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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.

## Cluster Theory is the Moduli Theory of A-branes in 4-manifolds

Jeudi mai 19, 2016
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We'll explain the slogan of the title: a cluster variety is a space associated to a quiver, and which is built out of algebraic tori.

They appear in a variety of contexts in geometry, representation theory, and physics. We reinterpret the definition as: from a quiver (and some additional choices) one builds an exact symplectic 4-manifold from which the cluster variety is recovered as a component in its moduli space of Lagrangian branes. In particular, structures from cluster algebra govern the classification of exact Lagrangian surfaces in Weinstein 4-manifolds.

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## The classification of chiral WZW models

Mardi avr 05, 2016
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I will explain how to axiomatize the notion of a chiral WZW model using the formalism of VOAs (vertex operator algebras). This class of models is in almost bijective correspondence with pairs (G,k), where G is a connected (not necessarily simply connected) Lie group and k in H^4(BG,Z) is a degree four cohomology class subject to a certain positivity condition. To my surprise, I have found a couple extra models which satisfy all the defining properties of chiral WZW models, but which don't come from pairs (G,k) as above.

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## Quiver varieties and elliptic quantum groups

Jeudi mar 17, 2016
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## Functorial field theories from factorization algebras

Jeudi mar 10, 2016
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## Quantization, reduction mod p, and the Weyl algebra

Jeudi mar 03, 2016
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## Towards a cluster structure on the trigonometric zastava - Michael Finkelberg

Jeudi fév 11, 2016

This is a joint work with A.Kuznetsov and L.Rybnikov.

We study a moduli problem on a nodal curve of arithmetic genus 1, whose solution is an open subscheme in the zastava space for projective line. This moduli space is equipped with a natural Poisson structure, and we compute it in a natural coordinate system. We compare this Poisson structure with the trigonometric Poisson structure on the transversal slices in an affine flag variety.

We conjecture that certain generalized minors give rise to a cluster structure on the trigonometric zastava.

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## 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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