Image © Andrew J. Hanson, Indiana University
Abstract: Starting with two non-birational derived equivalent K3 surfaces, one can ask whether their Hilbert schemes of points are birational. In this talk, we will show that in some cases they are but in most cases they are not. This is joint work with Giovanni Mongardi and Kota Yoshioka.
Abstract: It is a classical result of Beauville and Donagi that Fano varieties of lines on cubic fourfolds are hyper-Kahler. More recently, Lehn, Lehn, Sorger and van Straten constructed a hyper-Kahler eightfold out of twisted cubics on cubic fourfolds. In this talk, I will explain a new approach to these hyper-Kahler varieties via moduli of stable objects on the Kuznetsov components. Along the way, we will derive several properties of cubic fourfolds as consequences. This is based on a joint work with Chunyi Li and Laura Pertusi.
Abstract: I will talk about some basic facts about slope stable sheaves and the Bogomolov inequality. New techniques from stability conditions will imply new stronger bounds on Chern characters of stable sheaves on some special varieties, including Fano varieties, quintic threefolds and etc. I will discuss the progress in this direction and some related open problems.
Abstract: We provide a complete comparison between the localizations of the categories of dg categories, strictly unital A_\infty categories, and cohomological unital A_\infty categories by the corresponding classes of quasi-equivalences. A complete proof of this key result is missing in the literature. As an application, we can prove a claim of Kontsevich (and Drinfeld) that says that, over a field, the category of internal Homs between two dg categories is quasi-equivalent to the category of strictly unital A_\infty functors between them. We also deduce some results concerning the uniqueness of enhancements for 'geometric' derived categories. This is joint work with A. Canonaco and M. Ornaghi.
Abstract: Stability conditions on derived categories of algebraic varieties and their wall-crossings have recently been used extensively to study the geometry of moduli spaces of stable sheaves. In work in progress with Macri, Lahoz, Nuer, Perry and Stellari, we are extending this toolkit to the "relative" setting, i.e. for a family of varieties. Our construction comes with relative moduli spaces of stable objects; this gives additional ways of constructing new families of varieties from a given family, thereby potentially relating different moduli spaces of varieties. Our main application is for families of cubic fourfolds; in particular, this produces many new examples of algebraically constructed families of Hyperkaehler varieties over a base of maximal dimension 20.
Abstract: Let C be a general plane quartic curve. It follows from the Plucker formulas that C admits 24 inflection lines. We address the question of finding all the plane quartics D, having the same 24 inflection lines as C. In joint work with Marco Pacini, we show that, over fields of characteristic coprime with 6, the curve C is uniquely determined by its configuration of inflection lines.
Abstract: This talk is based on joint work with Sam Payne (arXiv:1703.10228). Kontsevich and Soibelman's motivic upgrade of Donaldson-Thomas theory produces refined curve counting invariants by means of motivic vanishing cycles of potential functions. In order to get a coherent theory, Kontsevich-Soibelman and Davison-Meinhardt have conjectured formulas for the motivic vanishing cycles of special types of functions. I will explain how one can deduce these formulas from a combination of Hrushovski-Kazhdan motivic integration and tropical geometry.
Abstract: I will begin by reviewing the geometry of a cyclic cover branched in a divisor. I will then explain how it gives the first ever example of a non-split P^n-functor. This is a joint work with Rina Anno (Kansas).
Abstract: We introduce X-stability conditions on Calabi-Yau-X categories and spaces of their specializations, the q-stability conditions. The motivating example comes from the Calabi-Yau-X category D(S) associated to a graded marked surface S, constructed from quivers with superpotential. We show that the cluster category of D(S) is Haiden-Katzarkov-Kontsevich's topological Fukaya category C(S) and Bridgeland-Smith type Calabi-Yau-N categories are the orbit quotients of D(S). Moreover, we show that stability conditions on C(S) induce q-stability conditions on D(S). Finally, we are constructing moduli space to realize the fiber of the spaces of q-stabilty conditions for given complex number s.
Abstract: In 1997 Hitchin proved that the Riemann Hilbert correspondence between Fuchsian systems and conjugacy classes of representations of the fundamental group of the punctured sphere is a Poisson map. Since then, some generalisations of this result to the case of irregular singularities have been proposed by Boalch and by Gualtieri, Li and Pym. In this talk we interpret irregular singularities as the result of collisions of boundaries in a Riemann surface and show that the Stokes phenomenon corresponds to the presence of "bordered cusps". We introduce the concept of decorated character variety of a Riemann surface with bordered cusps and construct a generalised cluster algebra structure and cluster Poisson structure on it. We define the quantum cluster algebras of geometric type and show that they provide an explicit canonical quantisation of this Poisson structure.
Abstract: The study of separating invariants is a new trend in Invariant Theory and a return to its roots: invariants as a classification tool. For a finite group acting linearly on a vector space, a separating set is simply a set of invariants whose elements separate the orbits o the action. Such a set need not generate the ring of invariants. In this talk, we give lower bounds on the size of separating sets based on the geometry of the action. These results are obtained via the study of the local cohomology with support at an arrangement of linear subspaces naturally arising from the action. (Joint with Jack Jeffries)
Abstract: I will introduce the isomonodromic deformation method for Painleve I and the corresponding Riemann-Hilbert problem in term of Stokes multipliers. I will then use a theory due to R. Nevanlinna, [Ueber Riemannsche Flaechen mit endlich vielen Windungspunkten, Acta Math 1932] to give an alternative construction of the monodromy manifold, and a proof of the surjectivity of the monodromy map. Finally, I will comment on some applications of the same method to other Painleve equations: in particular, I will show how to compute the numer of real roots of the rational solutions of the fourth Painleve equations.
Abstract: In 1997, Mukai introduced a geometric program to reconstruct a K3 surface from a curve on that surface. The idea is to first consider a Brill-Noether locus of vector bundles on the curve. Then the K3 surface containing the curve can be obtained uniquely as a Fourier-Mukai partner of the Brill-Noether locus. Mukai carried out this program for curves of genus 11. I will explain how wall-crossing with respect to Bridgeland stability conditions implies that the Mukai's strategy works for curves of higher genera.
Abstract: I will start describing the Gross-Hacking-Keel realization of mirror symmetry for log Calabi-Yau surfaces: the mirror variety is constructed by gluing elementary pieces together according to some gluing functions determined by counting rational curves in the original variety. I will then explain how to construct non-commutative deformations of these mirrors by including contributions of counts of higher genus curves in the original variety.
Abstract: Given a Fano manifold we will consider two ways of attaching a (usually infinite) collection of polytopes, and a certain combinatorial transformation relating them, to it. The first is via Mirror Symmetry, following a proposal of Coates-Corti-Kasprzyk-Galkin-Golyshev. The second is via symplectic topology, and comes from considering degenerating Lagrangian torus fibrations. We then relate these two collections using the Gross--Siebert program. I will also comment on the situation in higher dimensions, noting particularly that by 'inverting' the second method (degenerating Lagrangian fibrations) we can produce topological constructions of Fano threefolds.