Philip Boalch: Higgs bundles, connections and quivers
In string theory one is apparently supposed to replace a (Feynman) graph by a (Riemann) surface, to pass from a perturbative picture to a nonperturbative one. In the theory of hyperkahler manifolds there is a class of examples attached to graphs (and some data on the graph)--- the Nakajima quiver varieties, and a class of examples attached to Riemann surfaces (and some data on the surface, to specify the boundary conditions)---the wild Hitchin spaces. I will talk about these "nonperturbative" hyperkahler manifolds attached to surfaces, and how in some cases they are related to graphs. This yields a new theory of "multiplicative quiver varieties", and enables us to extend work of Okamoto and Crawley-Boevey to see the appearance of many non-affine Kac-Moody Weyl groups and root systems in the theory of connections on Riemann surfaces (in contrast to the usual, local, understanding of affine Kac-Moody algebras, in terms of loop algebras).Samuel Boissiere: Moduli spaces of automorphisms on hyperkaehler manifolds
I'll present recent results on the classification of non-symplectic automorphisms of prime order acting on irreducible holomorphic symplectic fourfolds deformation equivalent to the Hilbert scheme of two points on a K3 surface. This classification relates the isometry classes of two natural lattices associated to the action of the automorphism on the second cohomology group with integer coefficients with some topological invariants of the fixed locus. This is a joint work with Chiara Camere and Alessandra Sarti.Pierrick Bousseau: Around the GW/Kronecker correspondence
The GW/Kronecker correspondence, due to the combination of the work of Gross-Pandharipande-Siebert and Reineke, gives a relation between Donaldson-Thomas invariants of the m-Kronecker quivers and some relative Gromov-Witten invariants of some rational surfaces. The usual approach of this result uses the tropical vertex group: the two sides of the correspondence have the same algebraic structure. I will explain a more geometric approach, using as an intermediate step Donaldson-Thomas invariants of a local Calabi-Yau 3-fold.Renzo Cavalieri: Toric open invariants and Crepant Transformations
The question that the Crepant Resolution Conjecture (CRC) wants to address is: given an orbifold X that admits a crepant resolution Y, can we systematically compare the Gromov-Witten theories of the two spaces? That this should happen was first observed by physicists and the question was imported into mathematics by Y.Ruan, who posited as the search for an isomorphism in the quantum cohomologies of the two spaces. In the last fifteen years this question has evolved and found different formulations which various degree of generality and validity. Perhaps the most powerful approach to the CRC is through Givental's formalism. In this case, Coates, Iritani and Tseng propose that the CRC should consist of the natural comparison of geometric objects constructed from the GW potential fo the space. We explore this approach in the setting of open GW invariants. We formulate an open version of the CRC using this formalism, and verify it for the family of A_n singularities, for the Topological Vertex and for the (non Hard-Lefschetz) cyclic quotients resolving to the canonical of weighted projective planes. Our approach is well tuned with Iritani's approach to the CRC via integral structures. Further in the Hard-Lefschetz case, when combined with Givental's quantization formula, it naturally extends the genus 0 comparison of potentials to an all genera comparison.Kwokwai Chan: SYZ for local CY varieties
In this talk, I will explain the SYZ proposal for understanding mirror symmetry in the case of local Calabi-Yau (CY) manifolds/orbifolds. Along the way, we obtain a proof of a conjecture by Gross and Siebert which provides an enumerative meaning to (inverse) mirror maps in terms of virtual counts of holomorphic disks. This talk is based on a series of joint works with Cheol-Hyun Cho, Siu-Cheong Lau, Conan Leung and Hsian-HuaTseng.Ben Davison: A few ways to think about the Cohomological Hall algebra of Higgs bundles
In this talk I will introduce the Cohomological Hall algebra for the stack of all finite-dimensional representations of the fundamental group of a closed genus g Riemann surface S. This algebra A_g turns out to be a free supercommutative algebra, and I'll discuss a recent conjecture, identifying the generators of this algebra with the cohomology of the space of twisted representations of the fundamental group of S. The algebra structure on A_g respects Hodge structures, and so any conjecture you might have regarding the Hodge polynomial of twisted character varieties turns out to be (conjecturally) equivalent to one about untwisted character varieties. On the other hand, all of the above suggests that there should be a natural Hall algebra structure on the stack of semistable Higgs bundles, respecting a perverse filtration. I'll finish by giving some pointers as to how this should work.Mario Garcia-Fernandez: Conformal limits of irregular connections
We sketch the construction of isomonodromic families of irregular meromorphic connections \nabla(Z) on P^1, with values in the derivations of a class of infinite dimensional Poisson algebras. Our main results concern the limits of the families as we vary a scaling parameter R. In the R \to 0 'conformal limit' we recover a semi-classical version of the connections introduced by Bridgeland and Toledano Laredo (and so the Joyce holomorphic generating functions for DT invariants). The connections \nabla(Z) are a rough but rigorous approximation to the (mostly conjectural) four-dimensional tt*-connections introduced by Gaiotto-Moore-Neitzke. A precise comparison with these is established in a basic example. Joint work with Jacopo Stoppa and Sara A. Filippini.Tamas Hausel: Arithmetic of wild character varieties
I explain some conjectures on the mixed Hodge polynomials and perverse Hodge polynomials of tame character varieties and moduli of parabolic Higgs bundles on Riemann surfaces. We will then study the arithmetic of one class of Boalch's wild character varieties using the character theory of Yokonuma-Hecke algebras, and point out the relationship of the point count, and natural conjecture on their mixed Hodge polynomials to the tame case. This is joint work with Martin Mereb and Michael Wong.Manfred Lehn: Twisted cubics on cubic fourfolds
The moduli space M of generalised twisted cubics on a smooth cubic fourfold Y is a smooth 10-dimensional variety if Y does not contain a plane. M admits a contraction to an 8-dimensional holomorphic symplectic manifold Z that turns out to be deformation equivalent to a the Hilbert scheme of generalised four-tuples of points on some K3-surface. As M and Z vary with Y one obtains a 20-dimensional family of projective hyperkaehler manifolds. This is a report on joint work with Christian Lehn, Christoph Sorger, Duco van Straten and Nick Addington.Andrew MacPherson: Skeleta in non-Archimedean and tropical geometry
Non-Archimedean geometry is a tool to study degenerations of algebraic varieties (or complex manifolds). Tropical geometry is a way to capture leading order information from algebraic degenerations in combinatorial terms. It is therefore natural to try to associate tropical varieties to non-Archimedean analytic spaces. I'll introduce an algebro-geometric theory of skeleta, based on taking "Spec" of a semiring, designed to address this situation. The punchline (of the talk) will be the construction of a "universal tropicalisation" of a rigid analytic space. In deference to the fact that this is a mirror symmetry conference, I'll also try and fit in something about the relation to the SYZ conjecture.Hannah Markwig: Counting curls and hunting monomials - Tropical mirror symmetry for elliptic curves
Mirror symmetry relates Gromov-Witten invariants of an elliptic curve with certain integrals over Feynman graphs. We prove a tropical generalization of mirror symmetry for elliptic curves, i.e., a statement relating certain labeled Gromov-Witten invariants of a tropical elliptic curve to more refined Feynman integrals. This result easily implies the tropical analogue of the mirror symmetry statement mentioned above and, using the necessary Correspondence Theorem, also the mirror symmetry statement itself. In this way, our tropical generalization leads to an alternative proof of mirror symmetry for elliptic curves. We believe that our approach via tropical mirror symmetry naturally carries the potential of being generalized to more adventurous situations of mirror symmetry. Moreover, our tropical approach has the advantage that all involved invariants are easy to compute. Furthermore, we can use the techniques for computing Feynman integrals to prove that they are quasimodular forms. Joint work with Janko Boehm, Kathrin Bringmann and Arne Buchholz.Diego Matessi: On homological mirror symmetry of toric Calabi-Yau varieties
Using lagrangian torus fibrations on the mirror X of a 3-dimensional (open) toric variety X', I will show a construction of Lagrangian sections and Langrangian spheres in X. Then I will discuss a correspondence which sends sections to line bundles on X' and spheres to sheaves supported on the toric divisors of X'. I will give some evidence that this is a homological mirror symmetry correspondence. This is joint work with Mark Gross.Luca Migliorini: Support theorems for Hilbert schemes of families of planar curves
Given a versal family of curves with at worst planar singularities over a base A, we consider its associated relative Hilbert scheme of a fixed length: its total space is nonsingular, and the pushforward of the constant sheaf to A splits into a direct sum of intersection cohomology sheaves (plus shifts). It is an interesting problem to determine them or at least their supports. If the curves in the family are all integral, a result independently due to Maulik-Yun and Migliorini-Shende ensures that the sheaves are all supported on A (the support theorem). In case the family contains reducible curves this is no longer true and I will discuss a formula which takes the place of the support theorem. Joint work in progress with V. Shende and F. Viviani.Alessandra Sarti: Borcea-Voisin mirror Calabi-Yau threefolds and invertible potentials
Borcea and Voisin constructed a mirror symmetry for Calabi-Yau manifolds that arise as crepant resolution of the quotient of the product of a K3 surface and an elliptic curve by an involution. The mirror Calabi-Yau manifold is essentially obtained by considering the lattice mirror of the K3 surface and repeating the same construction. Recently Chiodo and Ruan, following an idea of the physicists Berglund and Hubsch, used non-degenerate invertible potentials to describe another construction of mirror Calabi-Yau manifolds. In the talk I will explain the connection between these two mirror symmetry constructions. This is a joint work in progress with M. Artebani and S. Boissiere.Vivek Shende: The link of an irregular connection
I will explain that the conjectured equality between the cohomology of a plane curve singularity and the Khovanov-Rozansky homology of its link follows (up to a spectral sequence) from the P=W conjecture in the case of irregular singularities. The key players are a formulation of the irregular Riemann-Hilbert correspondence which foregrounds what might be called the link of the connection, and a new sheaf-theoretic invariant of legendrian knots.Valentino Tosatti: Collapsing of Calabi-Yau metrics near a large complex structure limit
Motivated by the Strominger-Yau-Zaslow picture of mirror symmetry, in the late 90's Kontsevich-Soibelman, Gross-Wilson and Todorov formulated a conjecture describing the limit of unit-diameter Ricci-flat Kahler metrics on a polarized family of Calabi-Yau manifolds near a large complex structure limit. This conjecture was proved by Gross-Wilson for elliptically fibered K3 with I_1 singular fibers, but their approach relied on a delicate construction of explicit almost-Ricci-flat metrics which cannot be generalized to other cases. I will describe a more analytical approach, which allowed us to prove this conjecture for all elliptically fibered K3 and for some higher dimensional hyperkahler manifolds. (Joint work with M.Gross and Y.Zhang)Kazushi Ueda: Mirror symmetry and K3 surfaces
Mirror symmetry for K3 surfaces is not just a toy model for mirror symmetry for Calabi-Yau 3-folds, but an interesting subject in its own right. In the talk, we will discuss various aspects of mirror symmetry for K3 surfaces and their relation to other fields of mathematics.