# Mathematical Physics

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

## Seminar Series Events/Videos

Nov 26 2018 - 2:00pm
Room #: 394
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Dec 3 2018 - 2:00pm
Room #: 394
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Dec 10 2018 - 2:00pm
Room #: 394
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Jan 7 2019 - 2:00pm
Room #: 394
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Feb 25 2019 - 2:00pm
Room #: 400
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## Categories of line operators in 3d N=4 gauge theories

Monday Oct 29, 2018
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A 3d N=4 gauge theory admits two topological twists, which we'll simply call A and B. The two twists are exchanged by 3d mirror symmetry. It is known that local operators in the A (resp. B) twist include the Coulomb-branch (resp. Higgs-branch) chiral rings.  In this talk I will discuss the *line* operators preserved by the two twists, which in each case should have the structure of a braided tensor category.

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## Unification of integrability in supersymmetric gauge theories

Monday Oct 22, 2018
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The 8-vertex model and the XYZ spin chain have been found to emerge from gauge theories in various ways, such as 4d and 2d Nekrasov-Shatashvili correspondences, the action of surface operators on the supersymmetric indices of class-Sk theories, and correlators of line operators in 4d Chern-Simons theory.  I will explain how string theory unifies these phenomena.  This is based on my work with Kevin Costello [arXiv:1810.01970].

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## Positive geometries and the amplituhedron

Monday Oct 22, 2018
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Positive geometries are real semialgebraic spaces that are
equipped with a meromorphic canonical form" whose residues reflect
the boundary structure of the space.  Familiar examples include
polytopes and the positive parts of toric varieties.  A central, but
conjectural, example is the amplituhedron of Arkani-Hamed and Trnka.
In this case, the canonical form should essentially be the tree
amplitude of N=4 super Yang-Mills.

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## Yangians from Nakajima quiver varieties

Monday Oct 15, 2018
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One can associate to any finite graph Q the skew-symmetic Kac-Moody Lie algebra g_Q. While this algebra is always infinite, unless Q is a Dynkin diagram of type ADE, g_Q shares a lot of the nice features of a semisimple Lie algebra. In particular, the cohomology of Nakajima quiver varieties associated to Q gives a geometric representations of g_Q. Encouraged by this story, one could hope to define the Yangian of g_Q, for general Q, as a subalgebra of the algebra of endomorphisms of cohomology of quiver varieties.

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## Towards a categorification of a projection from the affine to the finite Hecke algebra in type A

Monday Sep 24, 2018

Work of Bezrukavnikov on local geometric Langlands correspondence and works of Gorsky, Neguţ, Rasmussen and Oblomkov, Rozansky on knot homology and matrix factorizations suggest that there should be a categorical version of a certain natural homomorphism from the affine Hecke algebra to the finite Hecke algebra in type A, sending basis lattice elements on the affine side to Jucys-Murphy elements on the finite side.

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## Langlands duality and self-duality for Hitchin systems

Monday Sep 24, 2018
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I will sketch why self-dual versions of the moduli of G-Higgs bundles are expected to arise physically from the study of 4d theories of class S. I will then describe an extension of the Langlands duality results of Hausel-Thaddeus (G=SL(n)) and Donagi-Pantev (arbitrary reductive G) that yields self-dual moduli spaces as a corollary.

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## Toward AGT for general algebraic surfaces

Thursday Sep 20, 2018
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The Alday-Gaiotto-Tachikawa correspondence connects gauge theory on a fourfold with conformal field theory. We are interested in a certain algebro-geometric incarnation of this framework, where the fourfold is an algebraic surface and instantons/differential geometry are replaced with sheaves/algebraic geometry. In this talk, we will present a certain approach to AGT that yields partial results for quite general surfaces, and ask questions about what still needs to be done to state and prove the full correspondence in the language of algebraic geometry.

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## Localization theory for W-algebras

Monday Sep 10, 2018
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The localization theorem, which has played a central role in representation theory since its discovery in the 1980s, identifies a regular block of Category O for a semisimple Lie algebra with certain D-modules on its flag variety. In this talk we will explain work in progress which produces a similar picture for the Virasoro algebra and more generally for affine W-algebras.

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## Quantized lattices and combinatorics - Zsuzanna Dancso

Monday Jul 09, 2018

Quantized lattices, or q-lattices, appear naturally through categorification constructions (for example from zigzag-algebras), but they haven't been studied from a lattice theory point of view. After establishing the necessary background, we'll explain the q-versions of various lattice theory concepts illustrated with examples from combinatorics and graph theory, and we'll list a number of open problems.

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## TBA

Wednesday Jul 04, 2018
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Abstract TBA.

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## LECTURES ON-DEMAND

### Jocelyn Bell Burnell: University of Oxford

Speaker: Jocelyn Bell Burnell