Autumn Term 2022

Jack Smith (University of Cambridge). Quantum, Hochschild, and Floer: a tale of three cohomologies. Friday 7th Oct, 1:30-2:30pm. Huxley 140.

Abstract: By counting holomorphic curves, symplectic topologists have amassed a dizzying array of invariants, including the quantum cohomology of a symplectic manifold, the Hochschild cohomology of its Fukaya category, and the Floer cohomologies of its Lagrangian submanifolds. Some of these objects are more geometric whilst others are more algebraic; some are easier to compute whilst others have more powerful properties. After introducing the basic constructions, I will explain a new source of connections between them, which enables one to translate easy geometric information into powerful algebraic information. I will then discuss some applications. A special role is played by Floer cohomology with local coefficients and, perhaps surprisingly, by matrix factorisations.

Alexander Petrov (Max Planck Institute). Hodge-to-de Rham (non)degeneration in positive characteristic and Steenrod operations. Friday 14th Oct, 1:30-2:30pm. Huxley 140.

Abstract: Hodge theory implies that the de Rham cohomology of a smooth projective complex algebraic variety is isomorphic to a direct sum of the cohomology of differential forms. Deligne and Illusie proved that the same holds for a smooth projective variety over a finite field F_p if it lifts to Z/p^2 and has dimension <=p. I will discuss how the situation changes if we consider varieties of arbitrary dimension. It turns out that the relevant obstruction to decomposing de Rham complex can be expressed in terms of other invariants of the variety such as the obstruction to lifting Frobenius and the Bockstein homomorphism. This allows us to give an example of a smooth projective variety over F_p that lifts to Z/p^2 but whose Hodge-to de Rham spectral sequence does not degenerate. The proof relies on the existence of prismatic cohomology but the key computation is a piece of homotopical algebra and is motivated by the properties of Steenrod operations from algebraic topology. Along the way, this allows to obtain a partial explicit formula for the Sen operator on de Rham cohomology, defined by Drinfeld and Bhatt-Lurie. On the positive side, it turns out that the de Rham complex of any smooth variety over F_p naturally decomposes as a direct sum of its cohomology after being pulled back along the Frobenius morphism. This implies that the Hodge-to-de Rham spectral sequence of a Frobenius-split smooth proper variety over F_p degenerates, regardless of its dimension.

Maxim Jeffs (Harvard University). Functorial mirror symmetry for very affine hypersurfaces. Friday 21st Oct, 1:30-2:30pm. Huxley 140.

Abstract: Mirror symmetry has allowed us to uncover many unexpected structures and results in algebraic geometry by translating them into ‘mirror’ statements about symplectic geometry. Even though the algebraic geometry of toric varieties is very well-understood, translating it under mirror symmetry into the symplectic geometry of very affine hypersurfaces (hypersurfaces in a complex torus) yields surprising new results in algebraic geometry. A very affine hypersurface and its complement (also a very affine hypersurface) have a number of natural symplectic operations relating them, as observed by Auroux. Even a seemingly trivial operation in symplectic geometry leads us from toric algebraic geometry into the world of derived schemes. I’ll explain how this appears in joint work with Benjamin Gammage where we prove Auroux’s conjectures about the toric analogs of these symplectic operations.

Baris Kartal (University of Edinburgh). Frobenius operators in symplectic topology. Friday 28th Oct, 1:30-2:30pm. Huxley 140.

Abstract: Given prime p, one can define Frobenius operators on the commutative rings of characteristic p. This notion generalizes to a larger class of rings and even to topological spaces and spectra. Spectra with circle actions and Frobenius operators are called cyclotomic spectra. Examples arise in algebraic and arithmetic geometry, as topological Hochschild homology of rings and categories, and many applications to these fields are found. By mirror symmetry, it is natural to expect the cyclotomic spectra to arise in symplectic topology. In this talk, we will explain how to obtain cyclotomic spectra using holomorphic cylinders in symplectic manifolds, i.e. by using Hamiltonian Floer theory. Joint work in progress with Laurent Cote.

Stephen Lynch (Imperial College). Ancient solutions of geometric flows. Friday 4th Nov, 1:30-2:30pm. Huxley 140.

Abstract: A one-parameter family of Riemannian submanifolds moves by mean curvature flow if its pointwise velocity equals its mean curvature vector. Mean curvature flow is a natural analogue of the heat equation for submanifolds, just as the Ricci flow is a heat equation for Riemannian metrics. This indicates that the flow can be used to tackle profound problems in geometry and topology, such as the Schoenflies conjecture. However, such applications can only follow from a detailed understanding of the singularities that form along the flow. Rescaling shows that solutions which extend arbitrarily far back in time (ancient solutions) model the behaviour of the flow close to a singularity. In recent years there has been enormous progress towards the classification of ancient solutions. I will survey some of these results, emphasising their importance for potential applications, and discuss a new characterisation of self-similarly shrinking convex solutions. This work should open the door to many further classification results in higher dimensions. I will make comparisons with the analogous story for Ricci flow, which has been unfolding simultaneously.

Bogdan Zavyalov (Institute for Advanced Study). Mod-p Poincare Duality in p-adic Analytic Geometry. Friday 11th Nov, 1:30-2:30pm. Huxley 140.

Abstract: Etale cohomology of F_p-local systems does not behave nicely on general smooth p-adic rigid-analytic spaces; e.g., the F_p-cohomology of the 1-dimensional closed unit ball is infinite. However, it turns out that the situation is much better if one considers proper rigid-analytic spaces. These spaces have finite F_p cohomology groups and these groups satisfy Poincare Duality if X is smooth and proper.
I will explain how one can prove such results using the concept of almost coherent sheaves that allows to “localize” such questions in an appropriate sense and actually reduce to some local computations.

Martin Taylor (Imperial College). The nonlinear stability of the Schwarzschild family of black holes. Friday 18th Nov, 1:30-2:30pm. Huxley 140.

Abstract: The Schwarzschild family of static black holes, discovered in 1915, constitutes the most famous family of solutions of the vacuum Einstein equations of general relativity. I will present a theorem on the full finite codimension nonlinear asymptotic stability of the Schwarzschild family. This is joint work with M. Dafermos, G. Holzegel and I. Rodnianski.

Denis Auroux (Harvard University). Fukaya categories of Landau-Ginzburg models and homological mirror symmetry. Friday 25th Nov, 1:30-2:30pm. Huxley 140.

Abstract: The main protagonist of this talk is the Fukaya category of a “Landau-Ginzburg model”, i.e., a symplectic fibration over the complex plane. (The unusual feature is that we allow fibrations “in stages”, i.e. the fibers may themselves be Landau-Ginzburg models). We will outline one possible definition of this category, and describe a pair of natural functors relating it to the Fukaya category of the fiber. These functors are of particular interest in homological mirror symmetry, where they correspond to inclusion and restriction functors between derived categories of coherent sheaves on a variety and a hypersurface inside it. This leads in particular to a proof of homological mirror symmetry for general hypersurfaces in toric varieties. The talk is partly based on joint work with Mohammed Abouzaid, and also connects with Maxim Jeffs’thesis.

Ben Davison (University of Edinburgh). Nonabelian Hodge theory for stacks. Friday 2nd Dec, 1:30-2:30pm. Huxley 140.

Abstract: The classical nonabelian Hodge correspondence provides a homeomorphism between the coarse moduli space of semistable Higgs bundles of fixed rank r and degree on a smooth complex projective curve C, and the coarse moduli space of (twisted) r-dimensional representations of the fundamental group of the same curve. Upgrading this statement to a homeomorphism of stacks, in the presence of strictly semistable objects, appears to be a hopeless task.
However, I will present a recent result, which states that the corresponding stacks have isomorphic Borel-Moore homology. This goes via a general result regarding 2-Calabi-Yau categories with strictly negative Euler form: their BPS Lie algebras (to be defined in this talk) are freely generated by the intersection cohomology of coarse moduli spaces of objects in the category. The result then follows from a Yangian-type PBW theorem relating BPS cohomology to BM homology of the stack.
This is joint work with Lucien Hennecart and Sebastian Schlegel Mejia.

Please note the unusual time and room:
Markus Upmeier (University of Aberdeen). Categorifications of Atiyah-Singer index theory with applications to calibrated geometry and gauge theory. Friday 9th Dec, 12:00-1:00pm. Huxley 139.

Abstract: The construction of enumerative invariants in algebra and geometry requires an understanding of the differential-topological properties of moduli spaces. In my talk, I will discuss a new approach to these questions in terms of categorifications of Atiyah-Singer index theory. The new applications include the construction of orientations for DT4-invariants and Floer-style gradings for G2-instantons. If time permits, I will explain how to compute the categorical index topologically.

Dmitry Kaledin (Independent University of Moscow). How to enhance categories, and why. Friday 16th Dec, 1:30-2:30pm. Huxley 140.

Abstract: It has become accepted wisdom by now that when you localize a category with respect to a class of morphisms, what you get is not just a category but a category “with a homotopical enhancement”. Typically, the latter is made precise through the machinery of “infinity-categories”, or “quasicategories”, but this is quite heavy technically and not really optimal from the conceptual point of view. I am going to sketch an alternative technique based on Grothendieck’s idea of a “derivator”.