Mathematical Physics

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

Seminar Series Events/Videos

Sep 26 2019 - 1:30pm
Room #: 394
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Sep 26 2019 - 4:30pm
Room #: 405
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Sep 30 2019 - 1:30pm
Room #: 394
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Thursday Sep 19, 2019
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I will discuss a theorem, joint work in progress with Constantin Teleman, in which we characterize which topological 3-dimensional Chern-Simons theories admit nonzero boundary theories.  Accepting some physical heuristics, it tells which gapped systems in 2+1 dimensions admit only gapless boundary systems.

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Thursday Sep 12, 2019
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I will discuss recent developments in describing the chiral algebras associated to 4d N=2 theories introduced by Beem et al. in terms of Omega backgrounds, and give a description of the class S chiral algebras following this perspective, in terms of boundary conditions, interfaces, and junctions in 4d N=4 SYM.

 

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Thursday Aug 22, 2019
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 In the 90s Turaev, Viro, Barrett and Westbury constructed a (2+1)D state sum TQFT associated to any fusion category. The associated phases of matter were popularized by Kiteav, Levin and Wen and are now central  examples in condensed matter physics and quantum information theory.

Despite the importance of these phases, many of the computational techniques for working with fusion categories have not percolated into condensed matter physics. Many of these techniques are "folk theorems"
and have not appeared in the literature in a digestible form.

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Tuesday May 21, 2019
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In 2012, Maulik proved a conjecture of Oblomkov-Shende relating: (1) the Hilbert schemes of a plane curve (alternatively, its compactified Jacobian), (2) the HOMFLY polynomials of the links of its singularities. We recast his theorem from the viewpoint of representation theory. For a split semisimple group G with Weyl group W, we state a stronger conjecture relating two virtual modules over Lusztig's graded affine Hecke algebra,  constructed from: (1) fibers of a parabolic Hitchin map, (2) generalized Bott-Samelson spaces attached to conjugacy classes in the braid group of W.

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Monday May 13, 2019
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Strominger-Yau-Zaslow explained mirror symmetry via duality between tori.  There have been a lot of recent developments in the SYZ program, focusing on the non-equivariant setting.  In this talk, I explain an equivariant construction and apply it to toric Calabi-Yau manifolds.  It has a close relation to the equivariant open GW invariants found by Aganagic-Klemm-Vafa and studied by Katz-Liu, Graber-Zaslow and many others.

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Monday May 06, 2019

In 1997, Kontsevich gave a universal solution to the "deformation quantization" problem in mathematical physics: starting from any Poisson manifold (the classical phase space), it produces a  noncommutative algebra of quantum observables by deforming the ordinary multiplication of functions.  His formula is a Feynma  expansion, involving an infinite sum over graphs, weighted by volume integrals on the moduli space of marked holomorphic disks.

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Monday Apr 22, 2019
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Let G be a split semi-simple algebraic group over Q. We introduce a natural cluster structure on moduli spaces of G-local systems over surfaces with marked points. As a consequence, the moduli spaces of G-local systems admit natural Poisson structures, and can be further quantized. We will study the principal series representations of such quantum spaces. It will recover many classical topics, such as the q-deformed Toda systems, quantum groups, as well as the modular functor conjecture for such representations, which should lead to new quantum invariants of threefolds.

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Monday Apr 15, 2019
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In the context of geometric quantisation, one starts with the data of a symplectic manifold together with a pre-quantum line bundle, and obtains a quantum Hilbert space by means of the auxiliary structure of a polarisation, i.e. typically a Lagrangian foliation or a Kähler structure. One common and widely studied problem is that of quantising Hamiltonian flows which do not preserve it.

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Monday Apr 15, 2019
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This is practice for a talk at Berkeley, so it will involve explaining stuff many people here probably already know.  I'll try to summarize what I've learned about 3 dimensional N=4 supersymmetric quantum field theories, their twists and how these manifest in terms of interesting objects in mathematics.  If nothing else, hopefully there will be some comedy value in my attempt.  

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Monday Apr 08, 2019
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In a synthetic approach to geometry and physics, one attempts to formulate an axiomatic system in purely logical terms, abstracting away from irrelevant "implementation details". In this talk, I will explain how intuitionistic logic and topos theory provide a synthetic theory of space, and then consider a (naive version of) Euclidean QFTs in terms of a conjectural elegant synthetic reformulation: a Euclidean QFT is nothing but a probability space in intuitionistic logic, extended by suitable modalities formalizing compact regions of space.

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