. The existence of this condensation procedure makes the collection of all topological operators into a semisimple higher category.

IV. Topological operators provide "noninvertible higher-form symmetries". These symmetries assign charges to operators of complementary dimension. This assignment is a version of what fusion category theorists call an "S-matrix".

V. The Tannakian formalism suggests a way to recognize higher gauge theories. It also suggests the existence of interesting higher versions of super vector spaces with more exotic tangential structures.

Talk 2: The Mathematical Foundations of Topological Quantum Computation: Anyons, Braids and Categories

Talk 3: Geometric Eisenstein series and the Fargues-Fontaine curve I, II, III, IV 

Talk 4: A mathematical approach towards Coulomb branches of 3d SUSY gauge theories and related topics

Talk 5: Topics in Field Theory and Topological Phases of Matter