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# Mathematical Physics

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

## Seminar Series Events/Videos

oct 17 2019 - 1:30pm
Room #: 294
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## Cohomology of hyperkahler manifolds

Jeudi oct 03, 2019
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I will talk about compact hyperkahler manifolds, which generalize the famous K3 surface to the higher dimensions. Given a compact simple hyperkahler manifold $M$, I will describe how the structure of cohomology algebra H*(M) is related with the so(b_2+2) Lie algebra action and the second cohomology group. I will explain how this is applied to the generalization of Kuga-Satake construction which allows us to assign for K3-type Hodge structure a Hodge structure of weight one (i.e. complex torus).

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## Tilting bundles and 3-dimensional field theory

Jeudi sep 26, 2019
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The category of coherent sheaves on an interesting variety X has an extremely annoying property: does not have enough projectives, so it cannot be equivalent to the category of modules over an algebra. However, if you pass to the derived category, this defect can be fixed in many interesting cases, by finding a tilting generator: that is, a vector bundle T such that any coherent sheaf can be resolved by a complex consisting of sums of copies of T, and Ext^i(T,T)=0 for all i>0.

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## Categorical Hikita Duality

Jeudi sep 26, 2019
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I will discuss joint work with Roman Bezrukavnikov on a categorical version of Hikita duality, which relates coherent sheaves on a symplectic resolution to constructible sheaves on the loop space of the dual resolution. I will focus on a basic case, where this can be made very explicit, and finish with some wild speculation on further generalisations.

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## Gapped boundary theories for 3d Chern-Simons

Jeudi 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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## Equivariant Localization in Factorization Homology and Vertex Algebras from Supersymmetric Gauge Theory

Jeudi 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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## Computing renormalization invariant properties of Levin-Wen phases

Jeudi aoû 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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## Algebraic Braids and Geometric Representation Theory

Mardi mai 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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## Equivariant SYZ mirror construction

Lundi mai 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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## Multiple zeta values in deformation quantization

Lundi mai 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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## Quantum geometry of moduli spaces of local systems

Lundi avr 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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