Palace of Catalan Music (S. Adams/GETTY)

## Geometry, Physics, and Representation Theory Seminar

### RTG Group Seminar

#### Fall 2017, Northeastern

 Meeting weekly on Thursdays 2:50-3:50pm in 509 Lake Hall at Northeastern. If you are not at Northeastern, but would like to recieve announcements, join the mailing list. If you have a question or would like to speak at seminar, email

### Schedule

 Date Speaker Title Sep 7 Vasily Krylov (Higher School of Economics, Moscow) On isomorphisms between quiver varieties of type A and slices in the affine Grassmannian. In my talk, I will discuss isomorphisms between quiver varieties of type A and transversal slices in the affine Grassmannian for $GL_d$. 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. Sep 14 Alex Perry (Columbia) Intersections of two Grassmannians in $\mathbf{P}^9$. I will discuss the intersection of two copies of $Gr(2,5)$ embedded in $\mathbf{P}^9$, 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. Sep 21 Rob Silversmith (SCGP, Stony Brook) Gromov-Witten Invariants of Symmetric Products of Projective Space. Through 3 general points and 6 general lines in $\mathbf{P}^3$, there are exactly 190 twisted cubics; 190 is a (genus-zero) Gromov-Witten invariant of $\mathbf{P}^3$. 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 $X=\mathrm{Sym}^d(\mathbf{P}^4)$, the symmetric product of projective space. By studying the equivariant geometry of $\mathrm{Sym}^d(\mathbf{P}^4)$, I extend the strategies of Givental/Lian-Liu-Yau to prove a mirror theorem for $\mathrm{Sym}^d(\mathbf{P}^4)$. Sep 28 Daniil Kliuev (Saint Petersburg State University) Deformations of pairs of Kleinian singularities. 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. Oct 5 Laura Pertusi (University of Milan) On the double EPW sextic associated to a Gushel-Mukai fourfold. A GM fourfold is a smooth dimensionally transverse intersection of the cone over the Grassmannian $\text{Gr}(2,5)$ with a quadric hypersurface in a eight-dimensional linear space over $\mathbb{C}$. 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. Oct 12 Lei Wu (Utah) Hyperbolicity Properties of Base Spaces of Families with Maximal Variation. 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. Oct 19 Zhiwei Yun (Yale) Spectral decomposition in Betti geometric Langlands. 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. Oct 26 Ben Webster (University of Waterloo and Perimeter Institute) Coherent sheaves on Coulomb branches. 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. Nov 2 CANCELLED. To be rescheduled. Betti spectral curves and Betti spectral 3-manifolds. . A Lagrangian $L$ in the cotangent bundle of $M$ can determine a family of local systems on $M$, by either Floer theory or microlocal sheaf theory. When $M$ is two-dimensional, there is a close analogy between that construction, and the family of Higgs bundles parametrized by a spectral curve in the cotangent of a Riemann surface. When $M$ is 3-dimensional you could do something similar, and the analogy is more compelling if you allow for the local system on $M$ to have "irregular singularities" along a boundary -- but you have to make up a definition of irregular singularities, in 3d. I'll try to explain starting from Deligne's old irregular Riemann-Hilbert correspondence. The talk is partly based on joint work with Linhui Shen and Eric Zaslow, and partly based on joint work with Xin Jin. Nov 9 John Christian Ottem (University of Oslo) The birational Torelli problem for Calabi-Yau 3-folds. 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. Nov 16 Noah Arbesfeld (Columbia) K-theoretic Donaldson-Thomas theory and tautological classes on the Hilbert scheme of points on a surface. 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. Mon, Nov 20 at 3:50 Ana Balibanu (Harvard) The wonderful compactification and the universal centralizer. Let $G$ be a complex semisimple algebraic group of adjoint type and $\overline{G}$ the wonderful compactification. We show that the closure in $\overline{G}$ of the centralizer $G^e$ of a regular nilpotent $e\in\text{Lie}(G)$ is isomorphic to the Peterson variety. We generalize this result to show that for any regular $x\in\text{Lie}(G)$, the closure of the centralizer $G^x$ in $\overline{G}$ is isomorphic to the closure of a general $G^x$-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. Nov 30 Simion Filip (Harvard) Hypergeometric Equations and Lyapunov Exponents 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. Dec 7 Thomas Nevins (UIUC) Cohomology of quiver varieties and other moduli spaces 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. End of Semester (Seminar Resumes Spring 2018)

### Schedule Spring 2018

#### Previous Semesters

 Spring 2017 Fall 2016 Spring 2016 Fall 2015 Spring 2015