Abstract: In this talk, I will talk about a connection between tropical geometry, representations of quantum affine algebras, and scattering amplitudes in physics. We give a systematic construction of prime modules (including prime non-real modules) of quantum affine algebras using tropical geometry. We propose a generalization of Grassmannian string integrals in physics, in which the integrand is a product indexed by prime modules of a quantum affine algebra. We give a general formula of u-variables using prime tableaux (corresponding to prime modules of quantum affine algebras of type A) and Auslander-Reiten quivers of Grassmannian cluster categories. This is joint work with Nick Early.

Abstract: I will present a compactification of Hitchin components of SL(n,R) representation varieties of surfaces, and show that boundary points are parametrizing some interesting, but still mysterious geometric objects. In particular the "rational" boundary points give rise to rational linear combinations of closed curves on the surface with certain restrictions on how curves can intersect. From this weighted multicurve we can construct an n-1 dimensional polyhedral complex with some building-like structure. In the case n=2, this compactification is Thurston's compactification by measured laminations, rational laminations are weighted simple closed multicurves, and the polyhedral complex is a tree. Using asymptotic analysis of the Hitchin equation, one can get a fairly good understanding of what happens for n=3. 

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Abstract: The main theorem of this talk will be that the affine closure of the cotangent bundle of the basic affine space (also known as the universal hyperkahler implosion) has symplectic singularities for any reductive group, where essentially all of these terms will be defined in the course of the talk. After discussing some motivation for the theory of symplectic singularities, we will survey some of the basic facts that are known about the universal hyperkahler implosion and discuss how they are used to prove the main theorem. Time permitting, we will also discuss a recent result, joint with Harold Williams, which identifies the universal hyperkahler implosion in type A with a Coulomb branch in the sense of Braverman, Finkelberg, and Nakajima, confirming a conjectural description of Dancer, Hanany, and Kirwan.

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Abstract: It is well known that the moduli space of coherent sheaves on a Calabi-Yau 3-fold locally can be realized as the critical locus of a regular function on a smooth ambient space. This structure can be obtained as a truncation of the -1-shifted symplectic structure on the derived moduli space, and has important applications in defining enumerative invariants for a Calabi-Yau 3-fold. In this work, we explore analogous local structure for the derived moduli space of coherent sheaves on a Calabi-Yau 4-fold, in particular, we show that a -2-shifted symplectic derived scheme etale locally is equivalent to the derived intersection of two Lagrangians inside a -1-shifted symplectic scheme. This is based on joint work with Nachiketa Adhikari.   

Abstract: Over 30 years ago, the work of Beauville and Deninger-Murre endowed the cohomology of an abelian scheme a (motivic) decomposition which splits the Leray filtration. This structure, now known as the Beauville decomposition, is induced by algebraic cycles obtained from the Fourier-Mukai coherent duality. In recent years, the cohomological study of Hitchin systems (e.g. the P=W conjecture) and compactified Jacobians associated with planar singularities suggests that there should exist an extension of the theory of Beauville decompositions for certain abelian fibrations with singular fibers, where the Leray filtration should be replaced by the perverse filtration. I will discuss some recent progress in this direction. In particular, I will present results in both the positive and the negative directions, where Lagrangian fibrations associated with hyper-Kähler manifolds and the tautological relations over the Deligne-Mumford moduli of stable curves play crucial roles. Based on joint work with Younghan Bae, Davesh Maulik, and Qizheng Yin.

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Abstract: Classically, symmetries are encoded by groups. In quantum field theory, there are more general "non-invertible" symmetries. In 2 dimensions, they are described by fusion categories. In 3 dimensions, the most familiar examples come from braided fusion categories. These fit into the broader class of fusion 2-categories. The aim of this talk is to introduce the definition of a fusion 2-category and to sketch a surprising classification result.

Abstract: The closure of a linear subspace of C^n in an n-fold product of CP^1 was first studied by Ardila and Boocher, who showed that the algebraic invariants of this closure beautifully reflect matroid combinatorics. This construction was later used by Huh and Wang to prove the top-heavy conjecture for realizable matroids and is central in the development of Kazhdan-Lusztig theory for matroids. This recent activity was inspired by an analogy with Schubert varieties, so this construction is sometimes referred to as a matroid Schubert variety.

I'll introduce matroid Schubert varieties and then share two results. The first characterizes matroid Schubert varieties as the class of equivariant compactifications of complex vector spaces which have certain properties. The second gives a generalized construction and proves that it also has one of the key properties of matroid Schubert varieties.

Contains joint work with Connor Simpson and Botong Wang.

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Abstract: A holomorphic differential induces a flat metric with saddle points such that the underlying Riemann surface can be realized as a polygon whose edges are pairwise identified by translation. Varying such flat surfaces by affine transformations induces an action on moduli spaces of differentials, called Teichmueller dynamics. Generic flat surfaces in an orbit closure of Teichmueller dynamics possess similar properties from the viewpoint of counting geodesics of bounded lengths, whose asymptotic growth rates satisfy a formula of Siegel–Veech type. In this talk, I will give an introduction to this topic and discuss some recent results about computing Siegel–Veech constants via intersection theory on moduli spaces.

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Abstract: Moduli spaces of logarithmic connections on Riemann surfaces have a rich geometric structure, and in particular can be made into symplectic spaces isomorphic to (complex) character varieties. The general wisdom is that the moduli spaces are obtained by gluing local pieces at each simple pole, involving choices of coadjoint orbits for a (dual) complex reductive Lie algebra: in particular quantising such orbits is a preliminary step towards quantising the moduli spaces themselves.

In this talk we will aim at a review of this story, and then describe a recent extension for irregular singular (= wild) meromorphic connections. The quantisation of the corresponding orbits is based upon a result of Alekseev--Lachowska, and is joint work with D. Calaque, G. Felder, and R. Wentworth.

(The main ingredient are the Shapovalov form for certain representations of truncated current Lie algebras, generalising the (generalised) Verma modules.)

If time allows, we will also recall how the moduli spaces can be deformed, and describe the universal space of local deformations: this is joint work with P. Boalch, J. Douçot, and M. Tamiozzo. 

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Abstract: In this talk we will be concerned with smooth, framed fiber bundles whose fibers are the standard d-dimensional disk, trivialized along the boundary. "Kontsevich's characteristic classes" are invariants defined for these bundles: given such a bundle \pi:E \to B, we can associate to it a collection of cohomology classes in H^*(B). On the other hand, there is a "bracket operation" for these bundles defined by Sander Kupers: namely, given two such bundles \pi_1 and \pi_2 as input, we can output a "bracket bundle" [\pi_1,\pi_2]. I will talk about this bracket bundle construction and a formula relating the Kontsevich's classes of [\pi_1,\pi_2] with those of \pi_1 and \pi_2. The main input of the proof is a novel but very natural configuration space generalizing the Fulton-MacPherson configuration spaces. This is a work in progress joint with Robin Koytcheff and Sander Kupers. 

Abstract: For any complex reductive algebraic group G, one can associate a completely integrable system J_G over the coadjoint quotient of the dual of the Lie algebra, which generalizes the cotangent bundle of a torus over its Lie algebra dual. The variety J_G plays an important role in geometric representation theory and mathematical physics. In particular, it is a Coulomb branch mathematically defined by Braverman-Finkelberg-Nakajima. I will present recent results on the (homological) mirror symmetry of J_G. I will also present a loop group version of the results, which is joint work with Zhiwei Yun. The latter mirror symmetry can be viewed as a Betti Geometric Langlands correspondence in the wild setting.

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Abstract: Stanley’s e-positivity conjecture is equivalent to the statement that the chromatic polynomial of a graph obtained as the incidence graph of a set of unit intervals on the real line is e-positive; this means that the symmetric polynomial rewritten as a polynomial in the elementary symmetric polynomials has nonnegative coefficients. I will present a proof of this conjecture for the coefficients corresponding to partitions of length 2, obtained in joint work with Alexandre Rok.

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