Spring Term 2022

There will be no seminar on 14th January because of this event.

John Nicholson (Imperial College). 4-manifolds and the homotopy type of a finite 2-complex. Friday 21st Jan, 1:30-2:30pm. Huxley 140.

Abstract: It has been known since the 1930s that two finite 2-dimensional CW-complexes with the same fundamental group and Euler characteristic become homotopy equivalent after wedging with a number of copies of S^2. However, it took until the 1970s to find examples which were not themselves homotopy equivalent. A long-standing problem has since been to determine for which k there exists examples with Euler characteristic k above the minimal value for a fixed fundamental group G. In the first part of this talk, I will discuss my complete resolution to this problem using techniques from integral representation theory.
A large part of the motivation for this work is a close analogy between the homotopy type of a finite 2-complex and the homeomorphism type of a smooth closed 4-manifold. I will explain this analogy and then discuss one aspect of the classification of 4-manifolds on which I hope my work on 2-complexes will eventually have an impact.

Please note the unusual time:
András Juhász (University of Oxford). Knot theory and machine learning. Friday 28th Jan, 3:15-4:15pm. Huxley 140.

Abstract: We introduce a new real-valued invariant called the natural slope of a hyperbolic knot in the 3-sphere, which is defined in terms of its cusp geometry. Our main result is that twice the knot signature and the natural slope differ by at most a constant times the hyperbolic volume divided by the cube of the injectivity radius. This inequality was discovered using machine learning to detect relationships between various knot invariants. It has applications to Dehn surgery and to 4-ball genus. We will also present a refined version of the inequality where the upper bound is a linear function of the volume, and the slope is corrected by terms corresponding to short geodesics that link the knot an odd number of times. This is joint work with Alex Davies, Marc Lackenby, and Nenad Tomasev.

Ailsa Keating (University of Cambridge). Symplectomorphisms of some Weinstein 4-manifolds. Friday 4th Feb, 1:30-2:30pm. Huxley 140.

Abstract: The goal is to introduce two new families of four-dimensional symplectomorphisms, inspired by mirror symmetry. The basic symplectic manifolds which support these are smoothings of cusp singularities. Their topology can be described explicitly, as can their mirrors, which are associated to the dual cusps. The new symplectomorphisms are a) `Lagrangian translations’, which we prove are mirror to tensors with line bundles; and b) `nodal slide recombinations’, which we prove are mirror to biholomorphisms. Together with spherical twists, these are expected to generate the symplectic mapping class groups of our manifolds. Joint work with Paul Hacking.

Alastair Craw (University of Bath). Partial crepant resolutions, punctual Hilbert schemes and ADE singularities. Friday 11th Feb, 1:30-2:30pm. Huxley 140.

Abstract: The Hilbert scheme of n points on the affine plane famously provides a projective symplectic resolution of the nth symmetric product of the affine plane. The aim of this talk is to introduce recent joint work with Gamelgaard, Gyenge and Szendr\H{o}i in which we study the Hilbert scheme of n points on any Kleinian singularity, i.e. on any surface of the form C^2/\Gamma, where \Gamma is a finite subgroup of SL(2,C). More generally, we introduce a class of Quot schemes associated to certain reflexive sheaves on the Kleinian singularity, and we show that these Quot schemes define all projective crepant (partial) resolutions of Hilb^[n](C^2/\Gamma).

Wicher Malten (University of Oxford). TBA Friday 18th Mar, 3:15-4:15pm. Huxley 140.