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# Mathematical Physics

This series consists of talks in the area of Mathematical Physics.

## Seminar Series Events/Videos

Currently there are no upcoming talks in this series.

## Symplectic duality and a presentation of the cohomology of Nakajima quiver varieties

Jeudi jan 28, 2016
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We will discuss a (conjectural) explicit presentation for the equivariant cohomology of Nakajima quiver varieties of type ADE. This presentation arises as a shadow of the expected symplectic duality between slices to Schubert varieties in the affine Grassmannian and Nakajima quiver varieties (a.k.a. the expected Coulomb and Higgs branches for a quiver gauge theory).

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## 2-associahedra and functoriality for the Fukaya category

Jeudi jan 21, 2016
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Categorical symplectic geometry studies an invariant of symplectic manifolds called the "Fukaya (A-infinity) category", which consists of the Lagrangian submanifolds and a symplectically-robust intersection theory of these Lagrangians.  Over the last two decades the Fukaya category has emerged as a powerful tool: for instance, it has produced inroads to Arnol'd's Nearby Lagrangians Conjecture, and it allowed Kontsevich to formulate the the Homological Mirror Symmetry conjecture.

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## Cluster Structures on Higher Teichmuller Spaces for Classical Groups - Ian Le

Jeudi jan 07, 2016

Let $S$ be a surface, $G$ a semi-simple group of type B, C or D. I will explain why the moduli space of framed local systems $A_{G,S}$ defined by Fock and Goncharov has the structure of a cluster variety, and fits inside a larger structure called a cluster ensemble. This was previously known only in type A. This gives a more direct proof of results of Fock and Goncharov for the symplectic and spin groups, and also allows one to quantize higher Teichmuller space in these cases.

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## Geometric Langlands and symplectic duality

Jeudi déc 03, 2015
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I will review the possible role in Geometric Langlands
of N=4 boundary conditions in four-dimensional supersymmetric Yang Mills theory.
The action of S-duality on such boundary conditions can be understood
in terms of symplectic duality.

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## Holomorphic Floer Quantization

Jeudi nov 19, 2015
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## The AGT conjecture for gauge theories on A-type ALE spaces, K-theoretic/Cohomological Hall algebras and sheaves on stacks

Jeudi nov 12, 2015
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K-theoretical/Cohomological Hall algebras, associated with the stack of zero-dimensional sheaves on $\mathbb{C}^2$, play a prominent role in the proof, given by Schiffmann and Vasserot, of the AGT conjecture for

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## Representations of truncated shifted Yangians and symplectic duality

Jeudi oct 29, 2015
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We study the representation theory of truncated shifted Yangians. These algebras arise as quantizations of slices to Schubert varieties in the affine Grassmannian. We will describe the combinatorics of their highest weights, which is encoded in Nakajima's monomial crystal. We also prove Hikita's conjecture in this context.

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## Basic aspects of 3d N=4 theories and symplectic duality

Jeudi oct 15, 2015
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I'll discuss the two topological twists of 3d N=4 theories, and explain how to understand them in the AKSZ/BV formalism, and how they relate to twists of 4d N =2 theories.  Symplectic duality then takes the form of an equivalence between 3d N=4 theories which interchanges the two topological twists.  I'll also introduce monopole operators and explain the role they play in symplectic duality.

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## Half-BPS boundary conditions in 3d N=4 theories

Jeudi oct 08, 2015
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## Boundaries and D-modules in 3d N=4 theories

Jeudi oct 01, 2015
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After a quick review of the Higgs and Coulomb branches of 3d N=4 theories, I'll introduce some simple classes of boundary conditions and explain how they lead to (pairs of) modules for certain (pairs of) quantum algebras. I will focus on abelian theories, for which the relevant boundary conditions/modules can be described using the geometry of (pairs of) hyperplane arrangements. From this, the simplest examples of symplectic-dual modules will arise.

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