Birational geometry of quiver varieties (Video, Slides)

Abstract: In this talk I will report on joint work in progress with A. Craw and T. Schedler on the birational geometry of quiver varieties. We give an explicit local description of the birational transformations that occur under variation of GIT for quiver varieties. The main consequence of this local picture is that one can show that all Q-factorial terminalizations of quiver varieties (excluding the (2,2) case) can be obtained by VGIT. I will try to explain what our results mean in two concrete classes of examples. Namely, for framed affine Dynkin quivers (corresponding to wreath product quotient singularities) and star shape quivers (corresponding to hyperpolygon spaces).

Elana Kalashnikov (Harvard)

Finding mirrors for Fano quiver ﬂag zero loci (Video, Notes)

Abstract: One interesting feature of the classiﬁcation of smooth Fano varieties up to dimension three is that they can all be described as certain subvarieties in GIT quotients; in particular, they are all either toric complete intersections (subvarieties of toric varieties) or quiver ﬂag zero loci (subvarieties of quiver ﬂag varieties). Fano varieties are expected to mirror certain Laurent polynomials; given such a Fano toric complete intersection, one can produce a Laurent polynomial via the Landau-Ginzburg model. In this talk, I’ll discuss ﬁnding mirrors of four dimensional Fano quiver ﬂag zero loci via ﬁnding degenerations of the ambient quiver ﬂag varieties. These degenerations generalise the Gelfand-Cetlin degeneration, which in the Grassmannian case has an important role in the cluster structure of its coordinate ring.

Masoud Kamgarpour (Queensland)

Geometric Langlands for hypergeometric sheaves (Video, Notes)

Abstract: Generalised hypergeometric sheaves are rigid local systems on the punctured projective line with remarkable properties. Their study originated in the seminal work of Riemann on the Euler--Gauss hypergeometric function and has blossomed into an active field with connections to many areas of mathematics. I will report on a joint work with Lingfei Yi, where we construct the Hecke eigensheaves whose eigenvalues are the irreducible hypergeometric local systems. This confirms a central conjecture of the geometric Langlands program for hypergeometrics. The key tool we use is the notion of rigid automorphic data due to Zhiwei Yun. This talk is based on the preprint arXiv:2006.10870.

Ivan Losev (Yale)

On unipotent Harish-Chandra bimodules

Abstract: The talk concerns an important class of Harish-Chandra bimodules over semisimple Lie algebras, unipotent ones. We give a general definition and establish several basic properties. Our approach is based on studying canonical quantizations of symplectic singularities. This is a joint work in progress with Lucas Mason-Brown and Dmytro Matvieievskyi.

Hiraku Nakajima (Kavli IPMU)

Euler numbers of Hilbert schemes of points on simple surface singularities (Video, Slides)

Abstract: We prove the conjecture by Gyenge, Némethi and Szendrői in arXiv:1512.06844, arXiv:1512.06848 giving a formula of the generating function of Euler numbers of Hilbert schemes of points Hilb^{n}(C^{2}/Γ) on a simple singularity C^{2}/Γ, where Γ is a finite subgroup of SL(2). This is based on my preprint arXiv:2001.03834.

Abstract: In this talk we will introduce generalized hyperpolygons, which arise as Nakajima-type representations of a comet-shaped quiver, following recent work joint with Steven Rayan (arXiv:2001.06911). After showing how to identify these representations with pairs of polygons, we shall associate to the data an explicit meromorphic Higgs bundle on a genus-g Riemann surface, where g is the number of loops in the comet. We shall see that, under certain assumptions on flag types, the moduli space of generalized hyperpolygons admits the structure of a completely integrable Hamiltonian system. Time permitting, we shall conclude the talk by mentioning some partial results on current work on the construction of triple branes (in the sense of Kapustin-Witten mirror symmetry), and dualities between tame and wild Hitchin systems (in the sense of Painlevé transcendents).

Filip Zivanovic (Oxford)

Exact Lagrangians in conical symplectic resolutions (Video, Slides)

Abstract: Conical symplectic resolutions are a vast family of holomorphic symplectic manifolds that appear in representation theory, algebraic and differential geometry, and also in theoretical physics. Their typical examples arise from the hyperkähler quotient construction (quiver and hypertoric varieties) but also from the representation theory of Lie algebras (resolutions of Slodowy varieties, slices in affine Grassmannians). In this talk, I will focus on their symplectic topology. In particular, we find families of non-isotopic exact Lagrangian submanifolds in them arising from different C*-actions. These Lagrangians have a very nice symplectic topology; in particular, we conjecture (work in progress) that all of their Floer-theoretic invariants are completely determined by their topology. At the end of the talk, I will discuss the special cases of Nakajima quiver varieties and resolutions of Slodowy varieties, where their count becomes feasible and interesting in its own.