Alternatively have a look at the program.

## Some applications of topology to physics I

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.

## Microlocal category of a symplectic manifold

Given a symplectic manifold $M$, one can consider its deformation quantization, i.e. an associative multiplication law on functions on $M$ that depends on a formal parameter $h$. When $M$ is the cotangent bundle of a manifold $X$, one essentially recovers the algebra of differential operators on $X$, or rather of $h$-differential operators $P(x, hd/dx)$. Modules over differential operators are well known to have interesting applications to PDE and other topics, so it is natural to hope that modules over deformation quantization would be interesting as well.

## Derived deformations and quantum cohomology

The simplest example of the interpretation of quantum cohomology as a deformation phenomenon is

served by the comparison of (genus zero) quantum cohomology ring of $P^n$ with miniversal unfolding space of the singularity

$A_n$. I will be discussing the vast generalization of the deformation formalism and speculating about its possible

applications to quantum cohomology theory.

## Fourier-Sato transform and Lusztig symmetries

In this talk I will explain how to compute, in terms of linear algebra data, the Fourier-Sato

transform of perverse sheaves over a complex affine space, smooth along a hyperplane arrangement.

As an example a geometric interpretation, and a generalization, of Lusztig's braid group action on

representations of quantum groups will be given. A joint work with M. Finkelberg and M. Kapranov

## Non-Archimedean Holography

We propose a discretization of the AdS/CFT correspondence with bulk space the $p$-adic Bruhat-Tits trees.

In addition to a consistent description of field theory on the boundary and gravity on the bulk, this approach

allows for a natural interpretation of the more recent viewpoint on holography, where bulk geometry emerges

from quantum entanglement on the boundary, in terms of a construction of holographic classical and quantum

codes on the Bruhat-Tits trees.

## Real models for the framed little $n$-disks operad

The little $n$-disks operad $D_n$ is a topological operad of rectilinear embeddings of a number of disjoint "little" disks into the unit disk. Its real homotopy type is known: Due to work of Kontsevich it is formal (over $\mathbb{R}$), i.e., there is a zigzag of (homotopy) Hopf cooperads relating the cooperad of differential forms $\Omega(D_n)$ with the cohomology cooperad $H(D_n)$.

## Universal tensor categories via representation theory of supergroups

For each of the four series of classical supergroups $GL(m|n)$, $OSP(m|n)$, $P(n)$ and $Q(n)$ we construct universal symmetric monoidal rigid categories by taking certain inverse limits . In the first two cases these categories are abelian envelopes of the Deligne categories $GL(t)$ and $O(t)$ (when t is an integer) while for $P$ and $Q$ we obtain some new tensor categories.

## Higgs bundles and mirror symmetry

The SYZ mirror of the moduli space of Higgs bundles for a group $G$ is interpreted as the moduli space for the Langlands dual group. The talk will discuss some of the outcomes and problems related to this, with particular reference to holomorphic Lagrangian submanifolds. Some instructive examples arise from a recent paper of Gaiotto.

## Brauer Groups in Algebraic Topology I

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.

## Quantum Hodge Field Theory

We introduce quantum Hodge correlators. They have the following format: Take a family $X/B$ of compact Kahler manifolds. Given an oriented topological surface $S$ with {special points} on the boundary, we assign to each interval between special points an irreducible local system on $X$, and to each special point an Ext between the neighboring local systems. A quantum Hodge correlator is assigned to this data and lives on the base $B$. It is a sum of finite dimensional convergent Feynman type integrals.

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