## Course on Frobenius Manifolds

## Brauer Groups in Algebraic Topology III

Let $k$ be a field. The collection of (isomorphism classes of) central division algebras over $k$ can be organized into an abelian group $\mathrm{Br}(k)$, called the Brauer group of $k$. In this series of talks, I'll describe some joint work with Mike Hopkins on a variant of the Brauer group which arises in algebraic topology, controlling the classification of certain cohomology theories known as Morava $K$-theories.

## Brauer Groups in Algebraic Topology II

Let $k$ be a field. The collection of (isomorphism classes of) central division algebras over $k$ can be organized into an abelian group $\mathrm{Br}(k)$, called the Brauer group of $k$. In this series of talks, I'll describe some joint work with Mike Hopkins on a variant of the Brauer group which arises in algebraic topology, controlling the classification of certain cohomology theories known as Morava $K$-theories.

## Some applications of topology to physics III

An axiom system for special quantum field theories was introduced over 25 years ago by Segal and Atiyah.

It has been much elaborated and developed, particularly in the topological case. In these three lectures I will

discuss aspects of this mathematical theory and applications to problems in physics.

## Some applications of topology to physics II

An axiom system for special quantum field theories was introduced over 25 years ago by Segal and Atiyah.

It has been much elaborated and developed, particularly in the topological case. In these three lectures I will

discuss aspects of this mathematical theory and applications to problems in physics.

## Lecture on Anabelian Geometry

## Characters of representations of Lie (super)algebras and (mock)theta functions

Prerequisits:

Understanding of the Weyl character formula, universal enveloping algebra and Verma modules.

Knowledge of some elements of Lie superalgebra theory.

Knowledge of some elements of Jacobi theta functions and modular forms.

Programm:

1. Lie superalgebras

2. Ane Lie (super)algebras: loop and KM constructions

3. Character formula for integrable and admissible representations in the Lie algebra case and Jacobi

theta functions

4. Character formula for tame integrable and admissible modules in the Lie superalgebra case and

mock theta functions

## Modular-invariance in rational conformal field theory: past, present and future

This is a tentative outline of my two lectures. No prior knowledge of vertex

rings is assumed, though the foundations will be covered only sketchily due to

time constraints. The main purpose of the lectures is to explain how modularity

enters into VOA theory, and to discuss an approach to the general problem of

modular-invariance in rational CFT based on a theory of vector-valued modular

forms, Fuchsian systems, and the Riemann-Hilbert problem.

Optional background reading on VOAs:

U. Heidelberg, downloadable lecture notes on connections between VOAs and the

## Mock theta functions and representation theory of affine Lie superalgebras and superconformal algebras

One of the beautiful properties in representation theory of ane Lie

algebras is the SL2pZq-invariance of the space of characters of integrable modules

discovered by Kac-Peterson in the early 1980's.

However, for ane Lie superalgebras, modular invariance had long been quite un-

clear except for only a few cases. Recently a remarkable breakthrough was brought

by Zwegers, who constructed a modular function from the supercharacter of the

ane slp2|1q-module of level 1 by adding non-holomorphic correction term, which

is called the "modication" procedure.

## Rogers-Ramanujan Identities and Moonshine

The Rogers-Ramanujan identities and Monstrous moonshine are among

the deepest results which occur at the interface of number theory and representation

theory. In these lectures the speaker will discuss these identities, and describe recent

work with Duncan, Griffin on Warnaar on their recent generalizations. This will include

a framework of Rogers-Ramanujan identities and singular moduli, and the theory of

umbral Moonshine.

## Rogers-Ramanujan Identities and Moonshine

The Rogers-Ramanujan identities and Monstrous moonshine are among

the deepest results which occur at the interface of number theory and representation

theory. In these lectures the speaker will discuss these identities, and describe recent

work with Duncan, Griffin on Warnaar on their recent generalizations. This will include

a framework of Rogers-Ramanujan identities and singular moduli, and the theory of

umbral Moonshine.

## Rogers-Ramanujan Identities and Moonshine

The Rogers-Ramanujan identities and Monstrous moonshine are among

the deepest results which occur at the interface of number theory and representation

theory. In these lectures the speaker will discuss these identities, and describe recent

work with Duncan, Griffin on Warnaar on their recent generalizations. This will include

a framework of Rogers-Ramanujan identities and singular moduli, and the theory of

umbral Moonshine.

## Kac-Moody superalgebras

I will review results of C. Hoyt and V. Serganova on the classication of Kac-Moody superalgebras.

In the supercase several Cartan matrices may determine the same superalgebra and our denition

of Kac-Moody superalgebra is based on this fact. Surprisingly, all indecomposable Kac-Moody

superalgebras with isotropic roots are nite-dimensional or ane.

## Kac-Wakimoto character formula for affine Lie superalgebras.

I will outline a proof of the Kac-Wakimoto character formula for some irreducible highest weight

modules over ane Lie superalgebras. This formula will be used in V. Kac's lectures.

## The arithmetic of Eisenstein cohomology classes

## Batalin-Vilkovisky formalism in topological quantum field theory

## Lecture on Classical and higher spherical polynomials

## Minicourse by Marco Maculan

## Advanced Geometry

## Minicourse on dendroidal topology

Dendroidal topology is an extension of simplical topology, geared towards the theory of operads and of infinity-operads. In the first part of the course, we will discuss topological operads and their algebras, and the Boardman-Vogt resolution. Next, we will introduce the category of dendroidal sets, and the corresponding notion of infinity operad. This will naturally include a discussion of the Joyal model structure on simplicial sets and the corresponding notion of infinity category. We will discuss several models for the homotopy theory of dendroidal sets, e.g.