Seminars this semester
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 coresandquotients formula to motivate the statement of an enriched version of Euler's product formula for partitions, that was conjectured by GuseinZade, Luengo, and MelleHernández 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 GuseinZade, Luengo and MelleHernández came to discover it by studying Hilbert schemes of points on orbifolds, and how to use ChenRuan 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 MayerVietoris theorem for GromovWitten theory  
Abstract: The GromovWitten 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 GromovWitten 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 MayerVietoris 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 NekrasovShatashvili 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 codimensionk singular locus of the SYZ fibration. 



Oct 29  Tue  Noah Arbesfeld (Imperial College London)  
12:00  J11  Ktheoretic DonaldsonThomas 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 DonaldsonThomas 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 skewsymmetric 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 BrillNoether theory  
Abstract: A classical method to study BrillNoether 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 2dimensional slice of the space of Bridgeland stability conditions on this category and sketch some of the possible applications of wallcrossing in BrillNoether theory. 



Dec 3  Tue  Dimitri Wyss (École Polytechnique Fédérale de Lausanne)  
12:00  J11  Nonarchimedean 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 nonarchimedean integrals on these moduli spaces, building on work of DenefLoeser 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 nonarchimedean 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  LTB  Intersection theory of derived stacks  
Abstract: I will discuss how various formalisms of intersection theory (Chow groups, Ktheory, 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 RiemannRoch 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 nonlinear partial differential equations for one function that describes the local structure of a Frobenius manifold. In enumerative geometry the WDVV equations control the GromovWitten invariants in genus zero. In his fundamental works, A. Givental interpreted solutions of the WDVV equations as cones in a certain infinitedimensional vector space. This allowed him to introduce a group action on solutions of the WDVV equations which proved to be a powerful tool in GromovWitten 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 GromovWitten theory. 



Jan 28  Tue  Helge Ruddat (Johannes Gutenberg University Mainz)  
12:00  J11  Smoothing toroidal crossing varieties  
Abstract: Friedman and KawamataNamikawa 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 GrossSiebert and ChanLeungMa. 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. 


