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).

*Egor Yasinsky (École Polytechnique de Paris). ***Birational involutions of the projective plane.** Friday 18th Feb, 1:30-2:30pm. Huxley 140.

**Abstract:** The study of birational involutions of the complex projective plane goes back to the works of E. Bertini. However, such involutions were completely classified only around 2000 by L. Bayle and A. Beauville (basically, using the methods of minimal model program in dimension 2). What if one replaces the complex numbers with some other field, for example the field of real numbers? It turns out that the classification becomes much more sophisticated. I will report on a joint work with I. Cheltsov, F. Mangolte and S. Zimmermann in which we discovered many interesting involutions of the real projective plane. Surprisingly, some of them are directly related to the classical works of Sophie Kowalevski.

*Claude Viterbo (Université de Paris-Saclay). ***Completions and coisotropy in symplectic topology.** Friday 25th Feb, 1:30-2:30pm. Huxley 140.

**Abstract:** We shall introduce the notion of γ-coisotropic set and show how this notion is relevant in various settings, for the study of Lagrangian submanifolds, singular Hamiltonian systems or singular support of sheaves.

*Adam Gal (University of Oxford). ***Hall categories and quantum group categorification.** Friday 4th Mar, 1:30-2:30pm. Huxley 140.

**Abstract:** I will describe an approach to geometric categorification of quantum groups using an underlying combinatorial structure controlling the higher algebra structure which I call a “Hall category”. I will then indicate how this could be used to categorify the bi-algebra structure on quantum groups which should lead to a construction of non-trivial 4D TQFT’s.

Michael McQuillan (Università di Roma “Tor Vergata”). **Thurston Vanishing.** Friday 11th Mar, 1:30-2:30pm. Huxley 140.

**Abstract:** A central result in the dynamics of a rational map f of $\mathbb{P}_\mathbb{C}$ of degree d is that the number of non-repelling cycles is bounded by 2d-2. This was conjectured by Fatou, and first proved by Mitsu Shishikura. Subsequently, however, a radically easier proof was given by Adam Epstein. Epstein’s proof cries out for a topus theoretic interpretation, and the object of the talk will be to present one. In particular Epstein’s appeal to what he refers to as Thurston rigidity, i.e. the key lemma in Thurston’s proof of the rigidity of post critically finite dynamical systems becomes the vanishing of a second Ext groups in the topus. The topus method also permits an optimal combination of the work of Epstein with that of Shishikura, albeit that everything should be considered a compliment to the former.

*Wicher Malten (University of Oxford). ***From braids to transverse slices in reductive groups.** Friday 18th Mar, 1:30-2:30pm. Huxley 140.

**Abstract:** We explain how group analogues of Slodowy slices arise by interpreting certain Weyl group elements as braids. Such slices originate from classical work by Steinberg on regular conjugacy classes, and different generalisations recently appeared in work by Sevostyanov on quantum group analogues of W-algebras and in work by He-Lusztig on Deligne-Lusztig varieties. Also building upon recent work of He-Nie, our perspective furnishes a common generalisation and a simple geometric criterion for Weyl group elements to yield strictly transverse slices. We finish by briefly discussing their Poisson structures and quantisations.

*Carolina Araujo (IMPA). ***Higher Fano manifolds.** Friday 25th Mar, 1:30-2:30pm. Huxley 140.

**Abstract:** Fano manifolds are complex projective manifolds having positive first Chern class. The positivity condition on the first Chern class has far reaching geometric and arithmetic implications. For instance, Fano manifolds are covered by rational curves, and families of Fano manifolds over one dimensional bases always admit holomorphic sections. In recent years, there has been great effort towards defining suitable higher analogues of the Fano condition. Higher Fano manifolds are expected to enjoy stronger versions of several of the nice properties of Fano manifolds. For instance, they should be covered by higher dimensional rational varieties, and families of higher Fano manifolds over higher dimensional bases should admit meromorphic sections (modulo Brauer obstruction). In this talk, I will discuss a possible notion of higher Fano manifolds in terms of positivity of higher Chern characters, and discuss special geometric features of these manifolds.