# Spring Term 2020

Sam Gunningham (King’s College). The finiteness conjecture for Skein modules. Friday 10th Jan, 1:30-2:30pm. Huxley 340.

Abstract: The Kauffman bracket skein module of an oriented 3-manifold M is a vector space (depending on a parameter q) which is generated by framed links in M modulo certain skein relations. The goal for the talk is the explain our recent proof (joint with David Jordan and Pavel Safronov) that the skein module of a closed 3 manifold is finite dimensional for generic q, confirming a conjecture of Witten.
Skein modules are known to give quantizations of (shifted) Poisson structures on character varieties. Our proof uses ideas from the theory of deformation quantization, in particular the notion of holonomic modules.

Sara Tukachinsky (Institute for Advanced Study, Princeton). Counts of pseudoholomorphic curves: Definition, calculations, and more. Friday 17th Jan, 1:30-2:30pm. Huxley 145.

Abstract: Genus zero open Gromov-Witten (OGW) invariants should count pseudoholomorphic disks in a symplectic manifold X with boundary conditions in a Lagrangian submanifold L, satisfying various constraints at boundary and interior marked points. In a joint work with Jake Solomon we developed an approach for defining OGW invariants using machinery from Fukaya A_\infty algebras. In a recent work, also joint with Solomon, we find that the generating function of OGW satisfies a system of PDE called open WDVV equations. For projective spaces, open WDVV give rise to recursions that, together with other properties, allow the computation of all OGW invariants.

Dustin Clausen (Copenhagen University and Max Planck Institute in Bonn). Liquid modules and complex geometry. Friday 24th Jan, 1:30-2:30pm. Huxley 145.

Abstract: I will talk about some joint work with Peter Scholze in which we provide new foundations for functional analysis. This leads to a natural notion of quasicoherent sheaf on complex manifolds, an infinite-dimensional generalization of the usual notion of coherent sheaf. The theory of quasicoherent sheaves has very nice formal properties which lead to simple proofs of foundational results such as Serre duality and the finiteness of coherent cohomology for proper maps.

Dan Waldram (Imperial College). G2 manifolds, moment maps and generalised geometry. Friday 31st Jan, 1:30-2:30pm. Huxley 145.

Abstract: String theory gives a natural extension of G2 holonomy manifolds to supersymmetric geometries that incorporate additional “flux” degrees of freedom. When the manifold is compact, this simply complexifies the three-form defining the G2 structure. Using an extension of Hitchin’s notion of generalised complex structure, we show how the conditions for supersymmetry can be recast into (1) the existence of an involutive sub-bundle of a generalised tangent space that combines vectors, two-forms and five-forms and (2) the vanishing of a moment map for the action of diffeomorphisms and gauge transformations of the flux. The space of extended G2 structures admits a natural Kahler metric implying that solving the moment map is equivalent to a GIT quotient. An extension of Hitchin’s G2 functional plays the role of of the norm functional and there is an analogue of the Futaki invariant.

Michael McQuillan (Università di Roma “Tor Vergata”) Very Functorial resolution of singularities. Friday 7th Feb, 1:30-2:30pm. Huxley 145.

Abstract: The basis of the talk is my new proof of the resolution of singularities in characteristic zero, which unlike Hironaka’s approach, and all derivatives thereof, constructs the centres in which the algorithm modifies the variety by building up, i.e. looking at increasingly fine infinitesimal data irrespective of the dimension, rather than building down, i.e. cutting with a maximal contact hyperplane and reducing dimension. The talk will aim to complement the preprint (GAFA to appear),
https://arxiv.org/pdf/1906.06745
by explaining several simplifications to the already very simple proof (most of the complication is to handle the non-geometric case of excellent rings) and how far these simplifications go to removing the hypothesis of characteristic 0.

Travis Schedler (Imperial College). Singularities of Moduli of 2-Calabi-Yau’s: Quiver and character varieties. Friday 14th Feb, 1:30-2:30pm. Huxley 145.

Abstract: Many moduli spaces appearing in algebraic geometry and topology admit a symplectic structure on the smooth locus, such as: character varieties and moduli of Higgs bundles on Riemann surfaces, moduli of sheaves on K3 surfaces, and Nakajima quiver varieties. At singularities they are often known to étale-locally have quiver models. This implies geometric properties such as normality and factoriality, or existence of symplectic resolutions. I will explain how this falls under a general framework of moduli of sheaves/modules over 2-Calabi-Yaus. A theorem of Bocklandt-Galluzzi-Vaccarino provides the local model, once one establishes the 2CY property. I will explain a theorem in joint work with Kaplan which carries this out for character varieties of open surfaces, or more generally multiplicative quiver varieties.

Simon Donaldson (Imperial College and Simons Center, Stony Brook). Associative submanifolds and gradient line graphs. Friday 21st Feb, 1:30-2:30pm. Huxley 145.

Abstract: We will discuss a construction, in part conjectural, of associative submanifolds in 7-manifolds with G_{2} structures having co-associative fibrations by K3 surfaces. The construction bears on the “adiabatic” regime, where the fibres are very small and is based on knowledge of complex curves in K3 surfaces. The input for the construction is a graph whose edges are integral curves of certain vector fields and there are some similarities with constructions in 3-dimensional Floer theories. We will explain how the fundamental transition phenomena, where co-asociative submanifolds develop singularities, are visible in the adiabatic picture. This is in part joint work with C. Scaduto.

Daniel Huybrechts (University of Bonn). Complex multiplication and twistor spaces. Friday 28th Feb, 1:30-2:30pm. Huxley 145.

Abstract: Despite the transcendental nature of the twistor construction, the algebraic fibres of the twistor space of a K3 surface share certain arithmetic properties. We prove that for a polarized K3 surface with complex multiplication, all algebraic fibres of its twistor space away from the equator have complex multiplication as well.

Gregory Sankaran (University of Bath). Blowups with log canonical singularities. Friday 6th Mar, 1:30-2:30pm. Huxley 145.

Abstract: We show that the minimum weight of a weighted blow-up of $\mathbf A^d$ with $\varepsilon$-log canonical singularities is bounded by a constant depending only on $\varepsilon$ and $d$. This was conjectured by Birkar. Using the recent classification of 4-dimensional empty simplices by Iglesias-Valino and Santos, we work out an explicit bound for blowups of $\mathbf A^4$ with terminal singularities: the smallest weight is always at most 32, and at most 6 in all but finitely many cases.