Abstract: The resolutions of Slodowy slices \tilde{S}_e are symplectic varieties that contain the Springer fiber (G/B)_e as a Lagrangian subvariety. In joint work with R. Bezrukavnikov, M. McBreen and Z. Yun, we construct analogues of these spaces for homogeneous affine Springer fibers. We further understand the categories of microlocal sheaves in these symplectic spaces supported on the affine Springer fiber as a localization of the category of representations of the small quantum group. 

This result provides a categorification of joint work with R.Bezrukavnikov, P. Shan and E. Vasserot and can be used to further understand the center of the small quantum group. 

If I have time I will then mention some recent project with C. Morton-Ferguson, giving a geometric interpretation of the appearance of the Kazhdan-Laumon category as a subquotient of the category of small quantum group representations.

Abstract: The (small) quantum connection is one of the simplest objects built out of Gromov-Witten theory, yet it gives rise to a repertoire of rich and important questions such as the Gamma conjectures and the Dubrovin conjectures. There is a very basic question one can ask about this connection: what is its formal singularity type? People's (e.g. Kontsevich-Katzarkov-Pantev, Iritani) expectation for this is packaged into the so-called exponential type conjecture, and the goal of this talk is to discuss a proof in the case of closed monotone symplectic manifolds.

My approach uses a reduction mod p argument, and I will start by introducing some basic ordinary differential equations (in particular in characteristic p) and Katz's local monodromy theorem. Then I will demonstrate the key idea of proof pretending we were working in a B-side mirror situation---matrix factorizations, where it is particularly simple. Finally, I will explain how to adapt the proof to the case of quantum connections using certain equivariant operations on quantum cohomology. 

Abstract: We give explicit, GL(V)-equivariant minimal free resolutions of natural submodules of a polynomial ring Sym(V) over its Veronese subalgebras. The free modules appearing in the resolutions are base changes of Schur modules associated to ribbon diagrams, and the differential comes from a simple degree lowering map on a tensor algebra. We use these resolutions to compute Hom and Tor between these modules. This is based on joint work with Ayah Almousa, Michael Perlman, Victor Reiner, and Keller VandeBogert.

Abstract: Let $G$ be a simple algebraic group and let $G^\vee$ be its Langlands dual group. Barbasch and Vogan based on earlier work of Lusztig and Spaltenstein, define a duality map $D$ that sends nilpotent orbits $\mathbb{O}_{e^\vee} \subset \mathfrak{g}^\vee$ to special nilpotent orbits $\mathbb{O}_e\subset \mathfrak{g}$. In a work by Losev, Mason-Brown and Matvieievskyi, an upgraded version $\tilde{D}$ of this duality is considered, called the refined BVLS duality. $\tilde{D}(\mathbb{O}_{e^\vee})$ is a $G$-equivariant cover $\tilde{\mathbb{O}}_e$ of $\mathbb{O}_e$. Let $S_{{e^\vee}}$ be the nilpotent Slodowy slice of the orbit $\mathbb{O}_{e^\vee}$. The two varieties $X^\vee= S_{e^\vee}$ and $X=$ Spec$(\mathbb{C}[\tilde{\mathbb{O}}_e])$ are expected to be symplectic dual to each other. In this context, a version of the Hikita conjecture predicts an isomorphism between the cohomology ring of the Springer fiber $\mathcal{B}_{e^\vee}$ and the ring of regular functions on the scheme-theoretic fixed point $X^T$ for some torus $T$. This conjecture holds when $G$ is of type A. In this talk, I will discuss the statuses of similar statements about the Hikita conjecture for general $G$. Part of the result is based on a joint work in preparation with Vasily Krylov and Dmytro Matvieievskyi.

Abstract:  Let X be a Fano variety with G action. The quantum GIT conjecture predicts a formula for the quantum cohomology of "anti-canonical" GIT quotients X//G in terms of the equivariant quantum cohomology of X. The formula is motivated by ideas from 3- dimensional gauge theory ("Coulomb branches") and provides a vast generalization of Batyrev's formula for the quantum cohomology of a toric Fano variety. I will explain describe our ongoing work with C. Teleman proving this conjecture. The strategy of proof involves ideas from Hamiltonian Floer theory

Abstract: Higher Teichmuller spaces are associated with pairs (G,S) consisting of a split semi-simple Lie group G and a marked surface S. The classical

case is the one where G is the group PGL(2,R). In the 2000s, Fock and Goncharov have developed a cluster-theoretic approach to higher Teichmuller

theory for groups of type A. Their work was extended to all classical groups by Ian Le and to arbitrary split semi-simple groups by Goncharov-Shen

in 2019 (preprint). In particular, they construct a cluster algebra A(G,S) for each pair (G,S). In this lecture series, we will give an introduction to the tools used in the ongoing efforts aiming at the additive categorification of these cluster algebras. For G=PGL(2,R), this aim was reached thanks to the work of Fomin--Shapiro--Thurston, Derksen--Weyman--Zelevinsky and Labardini-Fragoso. An alternative approach, based on Kapranov-Schechtman's idea of perverse schober and techniques from higher category theory, is due to Merlin Christ. The category he constructs is equivalent, as an extriangulated category, to the Higgs category (in the sense of Yilin Wu) associated to the ice quiver with potential constructed by Labardini-Fragoso (at least if the surface does not have punctures). Christ's construction admits a natural conjectural generalization to (simply-laced) groups G of higher rank. In the case where the surface is a triangle, recent work by Miantao Liu will likely confirm that Christ's conjectural category is the Higgs category associated with Goncharov-Shen's ice quiver endowed with a natural potential. In particular, it would contain a canonical cluster-tilting object with the expected endomorphism algebra. If this holds, the conjectural categorification will follow if one can 1) prove a glueing theorem for cluster-tilting objects and 2) check that the glued objects are infinitely mutable.

Abstract: Uglov's work on higher-level Fock spaces has given rise to an intricate framework of equivalences between blocks of cyclotomic rational Cherednik algebras, sometimes dubbed (categorical) level-rank duality. Ting Xue and I offer evidence for a generalization, involving the Hecke algebras attached to cuspidal pairs of arbitrary finite reductive groups G^F in the work of Deligne–Lusztig and Broué–Malle–Michel. The most striking feature is that our equivalences should arise from several different geometric realizations at once, related only by Bott–Samelson-type constructions. This lets us make precise an apparently new analogy: Hecke algebras at prime powers are to the coset GF as Hecke algebras at roots of unity are to the group G.

Abstract: Black hole thermodynamics is an increasingly relevant topic at the conjunction of classical and quantum field theory. The formulation of this topic by Wald (and others) relies heavily on conservation laws and symmetries as well as the geometry of interesting spacetime submanifolds: this suggests that the factorization algebra approach to the Batalin-Vilkovisky formalism will be useful in expanding our understanding of this subject. In this talk, I will describe developments in this direction based on joint work with Ryan Grady and Surya Raghavendran.

Abstract: This course concerns two classes of algebras, and the comparison between them.  The first are critical cohomological Hall algebras (CoHAs), with underlying vector space given by the vanishing cycle cohomology of moduli spaces of representations of Jacobi algebras.  These first appeared in the work of Kontsevich and Soibelman, and provide a kind of categorification of refined BPS invariants in physics, as well as a connection between BPS state counting and quantum groups.  The definition of the algebra depends on a choice of a quiver and potential (a linear combination of cyclic words in the arrows of the quiver).  For certain choices, it is possible to describe the resulting algebra, but most of the time it is not.  A very general result says that, for all choices of symmetric quiver with potential, there is a canonical subspace in the cohomological Hall algebra, closed under the Lie bracket, that generates it under the action of tautological classes and multiplication (a type of PBW theorem).  This is the so-called BPS Lie algebra

The second class of algebras arise in the work of Maulik and Okounkov, in their book [4].  These are so-called Yangian algebras, which really do generalise the Yangians considered by Drinfeld in finite type, to arbitrary quivers.  Via the theory of stable envelopes developed in [4], Maulik and Okounkov construct R matrices on tensor products of cohomologies of Nakajima quiver varieties.  Via these R matrices, they are able to "reverse engineer" a Yangian-type algebra, for which the cohomology of Nakajima quiver varieties becomes a category of modules with their given R matrix providing a braiding (with spectral parameters).  Just as in the case of CoHAs, Maulik and Okounkov prove a kind of PBW theorem, relating their Yangians to much "smaller" Lie subalgebras; in finite type, these Lie algebras are just the classical Lie algebras sitting inside the Yangian.  It turns out that these Lie algebras are isomorphic to doubles of certain BPS Lie agebras, allowing us to transport many facts that are known on the critical CoHA side to the Maulik-Okounkov Yangian side, and vice versa.

Abstract: This course concerns two classes of algebras, and the comparison between them.  The first are critical cohomological Hall algebras (CoHAs), with underlying vector space given by the vanishing cycle cohomology of moduli spaces of representations of Jacobi algebras.  These first appeared in the work of Kontsevich and Soibelman, and provide a kind of categorification of refined BPS invariants in physics, as well as a connection between BPS state counting and quantum groups.  The definition of the algebra depends on a choice of a quiver and potential (a linear combination of cyclic words in the arrows of the quiver).  For certain choices, it is possible to describe the resulting algebra, but most of the time it is not.  A very general result says that, for all choices of symmetric quiver with potential, there is a canonical subspace in the cohomological Hall algebra, closed under the Lie bracket, that generates it under the action of tautological classes and multiplication (a type of PBW theorem).  This is the so-called BPS Lie algebra

The second class of algebras arise in the work of Maulik and Okounkov, in their book [4].  These are so-called Yangian algebras, which really do generalise the Yangians considered by Drinfeld in finite type, to arbitrary quivers.  Via the theory of stable envelopes developed in [4], Maulik and Okounkov construct R matrices on tensor products of cohomologies of Nakajima quiver varieties.  Via these R matrices, they are able to "reverse engineer" a Yangian-type algebra, for which the cohomology of Nakajima quiver varieties becomes a category of modules with their given R matrix providing a braiding (with spectral parameters).  Just as in the case of CoHAs, Maulik and Okounkov prove a kind of PBW theorem, relating their Yangians to much "smaller" Lie subalgebras; in finite type, these Lie algebras are just the classical Lie algebras sitting inside the Yangian.  It turns out that these Lie algebras are isomorphic to doubles of certain BPS Lie agebras, allowing us to transport many facts that are known on the critical CoHA side to the Maulik-Okounkov Yangian side, and vice versa.

Abstract: The moduli space of hyperbolic metrics on a compact oriented surface without boundary has a well-known symplectic structure. In this talk, I will describe a generalization to the (infinite-dimensional) moduli space of conformally compact hyperbolic metrics on surfaces with boundary, up to diffeomorphisms fixing the boundary.  The action of boundary diffeomorphisms is Hamiltonian, making this space into an example of a Hamiltonian Virasoro space. (Joint work with Anton Alekseev.)