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.
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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.
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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.
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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.
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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.
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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.
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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.
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Nov 19 Tue Gwyn Bellamy (Glasgow)
12:00 J11 Resolutions of symplectic quotient singularities
 
  Abstract:
In this talk I will explain how one can explicitly construct all crepant resolutions of the symplectic quotient singularities associated to wreath product groups. The resolutions are all given by Nakajima quiver varieties. In order to prove that all resolutions are obtained this way, one needs to describe what happens to the geometry as one crosses the walls inside the GIT parameter space for these quiver varieties. This is based on joint work with Alistair Craw.
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Nov 21 Thu Soheyla Feyzbakhsh (Imperial College London)
12:00 J11
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Dec 3 Tue Dimitri Wyss (École Polytechnique Fédérale de Lausanne)
12:00 J11
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Dec 10 Tue Sira Gratz (Glasgow)
12:00 J11
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Dec 17 Tue Alexandr Buryak (Leeds)
12:00 J11
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