Cambridge

We consider a generalization of degeneracy loci of morphisms between vector bundles based on orbit closures of algebraic groups in their linear representations. Using a certain crepancy condition on the orbit closure we gain some control over the canonical sheaf in a preferred class of examples. This is notably the case for Richardson nilpotent orbits and partially decomposable skew-symmetric three-forms in six variables. We show how these techniques can be applied to construct Calabi-Yau manifolds and Fano varieties of dimension three and four.

This is a joint work with Vladimiro Benedetti, Laurent Manivel and Fabio Tanturri.

While many cubic fourfolds are known to be rational, it is expected that the very general cubic fourfold is irrational (although none have been
proven to be so). There is a conjecture for precisely which cubics are rational, which can be expressed in Hodge-theoretic terms (by work of Hassett)
or in terms of derived categories (by work of Kuznetsov). The conjecture can be phrased as saying that one can associate a `noncommutative K3 surface' to any cubic fourfold, and the rational ones are precisely those for which this noncommutative K3 is `geometric', i.e., equivalent to an honest K3 surface. It turns out that the noncommutative K3 associated to a cubic fourfold has a conjectural symplectic mirror (due to  Batyrev-Borisov). In contrast to the algebraic side of the story, the mirror is always `geometric': i.e., it is always just an honest K3 surface equipped with an appropriate Kähler form. After explaining this background, I will state a theorem: homological mirror symmetry holds in this context (joint work with Ivan Smith).

Lagrangian Floer cohomology groups are extremely hard compute in most situations. In this talk I’ll describe two ways to extract information about the self-Floer cohomology of a monotone Lagrangian possessing certain kinds of symmetry, based on the closed-open string map and the Oh spectral sequence. The focus will be on a particular family of examples, where the techniques can be combined to deduce some unusual properties.

It is well known that there is a finite colouring of the natural numbers such that there is no infinite set X with X+X (the pairwise sums from X, allowing repetition) monochromatic. It is easy to extend this to the rationals. Hindman, Leader and Strauss showed that there is also such a colouring of the reals, and asked if there exists a space 'large enough' that for every finite colouring there does exist an infinite X with X+X monochromatic. We show that there is indeed such a space. Joint work with Imre Leader.

The goal of this talk will be to give an overview of recent work, joint with Kim Moore, on a short time existence problem in Lagrangian mean curvature flow. More specifically, we consider a compact initial Lagrangian submanifold with a finite number of singularities, each asymptotic to a pair of transversely intersecting planes. We show it is possible to construct a smooth Lagrangian mean curvature flow, existing for positive times, that attains the singular Lagrangian as its initial condition in a suitable weak sense.  The construction uses a family of smooth solutions whose initial conditions approximate the singular Lagrangian. In order to appeal to compactness theorems and produce the desired solution, it is necessary to first establish uniform curvature estimates on the approximating family. As time allows I hope to focus in particular on the proof of these estimates, and their role in the proof of the main theorem.

Dynatomic curves parametrize n-periodic orbits of a one-parameter family of polynomial dynamical systems. These curves lack the structure of their arithmetic-geometric analogues (modular curves of level n) but can be studied dynamically.  Morton and Silverman conjectured a dynamical analogue of the uniform boundedness conjecture (theorems of Mazur, Merel), asserting uniform bounds for the number of rational periodic points for such a family.  I will discuss recent work towards the function field version of their conjecture, including results on the reduction mod p of dynatomic curves for the quadratic polynomial family z^2+c.

The technique of scanning, or the parameterised Pontrjagin--Thom construction, has been extraordinarily successful in calculating the cohomology of configuration spaces (McDuff), moduli spaces of Riemann surfaces (Madsen, Tillmann, Weiss), moduli spaces of graphs (Galatius), and moduli spaces of manifolds of higher dimension (Galatius, R-W, Botvinnik, Perlmutter), with constant coefficients. In each case the method also works to study the cohomology of moduli spaces of objects equipped with a "tangential structure". I will explain how choosing an auxiliary highly-symmetric tangential structure often lets one calculate the cohomology of these moduli spaces with large families of twisted coefficients, by exploiting the symmetries of the tangential structure and using a little representation theory.

 

Mirror symmetry is a duality from string theory that states that given a Calabi-Yau variety, there exists another Calabi-Yau variety so that various geometric and physical data are exchanged. The investigation of this mirror correspondence has its roots in enumerative geometry and hodge theory, but has been later interpreted by Kontsevich in a categorical setting. This exchange in data is very powerful, and has been shown to persist for zeta functions associated to Calabi-Yau varieties, although there is no rigorous statement for what arithmetic mirror symmetry would be. Instead of directly trying to state and prove arithmetic mirror symmetry, we will instead use mirror symmetry as an intuitional framework to obtain arithmetic results for special Calabi-Yau pencils in projective space from the Hodge theoretic viewpoint. If time permits, we will discuss work in progress in starting to find arithmetic implications of Kontsevich's Homological Mirror Symmetry.

Given two actions of a group $G$ on trees $T_1,T_2$, Guirardel introduced the "core", a $G$--cocompact CAT(0) subspace of $T_1\times
T_2$.  The covolume of the core is a natural notion of "intersection number" for the two tree actions (for example, if $G$ is a surface group
and $T_1,T_2$ are Bass-Serre trees associated to splittings along some curves, this "intersection number" is the one you'd expect).  We
generalise this construction by considering a fixed finitely-presented group $G$ equipped with finitely many essential, cocompact actions on
CAT(0) cube complexes $X_1,...,X_d$.  Inside $X=X_1\times ... \times X_d$, we find a $G$--invariant subcomplex $C$ which, although not convex
or necessarily CAT(0), has each component isometrically embedded with respect to the $\ell_1$ metric on $X$ (the key point is this change from
the CAT(0) to the $\ell_1$ viewpoint).  In the case where $d=2$ and $X_1,X_2$ are simplicial trees, $C$ is the Guirardel core.  Many
features of the Guirardel core generalise, and I will summarise these. For example, if the cubulations $G\to Aut(X_i)$ are "essentially
different", then $C$ is connected and $G$--cocompact.  Time permitting, I will discuss an application, namely a new proof of Nielsen realisation
for finite subgroups of $Out(F_n)$.  This talk is based on ongoing joint work with Henry Wilton.

A cornerstone of the study of mapping class groups is the
Nielsen--Thurston classification theorem. I will outline a
polynomial-time algorithm that determines the Nielsen--Thurston type and
the canonical curve system of a mapping class. Time permitting, I shall
describe a polynomial-time algorithm to compute the quotient orbifold of
a periodic mapping class, and I shall discuss the conjugacy problem for
the mapping class group. This is joint work with Mark Bell.