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.
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.
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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). TBA Friday 28th Feb, 1:30-2:30pm. Huxley 140.
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Aleksander Doan (UCL). TBA Friday 7th Mar, 1:30-2:30pm. Huxley 140.
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Nattalie Tamam (Imperial). TBA Friday 14 Mar, 1:30-2:30pm. Huxley 140.
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Angelo Vistoli (Scuola Normale Superiore di Pisa). TBA Friday 21 Mar, 1:30-2:30pm. Huxley 140.
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