Abstract: The study of Landau-Ginzburg/Calabi-Yau correspondence using linear sigma models was proposed by Witten in the early 90's of last century. A mathematical version of the correspondence using curve-counting theories has been realized by Chiodo, Ruan, Iritani, and many other people in the past decade. Namely, there is an equivalence between counting stable maps in the CY hypersurface of a weighted polynomial W (i.e. Gromov-Witten theory of the hypersurface) and counting $W$-spin structures (i.e. Fan-Jarvis-Ruan-Witten theory of W). So LG/CY correspondence can be realized as GW/FJRW correspondence.

I will talk about a generalization of such a GW/FJRW correspondence for non-CY hypersurfaces. A key ingredient here is the Gamma structures developed by Iritani, and Katzarkov-Kontsevich-Pantev. For Fano manifolds, Galkin-Golyshev-Iritani's Gamma conjectures predict the Gamma class of the quantum cohomology equals the asymptotic class from an irregular meromorphic connection. The irregular Riemann-Hilbert correspondence of the connection generates Stokes phenomenon. We can view the FJRW theory as a part of the GW theory in the Stoke decomposition. The similar phenomenon works for hypersurfaces of general type as well, where the GW theory is viewed as a part of the FJRW theory. Our GW/FJRW correspondence is compatible with Orlov's semi-orthogonal decomposition.

Abstract: The talk is based on the joint paper with Iva Halacheva, Joel Kamnitzer, and Alex Weekes https://arxiv.org/abs/1708.05105 . Solutions of the algebraic Bethe ansatz for quantum magnet chains are, generally, multivalued functions of the parameters of the integrable system. I will explain how to compute some monodromies of solutions of Bethe ansatz for the Gaudin magnet chain assigned to a semisimple finite-dimensional Lie algebra g in terms of Kashiwara crystals (which are combinatorial objects modeling finite-dimensional representations of g). Namely, the Bethe eigenvectors in the Gaudin model can be regarded as a covering of the Deligne-Mumford moduli space of stable rational curves, which is unramified over the real locus of the Deligne-Mumford space. The monodromy action of the fundamental group of the real Deligne-Mumford space (called cactus group) on the eigenvectors is naturally equivalent to the action of the same group by commutors (i.e. combinatorial analog of a braiding) on a tensor product of Kashiwara crystals.

If time allows, I will also discuss the generalization of this monodromy theorem to the XXX Heisenberg magnet chain which involves Kirillov-Reshetikhin crystals.

Abstract: The resolutions of Slodowy slices $\widetilde{\mathcal{S}}_e$ 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 some categories of coherent sheaves.

In this talk I will mostly focus on the case of the homogeneous element $ts$ for $s$ a regular semisimple element and will discuss some relations of these categories with the small quantum group providing a categorification of joint work with R.Bezrukavnikov, P. Shan and E. Vasserot.

Abstract: Cluster varieties come in pairs: for any X-cluster variety there is an associated Fock–Goncharov dual A-cluster variety. On the other hand, in the context of mirror symmetry, associated with any log Calabi–Yau variety is its mirror dual, which can be constructed using the enumerative geometry of rational curves in the framework of the Gross–Siebert program. I will explain how to bridge the theory of cluster varieties with the algebro-geometric framework of Gross–Siebert mirror symmetry, and show that the mirror to the X-cluster variety is a degeneration of the Fock–Goncharov dual A-cluster variety. To do this, we investigate how the cluster scattering diagram of Gross–Hacking–Keel–Kontsevich compares with the canonical scattering diagram defined by Gross–Siebert to construct mirror duals in arbitrary dimensions. This is joint work with Hülya Argüz.

Abstract: Let X and Y be compact hyper-Kahler manifolds deformation equivalent to the Hilbert scheme of n points on a K3 surface. A cohomology class in their product XxY is an analytic correspondence, if it belongs to the subalgebra generated by Chern classes of coherent analytic sheaves. Let f be a Hodge isometry of the second rational cohomologies of X and Y with respect to the Beauville-Bogomolov-Fujiki pairings. We prove that f is induced by an analytic correspondence. We furthermore lift f to an analytic correspondence F between their total rational cohomologies, which is a Hodge isometry with respect to the Mukai pairings, and which preserves the gradings up to sign. When X and Y are projective the correspondences f and F are algebraic. The case n=1 (when X and Y are K3 surfaces) was known as the Shafarevich conjecture and was solved by Nikolay Buskin in 2015 using work of Mukai. The higher dimensional case is based on recent results on equivalences of derived categories of hyperkahler varieties due to Taelman. We will also comment on the analogous result for hyperkahler manifolds of generalized Kummer deformation type (work in progress).

Abstract: Talk is based on the joint work with Gorsky, Mazin and Rozansky.

I will explain a construction of coherent sheaf on Hilb_n(C^2) whose global sections compute the triply graded homology of cabling of torus knot. I match the answer with the cohomology of the compactified Jacobian of the corresponding singularity.

Abstract: I will present recent results about Homological mirror symmetry (HMS) of algebraic symplectic varieties coming from representation theory. The first is about HMS of the universal centralizer of a complex reductive group G. The second, which is a joint work with Yun, is about HMS of a moduli space of Higgs bundles on P^1 with certain automorphic data. The latter variety is closely related to the first one as it is a symplectic compactification of the former. No prior knowledge about HMS is required.

Abstract: I will report on a joint work in progress with Dan Ciubotaru,

David Kazhdan and Yakov Varshavsky. The first result generalizes a

result of Aubert, Ciubotaru and Romano providing a description of the

unipotent cocenter of a p-adic group in terms of the Langlands dual group.

Time permitting I will discuss the relation of this result to characters

of representations and trace of Frobenius of affine character sheaves.

Abstract:

Abstract: We will discuss the general notion of symplectic duality between symplectic resolutions of singularities and give examples. Equivariant Hikita-Nakajima conjecture is a general conjecture about the relation between the geometry of symplectically dual varieties. We will consider the example of the Hilbert scheme of points on the affine plane and discuss the proof of the equivariant Hikita-Nakajima conjecture in this particular case. We will also briefly discuss the generalization of this proof to the case of ADHM spaces (moduli spaces of instantons on R^4). Time permitting, we will say about the possible approach towards the proof of Hikita-Nakajima conjecture for other symplectically dual pairs (such as Higgs and Coulomb branches of quiver gauge theories). The talk is based on the joint work with Pavel Shlykov arXiv:2202.09934.

Abstract: Attached to an integrable system (i.e. a proper holomorphic Lagrangian fibration associated with a symplectic varieties), we have perverse sheaves/Hodge modules that extend the local systems given by the variation of Hodge structures of the smooth fibers. In this talk, I will discuss some interesting symmetries carries by these sheaves, where Saito's theory of Hodge modules comes into play naturally. As an application, we will give a description of the "mirror" to the cotangent bundle of the symplectic variety using these sheaves. Based on joint work with Qizheng Yin, and joint work in progress with Davesh Maulik and Qizheng Yin.

Abstract: The Non Abelian Hodge Theory (NAHT) of Simpson, Corlette, et al. yields canonical diffeomorphisms between the moduli spaces of Higgs bundles, flat connections, and representations of the fundamental group of a curve. In positive characteristic, there is a ``twisted" version of NAHT, but it is not clear how to extract geometric information from it.

I will report on joint work with graduating student Siqing Zhang, where we prove a cohomological version of NAHT, i.e., we exhibit a canonical isomorphism between the etale cohomology rings of the moduli of Higgs bundles and of flat connections.

I will explain how this works via vanishing cycle theory.

TIme permitting, I will also discuss some perhaps unexpected corollaries relating cohomology rings of different moduli spaces, in equal and in mixed characteristic.