Benjamin Bakker |
Title: The geometric torsion conjecture for Hilbert modular varieties Abstract: A celebrated theorem of Mazur asserts that the order of the torsion part of the Mordell-Weil group of an elliptic curve over Q is absolutely bounded; it is conjectured that the same is true for abelian varieties over number fields, though very little progress has been made in proving it. The natural geometric analog where Q is replaced by the function field of a complex curve---dubbed the geometric torsion conjecture---is equivalent to the nonexistence of low genus curves in congruence towers of Siegel modular varieties. In joint work with J. Tsimerman, we prove the geometric torsion conjecture for abelian varieties with real multiplication. The proof uses the hyperbolic geometry of Hilbert modular varieties to produce new bounds on Seshadri constants of the canonical bundle along the boundary. |
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Title: Birational geometry of moduli spaces of stable rational curves Abstract: I will report on joint work with Jenia Tevelev on the birational geometry of the Grothendieck-Knudsen moduli space of stable rational curves with n markings. We prove that for n large, this space is not a Mori Dream Space, thus answering a question of Hu and Keel. | |
Ionut Ciocan-Fontanine |
Title: Quasimap theory Abstract: I will survey the theory of quasimap invariants of (a class of) GIT quotient targets and their relation via wall crossing to the Gromov-Witten invariants of these targets. This theory was developed in joint work with Bumsig Kim, and in part also with Daewoong Cheng and Davesh Maulik. |
Izzet Coskun |
Title: The birational geometry of moduli spaces of sheaves on the plane Abstract: I will discuss the birational geometry of moduli spaces of sheaves on the plane. I will first discuss joint work with Jack Huizenga and Matthew Woolf on the effective cone. Then I will describe joint work with Jack Huizenga on the ample cone. |
Duiliu-Emanuel Diaconescu |
Title: Donaldson-Thomas invariants and character varieties Abstract: A relation between the motivic Donaldson-Thomas invariants of Kontsevich and Soibelman and the cohomology of character varieties will be shown to arise naturally in string theory. This yields in particular a string theoretic derivation of a conjecture of Hausel, Letellier and Rodriguez-Villegas and generalizations. This is based on work with Wu-yen Chuang, Ron Donagi and Tony Pantev. |
Samuel Grushevsky |
Title: Shimura curves contained in the moduli space of Jacobians in low genus Abstract: We construct infinitely many Shimura curves (1-dimensional special subvarieties, which are in particularly totally geodesic) of the moduli space of abelian varieties, contained in the locus of hyperelliptic Jacobians in genus 3, and in the locus of Jacobians in genus 4 - only finitely many such examples were previously known. Joint work with Martin Moeller. |
Felix Janda |
Title: Relations in the tautological ring Abstract: The relations of Faber-Zagier form a (conjecturally full) set of relations between kappa classes in the cohomology of the moduli space of smooth curves. In 2012 Aaron Pixton gave a remarkable, conjectural extension of these relations to the Deligne-Mumford compactification of stable curves. So far there have been two proofs of the fact that what Pixton has written down are indeed relations. They use very different geometries. I want to discuss the relationship between the two proofs. |
Emanuele Macri |
Title: Stability conditions on abelian threefolds Abstract: I will present a new proof and a generalization a result by Maciocia and Piyaratne on the existence of Bridgeland stability conditions on any abelian threefold. As an application, we deduce the existence of Bridgeland stability conditions on a number of Calabi-Yau threefolds, namely Calabi-Yau threefolds of abelian type and Kummer threefolds. As in the work of Maciocia and Piyaratne, the idea is to show a Bogomolov-Gieseker type inequality involving Chern classes of certain stable objects in the derived category; this was conjectured by Bayer, Toda, and myself. Our approach uses the multiplication maps on abelian threefolds instead of Fourier-Mukai transforms. This is joint work with Arend Bayer and Paolo Stellari. |
Eyal Markman |
Title: Integral transforms from a K3 surface to a moduli space of stable sheaves on it Abstract: Let S be a K3 surface, v an indivisible Mukai vector, and M(v) the moduli space of stable sheaves on S with Mukai vector v. The universal sheaf gives rise to an integral functor F from the derived category of coherent sheaves on S to that on M(v). We show that the functor F is faithful. The bounded derived category of M(v) is rather mysterious at the moment. As a first step, we provide a simple conjectural description of its full subcategory whose objects are images of objects on S via the functor F. We verify that description whenever M(v) is the Hilbert scheme of points on S. This work is joint with Sukhendu Mehrotra. |
Georg Oberdieck |
Title: Gromov-Witten theory of Hilb(K3,n) and the Igusa cusp form Abstract: In this talk, I will first report on recent proven results about the genus 0 Gromov-Witten theory of the higher-dimensional hyperkaehler's Hilb(K3,n), the Hilbert schemes of points on a K3 surface. The results show that for several natural incidence conditions the GW counts are given by the Fourier coefficients of Jacobi forms of index n-1. In the second part, I will present several conjectures on the full (primitive case) two-point GW theory of Hilb(K3,n) and related topics. In particular, I will show how the Gromov-Witten theory of all the Hilb(K3,n)'s combined leads (conjecturally) to the inverse of a Siegel modular form of weight 10: the Igusa cusp form. |
Kieran O'Grady |
Title: Moduli and periods of double EPW-sextics Abstract: Smooth double EPW-sextics are the members of a locally complete family of projective HK fourfolds. Their moduli space has a GIT compactification and a compactification provided by Hodge Theory. We will report on results of the author on the relation between these two compactifications. I will also report on joint work in progress with Radu Laza which led us to reconsider moduli of quartic surfaces. |
Anand Patel |
Title: Current developments in the study of Hurwitz spaces Abstract: This talk is intended to survey some current progress and open problems in the study of branched covers. I will limit my attention to three fundamental aspects of the study of Hurwitz spaces: Their birational geometry, their divisor theory, and their Chow rings. |
Aaron Pixton |
Title: A conjectural formula for the double ramification cycle Abstract: Double ramification cycles parametrize curves that admit maps to the projective line with specified ramification profiles over zero and infinity. These cycles can be extended to the moduli space of stable curves by using the virtual class in relative Gromov-Witten theory. I will describe a conjectural formula for these extensions in terms of tautological classes. The formula is partially motivated by an analogy with a recently proven formula for Witten's r-spin class. |