Abstract: This talk is about singularities of Schubert varieties, studied via sheaf-theoretic invariants like intersection cohomology and parity sheaves. The motivation comes from the use of these sheaves in representation theory, which began with the celebrated proof of the Kazhdan-Lusztig conjectures by Beilinson-Bernstein localisation. We will present examples of poor behaviour (in particular exhibit non-perverse parity sheaves), which thwart historic overly-optimistic conjectures on the singularities of Schubert varieties.

Abstract: The study of admissible sub-objects of a certain object in an additive category relatively to a Quillen exact structure is an exciting subject that leads to some unaccepted characterizations. We propose new general notions of intersections and some of sub-objects to study the Jordan-Holder property of an exact category. We then generalize the length function and the Gabriel-Roiter measure to the realm of exact categories. We also initiate the study of weakly exact structures, a generalization of both Quillen exact structurees and the important and widely used notion of Abelian categories. We investigate when these structures form lattices. This talk is based on several joint works: [1] , [2] and [3] .

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Abstract: Let S be a compact oriented surface (possibly with boundary), and let G be a compact connected simply connected Lie group. I will describe classes in the K-theory of a moduli space of flat G-connections on S. In the case of a closed surface, these classes were introduced by Atiyah and Bott. When the boundary of the surface is non-empty, further investigation leads to a gauge theoretic version of a theorem of Teleman and Woodward.

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Abstract: We introduce a concept of 't-embeddings' of weighted bipartite planar graphs. We believe that these t-embeddings always exist and that they are good candidates to recover the complex structure of big bipartite planar graphs carrying a dimer model. We also develop a relevant theory of discrete holomorphic functions on t-embeddings; this theory unifies Kenyon's holomorphic functions on T-graphs and s-holomorphic functions coming from the Ising model. We provide a meta-theorem on convergence of the height fluctuations to the Gaussian Free Field.

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Abstract: General covariance is a crucial notion in the study of field theories on curved spacetimes. In our context, a generally covariant field theory is one whose dependence on a Riemannian (or Lorentzian) metric is equivariant with respect to the diffeomorphism group of the underlying manifold/spacetime. In this talk, we will make these notions precise by using stacks and the Batalin-Vilkovisky formalism, and will moreover recover the associated equivariant classical observables in the perturbative case.

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Abstract: Consider the space of Bridgeland stability conditions of a suitably nice triangulated category. Autoequivalences of the triangulated category act on the space of stability conditions. Fixing a stability condition imposes extra combinatorial structure on the category that can be used to study the group of autoequivalences in various different ways. This talk will showcase some of the fascinating structure that emerges via this idea, particularly for 2-Calabi-Yau categories associated to quivers. This is based on joint work with Anand Deopurkar and Anthony Licata.

Abstract: Structure often emerges from Hilbert schemes of points on surfaces when the underlying surface is fixed but the number of points parametrized is allowed to vary. One example of such structure comes from integrals of tautological bundles, which appear in physical and geometric computations. When compiled into generating series, these integrals display interesting functional properties. I will focus on the example of K-theoretic descendent series, certain series formed from holomorphic Euler characteristics of tautological bundles. Namely, I will explain how to use a Macdonald symmetry of Mellit to deduce that the K-theoretic descendent series are expansions of rational functions.

Abstract: A smooth genus 2 curve has a 6 dimensional family of possible complex structures, parametrized by the genus-2 Siegel space. We describe a generalized SYZ mirror family of symplectic manifolds and the mirror correspondence of Kahler cones with the Siegel space. We also describe the Fukaya category of the symplectic manifold (a Landau-Ginzburg model) with structure maps deformed by the B-field. This involves adapting Guillemin's Kahler potential to a toric variety of infinite type and computing monodromy of a symplectic fibration with critical locus given by the "banana manifold" of three P^1's attached at two points. Finally, we end with a homological mirror symmetry result between the genus 2 curves and their mirrors. This is joing work with H. Azam, C-C. M. Liu, and H. Lee.

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Abstract: In this talk we'll be considering the following situation: suppose we have a representation (V,\pi) of a Weyl group equipped with a canonical basis. Given an element g of the group, can we extract interesting information about the matrix of \pi(g) with respect to the basis? In general this is extremely difficult but in some situations there are beautiful answers to this question. The first results in this direction are due to Berenstein-Zelevinsky and Stembridge, who proved that the long element of the symmetric group acts on the Kazhdan-Lusztig basis by the Schutzenberger involution on tableau. I will explain vast generalizations of this theorem. The underlying ideas driving these results come from braid group actions on derived categories. This is based on joint work with Martin Gossow.

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Abstract: I will begin by giving an introduction to a special and widely studied class of Poisson manifolds: log symplectic manifolds. While these have degeneracies, they are sufficiently close to being symplectic that many properties and techniques from symplectic geometry extend. The main result I will present is my recent generalization of Lagrangian intersection Floer cohomology to log symplectic surfaces.

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Abstract: In the 60s and 70s, there was a flurry of activity concerning the question of whether or not various subgroups of homeomorphism groups of manifolds are simple, with beautiful contributions by Fathi, Kirby, Mather, Thurston, and many others. A funnily stubborn case that remained open was the case of area-preserving homeomorphisms of surfaces. For example, for balls of dimension at least 3, the relevant group was shown to be simple by work of Fathi from the 1970s, but the answer in the two-dimensional case was not known. I will explain recent joint work proving that the group of compactly supported area preserving homeomorphisms of the two-disc is in fact not a simple group, which answers the "Simplicity Conjecture" in the affirmative. Our proof uses a new tool for studying area-preserving surface homeomorphisms, called periodic Floer homology (PFH) spectral invariants; these recover the classical Calabi invariant via a kind of Weyl law. I will also briefly mention a generalization of our result to compact surfaces of any genus.

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Abstract: The Deligne tensor categories are defined as an interpolation of the categories of representations of groups GL_n, O_n, Sp_2n, or S_n to the complex values of the parameter n. One can extend many classical representation-theoretic notions and constructions to this context. These complex rank analogs of classical objects provide insights into their stable behavior patterns as n goes to infinity.

I will talk about some of my results on Harish-Chandra bimodules in the Deligne cateogories. It is known that in the classical case simple Harish-Chandra bimodules admit a classification in terms of W-orbits of certain pairs of weights. However, the notion of weight is not well-defined in the setting of the Deligne categories. I will explain how in complex rank the above-mentioned classification translates to a condition on the corresponding (left and right) central characters.

Another interesting phenomenon arising in complex rank is that there are two ways to define harish-Chandra bimodules. That is, one can either require that the center acts locally finitely on a bimodule M or that M has a finite K-type. The two conditions are known to be equivalent for a semi-simple Lie algebra in the classical setting, however in the Deligne categories, it is no longer the case. I will talk about a way to construct examples of Harish-Chandra bimodules of finite K-type using the ultraproduct realization of the Deligne categories.

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Abstract: Blohmann, Schiavina, and I have found a Lie-Rinehart algebra on a graded extension of the space of initial values for the Einstein equations whose bracket relations match those of the constraints on the initial values. This will be a follow-up to last year's talk, but I will not assume that anyone has heard it.

Abstract: Hessenberg varieties are subvarieties of the flag variety Flags(C^n), the study of which have rich interactions with symplectic geometry, representation theory, and equivariant topology, among other research areas, with particular recent attention arising from its connections to the famous Stanley-Stembridge conjecture in combinatorics. The special case of regular nilpotent Hessenberg varieties has been much studied, and in this talk I will describe some work in progress analyzing the local defining ideals of these varieties. In particular, using some techniques relating liason theory, geometric vertex decomposition, and the theory of Grobner bases (following work of Klein and Rajchgot), we are able to show that, for the coordinate patch corresponding to the longest word w_0, the local defining ideal for any indecomposable Hessenberg variety is geometrically vertex decomposable, and we find an explicit Grobner basis for a certain monomial order. This is a report on joint work in progress with Sergio Da Silva.

Abstract: One of the central foci of representation theory in the 20th century was the representation theory of Lie algebras, starting with finite dimensional algebras and expanding to a rich, but still mysterious infinite dimensional theory. In this century, we realized that this was only one special case of a bigger theory, with new sources of interesting non-commutative algebras whose representations we'd like to study, such as Cherednik algebras. In mathematical terms, we could connect these to symplectic resolutions of singularities, but a more intriguing explanation is that they arise from 3d quantum field theories. I'll try to provide an overview about what's known about this topic and what we're still confused about.

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