Autumn Term 2017

Elisa Postinghel (Loughborough University). Newton-Okounkov bodies and toric degenerations of Mori dream spaces. Friday 13th Oct., 1:30-2:30pm. Huxley 139.

Abstract: Building on work of Okounkov from the 1990s, in 2008 Kaveh and Khovanskii, Lazarsfeld and Mustata showed how to associate to an n-dimensional algebraic variety X and a line bundle a convex body in n-dimensional Euclidean space, the Newton-Okounkov body. In the first part of this talk we will revise construction and main properties of these bodies. In the second part of the talk we will see that for Mori dream spaces, Newton-Okounkov bodies are particularly nice and give rise to toric degenerations (using Anderson’s construction). This is joint work with Stefano Urbinati.

Nick Sheridan (University of Cambridge). Symplectic mapping class groups and homological mirror symmetry. Friday 20 Oct., 1:30-2:30pm. Huxley 139.

Abstract: A ​central object ​of ​study ​in symplectic topology ​is the topological group of symplectic automorphisms. I will give a brief overview of what is known about the topology of these groups in some specific cases, then explain how one can ​get new information about them​ by combining two recent results: a proof of homological mirror symmetry for a new collection of K3 surfaces (joint work with Ivan Smith), ​together with the computation of the derived autoequivalence group of a K3 surface of Picard​ rank one (Bayer–Bridgeland). For example, it is possible to give an example of a ​symplectic K3 whose smoothly trivial symplectic mapping class group (the group of isotopy classes of symplectic automorphisms which are smoothly isotopic to the identity) is infinitely-generated. This is joint work with Ivan Smith.

Paul Hacking (University of Massachusetts at Amherst). Mirror symmetry for open Calabi Yau manifolds. Friday 27 Oct., 1:30-2:30pm. Huxley 139.

Abstract: A log Calabi Yau manifold U is a complex manifold admitting a normal crossing compactification such that there is a holomorphic volume form on U with simple poles at the boundary. The Strominger–Yau–Zaslow formulation of mirror symmetry as duality of Lagrangian torus fibrations applies in this context. An algebraic version of the SYZ construction due to Gross and Siebert leads to an explicit description of the mirror as the spectrum of an algebra defined using tropical geometry, which we conjecture may be defined using Gromov–Witten invariants and coincides with the degree 0 symplectic cohomology of U. This algebra comes with a canonical vector space basis which in the special case that U is holomorphic symplectic solves conjectures of Fomin–Zelevinsky and Fock–Goncharov on cluster algebras. Joint work with Gross, Keel, Kontsevich, and Siebert.

Jakub Witaszek (Imperial College). Liftability of the Frobenius morphism and images of toric varieties. Friday 3 Nov., 1:30-2:30pm. Huxley 139.

Abstract: The celebrated proof of the Hartshorne conjecture by Shigefumi Mori allowed for the study of the geometry of higher dimensional varieties through the analysis of deformations of rational curves. One of the many applications of Mori’s results was Lazarsfeld’s positive answer to the conjecture of Remmert and Van de Ven which states that the only smooth variety that the projective space can map surjectively onto, is the projective space itself. Motivated by this result, a similar problem has been considered for other kinds of manifolds such as abelian varieties (Demailly-Hwang-Mok-Peternell) or toric varieties (Occhetta-Wiśniewski). In my talk, I would like to present a completely new perspective on the problem coming from the study of Frobenius lifts in positive characteristic. This is based on a joint project with Piotr Achinger and Maciej Zdanowicz.

Nicola Pagani (University of Liverpool). The indeterminacy of universal Abel-Jacobi sections. Friday 10 Nov., 1:30-2:30pm. Huxley 139.

Abstract: The (universal) Abel-Jacobi maps are the sections of the forgetful morphism from the universal Jacobian to the corresponding moduli space of smooth pointed curves. When the source and target moduli spaces are compactified, these sections are only rational maps, and it is natural to ask for the largest locus where each of them is a well-defined morphism. We explicitly characterize this locus, which depends on the chosen compactification of the universal Jacobian (for the source we fix the Deligne-Mumford compactification \bar{M}_{g,n} by means of stable curves). In particular, we deduce that for evey Abel-Jacobi map there exists a compactification of the universal Jacobian such that the map extends to a well-defined morphism on \bar{M}_{g,n}. We apply this to the problem of defining and computing several different extensions to \bar{M}_{g,n} of the double ramification cycle (= the locus of smooth pointed curves that admit a meromorphic function with prescribed zeroes and poles at the points). This is a joint work with Jesse Kass.

Michael Gröchenig (FU Berlin). Rigid local systems. Friday 17 Nov., 1:30-2:30pm. Huxley 139.

Abstract: An irreducible representation of an abstract group is called rigid, if it defines an isolated point in the moduli space of all representations.
Complex rigid representations are always defined over a number field. According to a conjecture by Simpson, for fundamental groups of smooth projective varieties one should expect furthermore that rigid representations can be defined over the ring of algebraic integers. I will report on joint work with H. Esnault where we prove this for so-called cohomologically rigid representations. Our argument is mostly arithmetic and passes through fields of positive characteristic and the p-adic numbers.

Vladimir Markovic (Caltech). Teichmueller flow and complex geometry of Moduli Spaces. Friday 24 Nov., 1:30-2:30pm. Huxley 139.

Steven Sivek (Imperial College). Khovanov homology detects the trefoils. Friday 1 Dec., 1:30-2:30pm. Huxley 139.

Abstract: Khovanov homology assigns to each knot in S^3 a bigraded abelian group whose graded Euler characteristic is the Jones polynomial. While it is not known whether the Jones polynomial detects the unknot, Kronheimer and Mrowka proved in 2010 that the Khovanov homology of K has rank 1 if and only if K is the unknot. Building on their work, I will outline a proof that Khovanov homology also detects the left and right handed trefoils, with an emphasis on the role played by contact geometry in this setting. This is joint work with John Baldwin.

Calum Spicer (Imperial College). Mori theory for foliations. Friday 15 Dec., 1:30-2:30pm. Huxley 139.

Abstract: In recent years there has been growing interest in the prospect of applying the ideas behind the Mori theory of varieties to foliations.
In the case of rank 1 foliations there has been a great deal of activity and work, but the case of higher rank of foliations remains relatively unexplored. In this talk I will explain some recent work toward developing a foliated Mori theory for co-rank 1 foliations, as well as indicating some applications of these ideas.