Matt Kerr (Washington University in St. Louis). K2 of curves and mirror symmetry Friday 10th Jan, 1:30-2:30pm. Huxley 140.
Abstract: In their simplest form, K2 classes on curves are just pairs of rational functions; what makes them interesting is the regulator map and its interaction with arithmetic of curves and differential equations. I’ll discuss a couple of applications to mirror symmetry: (i) via their close relationship to Hori-Vafa models, K2 cycles explain the asymptotic growth rate of local Gromov-Witten invariants of toric Fano surfaces; and (ii) they help to settle a conjecture on eigenvalues of quantum curves arising in the topological string/spectral theory correspondence.
Wendelin Lutz (University of Massachusetts). The Morrison Cone Conjecture under deformation Friday 17th Jan, 1:30-2:30pm. Huxley 140.
Abstract: Let Y be a Calabi—Yau variety. The Morrison Cone Conjecture is a fundamental conjecture in Algebraic Geometry on the geometry of the nef cone and the movable cone of Y: while these cones are usually not rational polyhedral, the cone conjecture predicts that the action of Aut(Y) on Nef(Y) admits a rational polyhedral fundamental domain, and that the action of Bir(Y) on Mov(Y) admits a rational polyhedral fundamental domain.
Even though the conjecture has been settled in special cases, it is still wide open in dimension at least 3.
We prove that if the cone conjecture holds for a smooth Calabi-Yau threefold Y, then it also holds for any smooth deformation of Y.
Samuel Muñoz Echániz (Cambridge). Mapping class groups of h-cobordant manifolds Friday 24th Jan, 1:30-2:30pm. Huxley 140.
Abstract: A cobordism W between compact manifolds M and M’ is an h-cobordism if the inclusions of M and M’ into W are both homotopy equivalences. This kind of cobordisms plays an important role in the classification of high-dimensional manifolds, as h-cobordant manifolds are often diffeomorphic.
With this in mind, given two h-cobordant manifolds M and M’, how different can their diffeomorphism groups Diff(M) and Diff(M’) be? The homotopy groups of these two spaces are the same “up to extensions” in a range of strictly positive degrees. Contrasting this fact, I will present examples of h-cobordant manifolds with different mapping class groups. In doing so, I will review the classical theory of h-cobordisms and introduce several moduli spaces of manifolds that help in answering this question.
Stephanie Baines (Imperial). Gradient Flows in Generalised Geometry Friday 31th Jan, 1:30-2:30pm. Huxley 140.
Abstract: The AdS/CFT correspondence in string theory predicts an equivalence between certain algebraic structures (“CFT”) and particular classes of solutions of Einstein’s equations (“AdS”). For generic supersymmetric theories, this implies a relation between the (non-)commutative Calabi-Yau algebras “with potentials” of Ginzburg and new extensions of conventional geometries, analogous to the “generalised complex geometry” of Hitchin and Gualtieri. In this talk we will introduce some aspects of these new geometries, and how supersymmetry implies that they are naturally encoded by a moment map for an extension of the diffeomorphism group, suggesting a new example of a Kobayashi-Hitchin-type correspondence. For simplicity we will primarily focus on a subclass of generalised structures that give a new description of Sasaki-Einstein spaces in terms of unstable objects. This picture provides a mathematical understanding of the dual of “a-maximisation” in the CFT in terms of maximising a Hilbert-Mumford slope, generalising a construction of Martelli-Sparks-Yau.
Heather Macbeth (Imperial). The state of the art in the formalisation of geometry Friday 7th Feb, 1:30-2:30pm. Huxley 140.
Abstract: The last ten years have seen extensive experimentation with computer formalisation systems such as Lean. It is now clear that these systems can express arbitrarily abstract mathematical definitions, and arbitrarily complicated mathematical proofs.
The current situation, then, is that everything is possible in principle — and comparatively little is possible yet in practice! In this talk I will survey the state of the art in geometry (differential and algebraic). I will outline the current frontier of what has been formalised, and I will try to explain the main obstacles to progress.
Benjamin Briggs (Imperial). Cohomological support varieties in commutative algebra Friday 14th Feb, 1:30-2:30pm. Huxley 140.
Abstract: Geometric methods have proven powerful in understanding representations of finite groups, going back to Quillen’s stratification of the spectrum of group cohomology, and leading to Benson, Iyengar, and Krause’s classification of representations “up to building” in terms of support varieties. Avramov borrowed these ideas to define support varieties for modules over a complete intersection ring, and, with Buchweitz, used them to establish remarkable properties of the homological algebra over a complete intersection singularity. This theory has since expanded in scope in several directions, and in particular, after work of Jorgensen and Pollitz, has also been applied profitably to arbitrary commutative local rings. I’ll talk about work with Grifo and Pollitz on what can be seen from these cohomological support varieties, and in particular what they tell you about the deformation theory of your ring.
Michael Schmalian (Oxford). Tight Contact Structures and Twisted Geodesics Friday 21st Feb, 1:30-2:30pm. Huxley 140.
Abstract: We will discuss a recent result relating phenomena from two well-established, yet largely unrelated, subfields of 3-manifold topology. Specifically, we will demonstrate that the tightness of certain contact structures on a hyperbolic 3-manifold can be detected by the length and torsion of associated geodesics. No prior knowledge of contact topology or hyperbolic geometry is expected and we will give a brief introduction to both these fields.
Ezra Getzler (Northwestern University). The Gauss-Manin connection in noncommutative geometry Friday 28th Feb, 1:30-2:30pm. Huxley 140.
Abstract: The noncommutative Gauss-Manin connection is a flat connection on the periodic cyclic homology of a family of dg algebras (or more generally, A-infinity categories), introduced by the speaker in 1991.
The problem now arises of lifting this connection to the complex of periodic cyclic chains. Such a lift was provided in 2007 by Tsygan, though without an explicit formula. In this talk, I will explain how this problem is simplified by considering a new A-infinity structure on the de Rham complex of a derived scheme, which we call the Fedosov product; in joint work with Jones in 1990, the speaker showed that this product plays a role in a multiplicative version of the Hochschild-Kostant-Rosenberg theorem, and the point of the present talk is that it seems to be the correct product on the de Rham complex for derived geometry.
Aleksander Doan (UCL). Curve counting and real Cauchy-Riemann operators Friday 7th Mar, 1:30-2:30pm. Huxley 140.
Abstract: It is a long-standing open problem to generalize sheaf-counting invariants of complex projective three-folds to symplectic manifolds of real dimension six. One approach to this problem involves counting J-holomorphic curves C, for a generic almost complex structure J, with weights depending on J. Various existing symplectic invariants can be expressed as such weighted counts. In this talk, based on joint work with Thomas Walpuski, I will discuss a new construction of weights associated with curves and a closely related problem about the structure of the space of Cauchy-Riemann operators on C.
Nattalie Tamam (Imperial). Closure of orbits of the pure mapping class group in the character variety Friday 14 Mar, 1:30-2:30pm. Huxley 140.
Abstract: The pure mapping class group of an oriented surface acts on the character variety of the surface. For certain surfaces, such orbits are connected to solutions of known problems, such as the Markoff triples and the sixth Painlevé equation. We will discuss these examples, as well as the possible orbits closure in the general case; the algebraic relations preserved under the action, the obstructions to ‘big’ orbits, and the exceptional cases. This is a joint work with Alireza Salehi-Golsefidy.
Song Sun (Hangzhou). Kahler-Ricci shrinkers and Fano fibrations Thursday 20 Mar, 1-2pm. Huxley 140. Special time
Abstract: In this talk I will discuss complete (possibly non-compact) gradient shrinking Kahler-Ricci solitons, also known as Kahler-Ricci shrinkers, which are differential geometric objects arising from the study of singularities of Kahler-Ricci flows. We will first connect Kahler-Ricci shrinkers to algebraic geometry by showing that they are naturally quasi-projective varieties and admit the structure of a polarized Fano fibration (in the sense of minimal model program). The proof uses the boundedness result of Birkar for Fano type varieties. Then we will explain a Yau-Tian-Donaldson type conjecture for the existence of Kahler-Ricci shrinkers and a 2-step degeneration picture for determining a Kahler-Ricci shrinker at a finite time singularity of Kahler-Ricci flow. The latter is similar to the setting of metric tangent cones for singular Kahler-Einstein metrics.
Angelo Vistoli (Scuola Normale Superiore di Pisa). On the Cremona dimension of a p-group Friday 21 Mar, 1:30-2:30pm. Huxley 140.
Abstract: If G is a finite group, its Cremona dimension is the least n such that G is a subgroup of the group of birational automorphisms Bir(X) of a rationally connected projective variety X. Until 2009 we had a good understanding of the groups of Cremona dimension at most 2, that is, the finite subgroups of the classical Cremona group Bir(P^2), but no examples where known of a finite group of Cremona dimension larger than 3. Since then the progress has been rapid.
I will review the work that has been done on this, particularly by Prokhorov and Shramov, and state a remarkable fixed point theorem due to Haution, which in particular implies that if p is a prime, a non-abelian p-group has Cremona dimension at least p-1.
Then I will explain an improvement of this last result, due to G. Bresciani, Z. Reichstein and myself, which gives more refined lower bounds for the Cremona dimension of a p-group. This relies on a new technique, connecting Cremona dimension with relative Brauer groups for projective varieties over non-algebraically closed fields.