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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.

## Positive Representations of Split Real Quantum Groups

Lundi mai 01, 2017
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The notion of Positive Representations is a new research program devoted to the representation theory of split real quantum groups, initiated in a joint work with Igor Frenkel. It is a generalization of the special class of representations considered by J. Teschner for Uq(sl(2,R)) in Liouville theory, where it exhibits a strong parallel to the finite-dimensional representation theory of compact quantum groups, but at the same time also serves some new properties that are not available in the compact case.

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## Toward the geometry of double affine flag varieties and double affine Grassmannians

Vendredi avr 28, 2017
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I will discuss some results on double loop groups that point to geometric phenomena about double affine flag varieties and double affine Grassmannian. One result of this study is a definition of double affine Kazhdan-Lusztig polynomials.

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## Traces of intertwiners for quantum affine sl_2, affine Macdonald conjectures, and Felder-Varchenko functions

Lundi avr 24, 2017
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This talk concerns a family of special functions common to the study of quantum conformal blocks and hypergeometric solutions to q-KZB type equations.  In the first half, I will explain two methods for their construction -- as traces of intertwining operators between representations of quantum affine algebras and as certain theta hypergeometric integrals we term Felder-Varchenko functions.  I will then explain our proof by bosonization the first case of Etingof-Varchenko's conjecture that these constructions are related by a simple renormalization.

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## Hamiltonian and Lagrangian perspectives on elliptic cohomology

Lundi avr 10, 2017

The physics proof of the Atiyah-Singer index theorem relates the Hamiltonian and Lagrangian approaches to quantization of N=1 supersymmetric mechanics.

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## F-fields

Lundi mar 20, 2017
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An F-field on a manifold M is a local system of algebraically closed fields of characteristic p.  You can study local systems of vector spaces over this local system of fields.  On a 3-manifold, they are are rigid, and the rank one local systems are counted by the Alexander polynomial.  On a surface, they come in positive-dimensional moduli (p

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## Braid group symmetries of Grassmannian cluster algebras

Lundi mar 13, 2017
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We define an action of the k-strand braid group on the set of cluster variables for the Grassmannian Gr(k,n), provided k divides n. The action sends clusters to clusters, preserving the underlying quivers, defining a homomorphism from the braid group to the cluster modular group for Gr(k,n). Our results can be translated to statements about clusters in Fock-Goncharov configuration spaces of affine flags, provided the number of flags is even.

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Vendredi fév 24, 2017
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The Feynman diagram expansion for a Wilson loop observable in Chern-Simons gauge theory generates an infinite series of topological invariants for framed knots.  In this talk, I will describe a new perturbative formalism which conjecturally generates the same invariants for Legendrian knots in the standard contact R^3.  The formalism includes a `perturbative' localization principle which drastically simplifies the structure of calculations.  As time permits, I will provide some examples and applications.  This talk is based upon joint work with Brendan McLellan and Ruoran Zhang.

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## A reason for representation theorists to play billiards

Lundi fév 06, 2017
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Representations of the fundamental groups of surfaces appear so often in geometry that it's tempting to see them primarily as geometric structures. In recent years, however, researchers have uncovered beautiful new features of these representations by thinking of them instead as dynamical systems. As an invitation to the dynamical point of view, I'll describe how geometric tools from the study of billiards can be used to build invariants of surface group representations.

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## Heat Kernel Renormalization on Manifolds with Boundary

Lundi jan 30, 2017
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I will discuss how Costello's inductive renormalization
procedure for the construction of effective field theories can be
extended to manifolds with boundary.

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## Positive representations of quantum groups and higher Teichmuller theory

Lundi jan 23, 2017
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Positive representations are infinite-dimensional bimodules for the quantum group and its modular dual $\dpi{300}\inline U_{q^\vee}(\mathfrak{g}^\vee)$where both act by positive essentially self-adjoint operators.

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## LECTURES ON-DEMAND

### Gabriela González, Louisiana State University

Speaker: Gabriela Gonzalez