Spring Term 2019

Cheuk Yu Mak (University of Cambridge). Tropically constructed Lagrangians in mirror quintic threefolds. Friday 18th Jan, 1:30-2:30pm. Huxley 139.

Abstract: In this talk, we will explain how to construct embedded closed Lagrangian submanifolds in mirror quintic threefolds using tropical curves and the toric degeneration technique. As an example, we will illustrate the construction for tropical curves that contribute to the Gromov–Witten invariant of the line class of the quintic threefold. The construction will in turn provide many homologous and non-Hamiltonian isotopic Lagrangian rational homology spheres, and a geometric interpretation of the multiplicity of a tropical curve as the weight of a Lagrangian. This is a joint work with Helge Ruddat.

Christian Urech (Imperial College). Simple subgroups of the plane Cremona group. Friday 25th Jan, 1:30-2:30pm. Huxley 139.

Abstract: The plane Cremona group is the group of birational transformations of the projective plane. It was already studied intensely in the 19th century, but many classical problems have only been solved in the last decades. In this talk, I will give a description of all simple subgroups of this group using techniques from birational geometry, geometric group theory and dynamics.

Alan Thompson (Loughborough University). ADE surfaces. Friday 1st Feb, 1:30-2:30pm. Huxley 139.

Abstract: I will present joint work with V. Alexeev on the classification of rational surfaces equipped with an involution. Along the way, we will find that the classification of such surfaces seems to be governed by the representation theory of simple Lie algebras, in a way that is still not fully understood. I will conclude by arguing that a complete understanding of this link could have significant ramifications for a number of open problems in algebraic geometry.

Dhruv Ranganathan (University of Cambridge). Semistable reduction for Gromov-Witten theory and tropical curves. Friday 8th Feb, 1:30-2:30pm. Huxley 139.

Abstract: The Gromov-Witten invariants of an algebraic variety X form a system of subtle numerical invariants that are typically very difficult to compute. The degeneration formula is a kind of Mayer-Vietoris sequence for this theory, and promises to calculate the Gromov-Witten invariants of a variety by using “open sets’’ coming from components of a nice degeneration. The formula is expected to combine the invariants of the components combinatorially, to recover the invariants of X. I will give an introduction to these ideas, discuss such a formula, and explain where tropical geometry provides the key new insights.

Raphael Zentner (Universität Regensburg). SU(2)-representations of 3-manifolds, the pillowcase, and Floer homology. Friday 15th Feb, 1:30-2:30pm. Huxley 139.

Abstract: The pillowcase is topologically a 2-sphere with a flat orbifold metric with four singular points. It arises, for instance, as the SU(2)-character variety of the 2-torus. There is a natural class of area-preserving maps depending on a periodic function of one real variable, called shearing maps. Such maps appear naturally when one perturbs the Chern-Simons function whose Morse homology is instanton Floer homology. We prove an approximation result for such maps in the space of all area-preserving maps of the pillowcase in the C^0-topology, and in the C^k-topology for k>=1 together with Steven Sivek. Applications include:
1. All integral homology spheres have irreducible representations of their fundamental group in SL(2,C)
2. On any non-trivial knot in S^3 all Dehn-surgeries with sufficiently large slope admit irreducible SU(2)-representations (jt with Steven Sivek).

Mattia Talpo (Imperial College). Towards logarithmic degeneration for DT invariants. Friday 22th Feb, 1:30-2:30pm. Huxley 139.

Abstract: Donaldson-Thomas invariants count algebraic curves on Calabi-Yau 3-folds via counting their sheaves of ideals. As on the Gromov-Witten side, there are techniques to reduce computations to simpler geometries, by degenerating the manifold to a certain kind of singular varieties. Logarithmic geometry is a modern language that helps in handling these kinds of degenerations, and is being fruitfully applied to degeneration of GW invariants. After some review of these topics, I will speculate about possible applications of these new techniques to degeneration of DT invariants.

Michael Wong (UCL). Dimer models and Hochschild cohomology. Friday 1th Mar, 1:30-2:30pm. Huxley 139.

Abstract: A dimer model is a type of quiver embedded in a Riemann surface. It gives rise to an associative, generally noncommutative algebra called the Jacobi algebra. In the version of mirror symmetry proved by R. Bocklandt, the wrapped Fukaya category of a punctured surface is equivalent to the category of matrix factorizations of the Jacobi algebra of a dimer, equipped with its canonical potential. For the purposes of deformation theory, we explicitly describe the Hochschild cohomologies of the Jacobi algebra and the associated matrix factorization category in terms of dimer combinatorics.

Soheyla Feyzbakhsh (Imperial College). Wall-crossing and Brill-Noether theory. Friday 8th Mar, 1:30-2:30pm. Huxley 139.

Abstract: I will explain some of the recent applications of wall-crossing with respect to Bridgeland stability conditions in Brill-Noether theory. As an example, I will explain in details how one can apply the wall-crossing technique to reprove Jensen-Ranganathan’s result on the Brill-Noether theory of line bundles for curves of fixed gonality.

Susanna Zimmermann (Université d’Angers). Higher dimensional Cremona groups. Friday 15th Mar, 1:30-2:30pm. Huxley 139.

Abstract: Over an algebraically closed field, the Noether-Castelnuovo theorem implies that plane Cremona group does not have any finite (nontrivial) quotient. This is quite different for higher dimensional Cremona groups and I would like to present how to construct some (nontrivial) quotients. This is an application of the boundedness of Fano varieties and the decomposition theorem of birational maps between Mori fibre spaces into Sarkisov links.

Hannah Markwig (Universität Tübingen). Tropical mirror symmetry for elliptic curves and beyond. Friday 22th Mar, 1:30-2:30pm. Huxley 139.

Abstract: Mirror symmetry for elliptic curves relates the generating series of Hurwitz numbers of the elliptic curve (i.e. counts of covers with fixed genus and simple branch points) to Feynman integrals. This is
interesting, because quasimodularity properties of the generating series can be deduced, which is desirable to make statements about the asymptotics of the series. When passing to the tropical world, the relation works on a “fine level”, i.e. summand by summand. We review the known results and then talk about generalizations: we can count covers in a broader context, and we can also count tropical curves in a product of an elliptic curve with the projective line. In both cases, the generating series in question can be related to Feynman integrals summand by summand.
Joint work with Boehm, Bringmann, Buchholz, resp. with Boehm, Goldner.