Presentation Details 

Boalch, Philip 
Wild character varieties, meromorphic Hitchin systems and Dynkin diagrams 
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In 1987 Hitchin discovered a new family of algebraic integrable systems, solvable by spectral curve methods. One novelty was that the base curve was of arbitrary genus. Later on it was understood how to extend Hitchin's viewpoint, allowing poles in the Higgs fields, and thus incorporating many of the known classical integrable systems, which occur as meromorphic Hitchin systems when the base curve has genus zero. However, in a different 1987 paper, Hitchin also proved that the total space of his integrable system admits a hyperkahler metric and (combined with work of Donaldson, Corlette and Simpson) this shows that the differentiable manifold underlying the total space of the integrable system has a simple description as a character variety $$Hom(pi_1(Sigma), G)/G$$ of representations of the fundamental group of the base curve $Sigma$ into the structure group G. This misses the main cases of interest classically, but it turns out there is an extension. In work with Biquard from 2004 Hitchin's hyperkahler story was extended to the meromorphic case, upgrading the speakers holomorphic symplectic quotient approach from 1999. Using the irregular RiemannHilbert correspondence the total space of such integrable systems then has a simple explicit description in terms of monodromy and Stokes data, generalising the character varieties. The construction of such "wild character varieties", as algebraic symplectic varieties, was recently completed in work with D. Yamakawa, generalizing the author's construction in the untwisted case (20022014).
For example, by hyperkahler rotation, the wild character varieties all thus admit special Lagrangian fibrations.
The main aim of this talk is to describe some simple examples of wild character varieties including some cases of complex dimension 2, familiar in the theory of Painleve equations, although their structure as new examples of complete hyperkahler manifolds (gravitational instantons) is perhaps less wellknown. The language of quasiHamiltonian geometry will be used and we will see how this leads to relations to quivers, Catalan numbers and triangulations, and in particular how simple examples of gluing wild boundary conditions for Stokes data leads to duplicial algebras in the sense of Loday.
The new results to be discussed are joint work with R. Paluba and/or D. Yamakawa. 
Bullimore, Mathew 
Twisted Hilbert Space of 3d N = 2 Gauge Theories 
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I will talk about the description of 3d N = 2 gauge theories on R x C with a topological twist on a Riemann surface C as a supersymmetric quantum mechanics on R. I will focus on a mathematical description of the Hilbert space of supersymmetric ground states in U(1) gauge theories, demonstrating invariance under threedimensional mirror symmetry and reproducing the twisted index on S^1 x C. 
Galakhov, Dimitrii 
The twodimensional LandauGinzburg approach to link homology 
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I will give a very basic introduction to the twodimensional LandauGinzburg approach to link homology through the webbased formalism. Some peculiarities and relation to Khovanov homology will be discussed. 
Gukov, Sergei 
The sound of knots and 3manifolds 
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Barclay Prime is a restaurant in Philadelphia that became famous for its exquisite $100 cheesesteak. Served with a small bottle of Veuve Clicquot champagne, Barclay Prime’s cheesesteak is made of sliced Kobe beef, melted Taleggio cheese, shaved truffles, sauteed foie gras, caramelized onions and heirloom shaved tomatoes on a homemade brioche roll brushed with truffle butter and squirted with homemade mustard. Yet, in essence, it still is just a sandwich. One of the recent developments at the interface of physics and mathematics involves a number of very sophisticated ingredients from both fields. Yet, at its basic, it is just a familiar quantum mechanics that I will present to you on February 27. 
Korff, Christian 
From Dimers to Quantum Ktheory 
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Considering dimer configurations on the honeycomb lattice we define two different solutions of the quantum YangBaxter equation. Using techniques from quantum integrable systems, such as Baxter’s Qoperator and the Bethe ansatz, we define a generalised quantum Schubert calculus which describes several known cases in the literature, for example the equivariant Ktheory ring of Grassmannians. We also explicitly construct a hitherto unknown ring, which we conjecture to be the quantum equivariant Ktheory of Grassmannians. (This has recently been confirmed  using different methods  by Buch, Chaput, Mihalcea and Perrin who consider all cominuscule varieties.) Our description via a quantum integrable system has the advantage of revealing an underlying “quantum group structure”. This structure is akin to the Yangian structures appearing in the work by Maulik and Okounkov on general Nakajima varieties.
This is joint work with Vassily Gorbounov, Aberdeen. 
Mariño, Marcos 
The spectral theory of quantum mirror curves 
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I will present a correspondence between open and closed topological strings on toric CalabiYau manifolds, on one hand, and the spectral theory of a new family of trace class operators on the real line, on the other hand. These operators are obtained by quantizing the mirror curves to the toric CYs, and they are exactly solvable by using open and closed BPS invariants of the CY. Conversely, the spectral problem makes it possible to define a nonperturbative completion of the topological string. If time permits, I will explain the relation of this framework to resurgent transseries. 
Maruyoshi, Kazunobu 
Surface defects in 4d supersymmetric field theories via integrable lattice model 
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In this talk, we consider surface defects in a certain class of 4d N=1 theories in relation with the integrable lattice model. The supersymmetric index of an N=1 theory realized by a brane tiling coincides with the partition function of an integrable 2d lattice model. We argue that a class of halfBPS surface defects in the 4d theory are represented in the lattice model as transfer matrices constructed from Loperators. For a surface defect labeled by the fundamental representation of SU(2) in the 4d theory with SU(2) gauge groups, we identify the relevant Loperator as that discovered by Sklyanin in the context of the eightvertex model. We perform nontrivial checks against the residue computation in class S and class S_k theories. The corresponding transfer matrix unifies 2k difference operators obtained for class S_k theories into a oneparameter family of difference operators.
This talk is based on the collaboration with Junya Yagi, and on arXiv:1606.01041. 
Nawata, Satoshi 
Representations of DAHA from Hitchin moduli space 
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I will talk about physics approach to understand representation theory of double affine Hecke algebra (DAHA). DAHA can be realized as an algebra of line operators in 4d N=2* theory and therefore it appears as quantization of coordinate ring of Hitchin moduli space over oncepunctured torus. Using 2d Amodel on the Hitchin moduli space, I will explain relationship between representation category of DAHA and Fukaya category of the Hitchin moduli space. 
Neitzke, Andrew 
Abelianization in ChernSimons theory 
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I will describe the notion of "abelianization" of flat connections over 2 and 3manifolds, and an application of this notion to the topological field theory of classical ChernSimons invariants. The notion of abelianization arose in joint work with Davide Gaiotto and Greg Moore, and the work on ChernSimons is joint work in progress with Dan Freed. 
Okounkov, Andrei 
Geometric construction of bethe eigenfunctions 
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A fundamental discovery of Nekrasov and Shatashvili equates quantum Ktheory of a Nakajima quiver variety (as a commutative ring) with Bethe equations for a certain quantum affine Lie algebra. I will explain how to go make the next step and find the corresponding Bethe eigenfunctions and, more generally, solutions to qKZ and dynamical difference equations. This is a joint work with Mina Aganagic. 
Pestun, Vasily 
Quiver gauge theories and quantum integrable systems 
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TBC 
Rasmussen, Jacob 
Knot homologies 
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I'll give an overview of knot homologies from a mathematical perspective. Exactly which topics I focus on will depend on what seems most interesting to the organizer and other participants. 
Shende, Vivek 
A skeletal introduction to homological mirror symmetry 
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Guided by structural features of the categories of coherent sheaves and exact lagrangians, I'll sketch a proof of homological mirror symmetry at the large complex structure limit. 
Simpson, Carlos 
Introduction to moduli spaces of flat connections 
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We give an overview of the moduli spaces parametrizing flat
connections over complex algebraic varieties: the character variety,
the Hitchin moduli space of Higgs bundles, and the moduli space of
algebraic bundles with integrable connection. We discuss the correspondences
between them, the hyperkahler structure, the Hitchin map,
the role of variations of Hodge structure. Further topics will include
cohomology jump loci, quasiprojective varieties and parabolic structures,
and current directions on the study of compactifications. 
Teschner, Jörg 
Relating the geometric Langlands correspondence to AGT 
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TBC 
Yamazaki, Masahito 
Integrable lattice models from gauge theory 
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In a celebrated paper in 1989, E. Witten discovered a beautiful connection between knot invariants (such as the Jones polynomial) and threedimensional ChernSimons theory. Since there are similarities between knot theory and integrable models, it is natural to ask if there is also a gauge theory explanation for integrable models. The answer to this question was recently given by K. Costello in 2013. In this talk I will describe my ongoing work with K. Costello and E. Witten, which explains many results in integrable models from standard quantum field theory analysis of Costello's theory. 