# 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.

## Beyond Geometric Invariant Theory

Wednesday Oct 04, 2017

Geometric invariant theory (GIT) is an essential tool for constructing moduli spaces in algebraic geometry. Its advantage, that the construction is very concrete and direct, is also in some sense a draw-back, because semistability in the sense of GIT is often more complicated to describe than related intrinsic notions of semistability in moduli problems. Recently a theory has emerged which treats the results and structures of geometric invariant theory in a broader context.

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## Moore-Tachikawa conjecture, affine Grassmannian and Coulomb branches of star-shaped quivers

Monday Oct 02, 2017
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Moore and Tachikawa conjecture that there exists a functor from the category of 2-bordisms to a certain category whose objects are algebraic groups and morphisms between $G$ and $H$ are given by affine symplectic varieties with an action of $G\times H$.  I will explain a proof of this conjecture due to Ginsburg and Kazhdan, and its relation to Coulomb branches of certain quiver gauge theories which allows to make interesting calculations.

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## An extended category for the BFN construction

Monday Sep 25, 2017
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I'll do my best to explain my approach to the BFN construction of (quantum) Coulomb branches.  This approach is based on viewing the BFN algebra as an endomorphism algebra in a larger category that's easier to present (and which we can draw some pretty pictures for).  In particular, this approach is helpful in understanding the representation theory of this algebra, and in constructing and analyzing tilting generators on Coulomb branches.

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## Mathematical Seminar - Davide Gaiotto

Monday Sep 25, 2017
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He will discuss relations between Virasoro and Kac-Moody conformal blocks, character varieties and quantum groups, and AGT.

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## Braided algebra and dual bases of quantum groups

Wednesday Jul 19, 2017
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The talk is based on my recent work with Ryan Aziz. We find a dual version of a previous double-bosonisation theorem whereby each finite-dimensional braided-Hopf algebra in the category of corepresentations of a coquasitriangular Hopf algebra gives a new larger coquasitriangular Hopf algebra, for example taking c_q[SL_2] to c_q[SL_3] for these quantum groups reduced at certain odd roots of unity.

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## Cluster duality and mirror symmetry for Grassmannians

Monday Jun 12, 2017
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We use the cluster structure on the Grassmannian and the combinatorics of plabic graphs to exhibit a new aspect of mirror symmetry for Grassmannians in terms of polytopes.

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## Donaldson-Thomas transformations for moduli spaces of local systems on surfaces

Wednesday May 31, 2017
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Kontsevich and Soibelman defined Donaldson-Thomas invariants of a 3d Calabi-Yau category with a stability condition. Any cluster variety gives rise to a family of such categories. Their DT invariants are encapsulated in single formal automorphism of the cluster variety, called the DT-transformation. An oriented surface S with punctures, and a finite number of special points on the boundary give rise to a moduli space, closely related to the moduli space of PGL(m)-local systems on S, which carries a canonical cluster Poisson variety structure.

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## Moduli of Vacua and Categorical representations

Friday May 19, 2017
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I will present some results on three-dimensional gauge theory from the point of view of extended topological field theory. In this setting a theory is specified by describing its collection of boundary conditions - in our case, a collection of categories (standing in for 2d TFTs) with a prescribed symmetry group G. We will apply ideas from Seiberg-Witten geometry to construct a new commutative algebra of symmetries for categorical representations (or line operators in the gauge theory) -  a categorification of Kostant's description of the center of the enveloping algebra.

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## Vacua and Singular Supports

Monday May 15, 2017
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The notion of singular support for coherent sheaves was introduced by Arinkin and Gaitsgory in order to carefully state the geometric Langlands conjecture.  This is a conjectural equivalence of categories of sheaves on certain moduli spaces: in order to make the conjecture reasonable one needs to restrict to sheaves which satisfy a certain "singular support condition".  In this talk I'll explain how to think about this singular support condition from the point of view of boundary conditions in twisted N=4 gauge theory.  Specifically, Arinkin and Gaitsgory's singular support condition arises

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## Symplectic realization of 4D Wall-Crossing Formula on HyperKahler Surfaces

Thursday May 04, 2017
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In this talk, we will discuss an open Gromov-Witten invariant on hyperKahler surfaces, including K3 surfaces and certain Hitchin moduli spaces. The invariant is defined via the Lagrangian Floer theory and satisfy the Kontsevich-Soibelman wall-crossing formula and are expect to recover the generalized Donaldson-Thomas invariants studied in the work of Gaiotto-Moore-Neitzke.

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## LECTURES ON-DEMAND

### Roger Melko: Perimeter Institute and University of Waterloo

Speaker: Roger Melko