Abstract: In my talk, I will discuss isomorphisms between quiver varieties of type A and transversal slices in the affine Grassmannian for GLd. Such an isomorphism was first constructed by Mirković and Vybornov. Their proof and construction are rather combinatorial. We will explain a geometric construction of isomorphisms between quiver varieties and transversal slices that follows by combining ideas of Braverman-Finkelberg and Nakajima. We will compute these geometric isomorphisms and time permitting explain why they coincide with Mirkovic-Vybornov's.

Abstract: I will discuss the intersection of two copies of Gr(2,5) embedded in P9, and the intersection of the two projectively dual Grassmannians in the dual projective space. These intersections are deformation equivalent Calabi-Yau threefolds. I will explain why they are derived equivalent but generically not birational, and use this to obtain a counterexample to the birational Torelli problem for Calabi-Yau threefolds, as well as new examples of zero divisors in the Grothendieck ring of varieties. This is joint work with Lev Borisov and Andrei Caldararu.

Abstract: Through 3 general points and 6 general lines in P3, there are exactly 190 twisted cubics; 190 is a (genus-zero) Gromov-Witten invariant of P3. I will introduce Gromov-Witten invariants of a smooth complex projective variety X, and discuss how a torus action on X can help us compute its Gromov-Witten invariants. Applying this to a topic variety X, Kontsevich, Givental, and Lian-Liu-Yau proved the “quintic mirror theorem” predicted by string theorists. I will discuss the difficulties that arise when X is not toric. In particular, I will talk about a concrete nontoric orbifold X=Symd(P4), the symmetric product of projective space. By studying the equivariant geometry of Symd(P4), I extend the strategies of Givental/Lian-Liu-Yau to prove a mirror theorem for Symd(P4).

Abstract: Kleinian singularities, i.e., the varieties corresponding to the algebras of invariants of Kleinian groups are of fundamental importance for algebraic geometry, representation theory and singularity theory. The filtered deformations of these algebras of invariants were classified by Slodowy (the commutative case) and Losev (the general case). To an inclusion of Kleinian groups, there is the corresponding inclusion of algebras of invariants. We classify deformations of these inclusions when a smaller subgroup is normal in the larger.

Abstract: A GM fourfold is a smooth dimensionally transverse intersection of the cone over the Grassmannian Gr(2,5) with a quadric hypersurface in a eight-dimensional linear space over C. These Fano fourfolds have a lot of similarities with cubic fourfolds. Debarre and Kuznetsov constructed an associated EPW sextic hypersurface, whose double cover, when smooth, is a hyperkähler fourfold deformation equivalent to the Hilbert square of a K3 surface.

The aim of this talk is to study the double EPW sextic associated to a GM fourfold as a moduli space of (twisted) stable sheaves on a K3 surface, as done by Addington for the Fano variety of lines of a cubic fourfold. To this end, we discuss the problem of characterizing Hodge-special GM fourfolds with an associated K3 surface in terms of their Mukai lattice. Then we prove a necessary and sufficient condition in order to have the double EPW sextic birational to the Hilbert square of a K3 surface.

Abstract: Hyperbolicity is an interesting but difficult property for both analytic and algebraic varieties. I will recall some related conjectures from both the analytic and algebraic points of view. Then I will introduce a new hyperbolic result for the base space of families with maximal variation and deduce Brody hyperbolity for moduli stacks of polarized varieties of general type from it. If time permits, I will also explain how Hodge theory comes into this story. This is a recent work joint with Mihnea Popa and Behrouz Taji.

Abstract: This is joint work with David Nadler, continuing a program initiated by Ben-Zvi and Nadler. We consider a variant of the geometric Langlands conjecture, which is expected to be of topological nature. It relates constructible sheaves on the moduli space of G-bundles on an algebraic curve (with an important condition on the singular support of the constructible sheaves) to quasi-coherent sheaves on certain character varieties of the dual group. We show that the latter category acts on the former, hence establishing a spectral decomposition of automorphic sheaves in this setting. There is an analogy of our result with Vincent Lafforgue's work on the classical Langlands correspondence over a function field.

Abstract: Work of Bezrukavnikov and Kaledin showed that the category of coherent sheaves on certain special conic symplectic resolutions (a special class of quasi-projective varieties) has a very special structure: it is derived equivalent to the representations of a noncommutative algebra arising from deformation quantization, and in fact, these derived equivalences stitch together into D-equivalences between the different crepant resolutions of a single singular affine variety. Together, these equivalences to compose to give an action of a generalization the braid group on this category.

While this picture is quite beautiful, the general implementation of it is hard to make explicit. I'll discuss a special case where this is more tractable: the Coulomb branches, recently defined mathematically by work of Braverman, Finkelberg and Nakajima, in particular for quiver gauge theories. In this case, the non-commutative algebras underlying this picture have a very concrete realization: they are versions of KLR algebras drawn on cylinders. Using this realization, one can, for example, prove the (recently proven) conjecture of Bezrukavnikov and Okounkov relating the group action above to the monodromy of the quantum connection for quiver varieties/Slodowy slices in type A.

Abstract: The intersection of two general PGL(10)-translates of the Grassmannian Gr(2,5) is a Calabi-Yau 3-fold X, and the intersection of the projective duals of the two translates is another Calabi-Yau 3-fold Y. We show that X and Y provide counterexamples to a certain ”birational” Torelli problem for Calabi-Yau 3-folds, namely, they are deformation equivalent, derived equivalent, and have isomorphic Hodge structures, but they are not birational. This is joint work with Jørgen Vold Rennemo.

Abstract: The integrals of characteristic classes of tautological sheaves on the Hilbert scheme of points arise in enumerative problems. I'll explain an approach to the K-theoretic versions of such expressions.

Namely, I will explain how to compute the K-theoretic Donaldson-Thomas partition functions of toric Calabi-Yau threefolds, and will deduce from this computation certain symmetries satisfied by generating functions of equivariant Euler characteristics of tautological classes on the Hilbert scheme.

Abstract: Let G be a complex semisimple algebraic group of adjoint type and G¯ the wonderful compactification. We show that the closure in G¯ of the centralizer Ge of a regular nilpotent e∈Lie(G) is isomorphic to the Peterson variety. We generalize this result to show that for any regular x∈Lie(G), the closure of the centralizer Gx in G¯ is isomorphic to the closure of a general Gx-orbit in the flag variety. We consider the family of all such centralizer closures, which is a partial compactification of the universal centralizer. We show that it has a natural log-symplectic Poisson structure that extends the usual symplectic structure on the universal centralizer.

Abstract: Most integrals of algebraic differential forms cannot be computed explicitly, however when they depend on a parameter the integrals satisfy explicit differential equations and this naturally leads to variations of Hodge structures. Hypergeometric equations are some of the simplest non-trivial examples and I will use them as an example to explain some relations between Hodge theory and invariants coming from dynamical systems called Lyapunov exponents. The necessary background will be provided.

Abstract: Nakajima's quiver varieties form an important class of algebraic symplectic varieties. A quiver variety comes naturally equipped with certain “tautological vector bundles”; I will explain joint work with McGerty that shows that the cohomology ring of the quiver variety is generated by the Chern classes of the tautological bundles. Analogous results (work in preparation with McGerty) also hold for the Crawley-Boevey—Shaw “multiplicative quiver varieties,’’ in particular for twisted character varieties; and the cohomology results in both cases generalize to other cohomology theories, derived categories, etc. I hope to explain the main ideas behind the proofs of such theorems and how they form part of a general pattern in noncommutative geometry.