# Autumn Term 2019

Jeff Hicks (University of Cambridge). The support of a Lagrangian. Friday 11th Oct, 1:30-2:30pm. Huxley 308.

Abstract: An SYZ fibration is a fibration of a symplectic manifold $X$ whose fibers are Lagrangian tori. If $Q$ is the base of the fibration, we can associate to each Lagrangian submanifold $L$ a subset of of $Q$ called the SYZ support. This is defined as the points of $Q$ for which the associated Lagrangian torus fiber $F_q$ has non-trivial Lagrangian intersection Floer homology with $L$. For example, when $L=F_q$ is some torus fiber, the support is simply $q$.
We extend this to the setting where $L$ is a Lagrangian submanifold determined by the data of a tropical curve in $Q$. In specific examples we compute this support explicitly, and make some connections to mirror symmetry for toric varieties.

Thomas Prince (University of Oxford). Smoothing Calabi-Yau toric hypersurfaces using the Gross-Siebert program. Friday 18th Oct, 1:30-2:30pm. Huxley 308.

Abstract: We describe how to form a novel dataset of Calabi-Yau threefolds via an application of the Gross-Siebert algorithm to a reducible union of toric varieties obtained by degenerating anti-canonical hypersurfaces in a class of (around 1.5 million) Gorenstein toric Fano fourfolds. Many of these constructions correspond to smoothing such a hypersurface; in contrast to the famous construction of Batyrev-Borisov which performs a crepant resolution. In addition we describe a mirror construction, which allows us to describe the Picard-Fuchs operators associated to these examples. We focus particularly on a class of examples related to joins of elliptic curves, including examples recently considered by Inoue and Knapp-Sharpe.

Benoit Daniel (Université de Lorraine). On the area of minimal surfaces in a slab. Friday 25th Oct, 1:30-2:30pm. Huxley 308.

Abstract: Consider a non-planar orientable minimal surface S in a slab which is possibly with genus or with more than two boundary components. We show that there exists a catenoidal waist W in the slab whose flux has the same vertical component as S such that Area(S) >= Area(W), provided the intersections of S with horizontal planes have the same orientation.This is joint work with Jaigyoung Choe.

John Ottem (University of Oslo). Enriques fibrations with non-algebraic integral Hodge classes. Friday 1st Nov, 1:30-2:30pm. Huxley 308.

Abstract: I will explain a construction of a certain pencil of Enriques surfaces with non-algebraic integral Hodge classes of non-torsion type. This gives the first example of a threefold with trivial Chow group of zero-cycles on which the integral Hodge conjecture fails. If time permits, I will explain an application to a classical question of Murre on the universality of the Abel-Jacobi maps in codimension three. This is joint work with Fumiaki Suzuki.​

No seminar on Friday 8th November.

Stergios Antonakoudis (Imperial College). On totally geodesic submanifolds of Teichmüller space. Friday 15th Nov, 1:30-2:30pm. Huxley 308.

Abstract: We will discuss recent results and progress on the study of totally geodesic submanifolds of Teichmüller space of Riemann surfaces, starting with explaining and presenting a new proof of Royden’s theorem.

Lenny Taelman (University of Amsterdam). Derived equivalences of hyperkähler varieties. Friday 22nd Nov, 1:30-2:30pm. Huxley 308.

Abstract: In this talk we consider auto-equivalences of the bounded derived category D(X) of coherent sheaves on a smooth projective complex variety X. By a result of Orlov, any such auto-equivalence induces an (ungraded) automorphism of the singular cohomology H(X,\Q).
If X is a K3 surface, then work of Mukai, Orlov, Huybrechts, Macrì and Stellari completely describes the image of the map \rho_X : \Aut D(X) –> Aut(H(X, \Q)). We will study the image of \rho_X for higher-dimensional hyperkähler varieties. An important tool is a certain Lie algebra acting on H(X, Q), introduced by Verbitsky, Looijenga and Lunts. We show that this Lie algebra is a derived invariant, and use this to study the image of \rho_X.

Lorenzo Foscolo (University College London)G2 manifolds from nodal Calabi-Yau 3-folds. TBA. Friday 29th Nov, 1:30-2:30pm. Huxley 308.

Abstract: I will discuss joint work with Mark Haskins and Johannes Nordström on the construction of families of Ricci-flat 7-dimensional manifolds with holonomy G2 close to a limiting Calabi-Yau 3-fold (modulo an antiholomorphic involution) with nodal singularities. In the first part of the talk, based on arXiv:1805.02612, I will describe the non-compact situation, where the limiting Calabi-Yau 3-fold is the conifold itself or its smoothing or small resolution. In this context our results provide a precise metric realisation of work by Atiyah-Maldacena-Vafa and Acharya in the early 2000’s on a large N duality in Type IIA String theory and its lift to M-theory. In the second part of the talk, I will report on work in progress in the compact case. I will explain the central role in our construction of a topological constraint on the nodal Calabi-Yau 3-fold that is analogous to Friedman’s necessary and sufficient condition for smoothing the nodes. I will also describe how our construction could lead to the first known compact G2 spaces with isolated conical singularities.

Kevin Buzzard (Imperial College). Doing algebraic geometry in a computer proof system. Friday 6th Dec, 1:30-2:30pm. Huxley 308.

Abstract: I’ll tell the story of how me and a team of undergraduates ended up being the first people in the world to tell a computer what a scheme was, and how it turned out to be more difficult than we thought. I’ll go through the definition we all thought we knew and point out some subtleties. I’ll then talk about how I went on to define adic spaces and perfectoid spaces.

Tom Leinster (University of Edinburgh). New invariants of metric spaces: magnitude and maximum entropy. Friday 13th Dec, 1:30-2:30pm. Huxley 308.

Abstract: This is the story of some of the geometrical fruits of a large-scale categorical programme to investigate invariants of size. One such fruit is magnitude, a (newish) real invariant of compact metric spaces, whose asymptotic behaviour determines classical invariants such as volume, surface area, dimension, etc. Another is a suite of new measures of entropy, generalizing classical quantities from information theory and closely related to measures of biological diversity. We’ll see that every compact metric space carries a canonical probability measure, which maximizes entropy in infinitely many senses at once (a result joint with Emily Roff). I will explain all this, starting from the beginning.