# Seminars this semester

Series:

Oct 1 Tue Paul Johnson (Sheffield)
12:00 J11 Partitions and Hilbert Schemes of Points

Abstract:
This will be a gentle, expository talk explaining some connections between the two objects in the title. I will begin with partitions: using the cores-and-quotients formula to motivate the statement of an enriched version of Euler's product formula for partitions, that was conjectured by Gusein-Zade, Luengo, and Melle-Hernández in 2009, and that I proved this summer with Jørgen Rennemo. Most of the talk will be giving the geometric context for this combinatorial formula, namely how Gusein-Zade, Luengo and Melle-Hernández came to discover it by studying Hilbert schemes of points on orbifolds, and how to use Chen-Ruan cohomology to generalise it and connect it to existing results on Hilbert schemes. I will vaguely gesture toward the proof in the last five minutes for the experts, but most of the talk should be accessible to the whole audience.

Oct 8 Tue Dhruv Ranganathan (Cambridge)
12:00 J11 A Mayer-Vietoris theorem for Gromov-Witten theory

Abstract:
The Gromov-Witten theory of a smooth variety X is a collection of invariants, extracted from the topology of the space of curves in X. I will explain how the Gromov-Witten theory of X can be computed algorithmically from the components of a simple normal crossings degeneration of X. The combinatorics of the geometry and complexity of the algorithm are both controlled by tropical geometry. The formula bears a strong resemblance to the Mayer-Vietoris sequence in elementary topology, and I will try to give some indication of how deep this analogy runs. Part of this story is still work in progress, joint with Davesh Maulik.

Oct 15 Tue Jenny August (Max Planck Institute for Mathematics in Bonn)
12:00 J11 The Stability Manifold of a Contraction Algebra

Abstract:
For a finite dimensional algebra, Bridgeland stability conditions can be viewed as a continuous generalisation of tilting theory, providing a geometric way to study the derived category. Describing this stability manifold is often very challenging but in this talk, I’ll look at a special class of symmetric algebras whose tilting theory is determined by a related hyperplane arrangement. This simple picture will then allow us to describe the stability manifold of such an algebra.

Oct 17 Thu Pierrick Bousseau (ETH Zurich)
15:00 J11 Quasimodular forms from Betti numbers

Abstract:
I will explain how to construct quasimodular forms starting from Betti numbers of moduli spaces of dimension 1 coherent sheaves on P2. This gives a proof of some stringy predictions about the refined topological string theory of local P2 in the Nekrasov-Shatashvili limit. Partly based on work in progress with Honglu Fan, Shuai Guo, and Longting Wu.

Oct 22 Tue Nick Sheridan (Edinburgh)
12:00 J11 The Gamma and SYZ conjectures: a tropical approach to periods

Abstract:
I'll start by explaining a new method of computing asymptotics of period integrals using tropical geometry, via some concrete examples. Then I'll use this method to give a geometric explanation for a strange phenomenon in mirror symmetry, called the Gamma Conjecture, which says that mirror symmetry does not respect integral cycles: rather, the integral cycles on a complex manifold correspond to integral cycles on the mirror multiplied by a certain transcendental characteristic class called the Gamma class. We find that the appearance of zeta(k) in the asymptotics of period integrals arises from the codimension-k singular locus of the SYZ fibration.

Oct 29 Tue Noah Arbesfeld (Imperial College London)
12:00 J11 K-theoretic Donaldson-Thomas theory and the Hilbert scheme of points on a surface

Abstract:
Tautological bundles on Hilbert schemes of points often enter into enumerative and physical computations. I'll explain how to use the Donaldson-Thomas theory of threefolds to produce certain combinatorial identities involving Young diagrams. The resulting identities can be expressed geometrically in terms of tautological bundles over the Hilbert scheme of points on the plane. I'll also explain how these identities can be used to study Euler characteristics of tautological bundles over Hilbert schemes of points on general surfaces.

Nov 5 Tue Ben Davison (Edinburgh)
12:00 J11 Strong positivity for quantum cluster algebras

Abstract:
I will discuss the positivity for quantum theta functions, a result of joint work with Travis Mandel: For a given skew-symmetric quantum cluster algebra, these functions provide a basis of a larger algebra, for which the structure constants are Laurent polynomials with positive coefficients. I will explain how the proof of this result follows from scattering diagram techniques and a very special case of the cohomological integrality theorem, joint work with Sven Meinhardt.

Nov 21 Thu Soheyla Feyzbakhsh (Imperial College London)
12:00 J11 Stability conditions on the derived category of coherent systems and Brill-Noether theory

Abstract:
A classical method to study Brill-Noether locus of higher rank semistable vector bundles on curves is to examine the stability of coherent systems. To have an abelian category we enlarge the category of coherent systems by the category $A(C)$ which consists of triples $(E_1, E_2, f)$ where $E_1$ is a direct sum of the structure sheaf of $C, E_2$ is a coherent sheaf on $C$, and $f$ is a sheaf morphism from $E_1$ to $E_2$. In this talk after a short description of the derived category of $A(C)$, I will describe a 2-dimensional slice of the space of Bridgeland stability conditions on this category and sketch some of the possible applications of wall-crossing in Brill-Noether theory.

Dec 3 Tue Dimitri Wyss (École Polytechnique Fédérale de Lausanne)
12:00 J11 Non-archimedean and motivic integrals on the Hitchin fibration

Abstract:
Based on mirror symmetry considerations, Hausel and Thaddeus conjectured an equality between 'stringy' Hodge numbers for moduli spaces of $SL_n$/$PGL_n$ Higgs bundles. With Michael Groechenig and Paul Ziegler we prove this conjecture using non-archimedean integrals on these moduli spaces, building on work of Denef-Loeser and Batyrev. This approach reduces their conjecture essentially to the duality between generic Hitchin fibers. Similar ideas also lead to a new proof of the geometric stabilization theorem for anisotropic Hitchin fibers, a key ingredient in the proof of the fundamental lemma by Ngô. In my talk I will outline the main arguments of the proofs and discuss the adjustments needed, in order to replace non-archimedean integrals by motivic ones. The latter is joint work with François Loeser.

Dec 10 Tue Sira Gratz (Glasgow)
12:00 J11 Higher SL(k)-friezes

Abstract:
Classical frieze patterns are combinatorial structures which relate back to Gauss' Pentagramma Mirificum, and have been extensively studied by Conway and Coxeter in the 1970's.

A classical frieze pattern is an array of numbers satisfying a local (2 x 2)-determinant rule. Conway and Coxeter gave a beautiful and natural classification of SL(2)-friezes via triangulations of polygons. This same combinatorics occurs in the study of cluster algebras, and has revived interest in the subject. From this point of view, a natural way to generalise the notion of a classical frieze pattern is to ask of such an array to satisfy a (k x k)-determinant rule instead, for k bigger than 2, leading to the notion of higher SL(k)-friezes. While the task of classifying classical friezes yields a very satisfying answer, higher SL(k)-friezes are not that well understood to date.

In this talk, we'll discuss how one might start to fathom higher SL(k)-frieze patterns. The links between SL(2)-friezes and triangulations of polygons suggests a link to Grassmannian varieties under the Plücker embedding. We find a way to exploit this relation for higher SL(k)-friezes, and provide an easy way to generate a number of SL(k)-friezes via Grassmannian combinatorics, and suggest some ideas towards a complete classification using the theory of cluster algebras. This talk is based on joint work with Baur, Faber, Serhiyenko and Todorov.

Dec 11 Wed Adeel Khan (University of Regensburg)
15:00 LT-B Intersection theory of derived stacks

Abstract:
I will discuss how various formalisms of intersection theory (Chow groups, K-theory, cobordism) can be extended to the setting of derived schemes and stacks. This gives a new approach to virtual phenomena such as the virtual fundamental class and virtual Riemann-Roch formulas.

Dec 18 Wed Alexandr Buryak (Leeds)
14:00 J11 Generalization of the Givental theory for the open WDVV equations

Abstract:
The WDVV equations, also called the associativity equations, is a system of non-linear partial differential equations for one function that describes the local structure of a Frobenius manifold. In enumerative geometry the WDVV equations control the Gromov-Witten invariants in genus zero. In his fundamental works, A. Givental interpreted solutions of the WDVV equations as cones in a certain infinite-dimensional vector space. This allowed him to introduce a group action on solutions of the WDVV equations which proved to be a powerful tool in Gromov-Witten theory. I will talk about a generalization of the Givental theory for the open WDVV equations that appeared in a work of A. Horev and J. Solomon in the context of open Gromov-Witten theory.

Jan 28 Tue Helge Ruddat (Johannes Gutenberg University Mainz)
12:00 J11 Smoothing toroidal crossing varieties

Abstract:
Friedman and Kawamata-Namikawa studied smoothability of normal crossing varieties. I present the proof of a significantly more general smoothing result that also works for toroidal crossing spaces and relates to work on mirror symmetry by Gross-Siebert and Chan-Leung-Ma. The key technologies are the construction of log structures, a proof of a degeneration of the log Hodge to de Rham spectral sequence as well as deformation theory governed by Gerstenhaber algebras. This project is joint with Simon Felten and Matej Filip.