Cambridge

This talk will be three short stories on the general theme of elastic
instabilities in soft solids. First I will discuss the inflation of a
cylindrical cavity through a bulk soft solid, and show that such a
channel ultimately becomes unstable to a finite wavelength peristaltic
undulation. Secondly, I will introduce the elastic Rayleigh Plateau
instability, and explain that it is simply 1-D phase separation, much
like the inflationary instability of a cylindrical party balloon. I will
then construct a universal near-critical analytic solution for such 1-D
elastic instabilities, that is strongly reminiscent of the
Ginzberg-Landau theory of magnetism. Thirdly, and finally, I will
discuss pattern formation in layer-substrate buckling under equi-biaxial
compression, and argue, on symmetry grounds, that such buckling will
inevitably produce patterns of hexagonal dents near threshold.

A uniform spanning tree of $\mathbb{Z}^4$ can be thought of as the "uniform measure" on trees of $\mathbb{Z}^4$. The past of 0 in the uniform spanning tree is the finite component that is disconnected from infinity when 0 is deleted from the tree. We establish the logarithmic corrections to the probabilities that the past contains a path of length $n$, that it has volume at least $n$ and that it reaches the boundary of the box of side length $n$ around 0. Dimension 4 is the upper critical dimension for this model in the sense that in higher dimensions it exhibits "mean-field" critical behaviour. An important part of our proof is the study of the Newtonian capacity of a loop erased random walk in 4 dimensions. This is joint work with Tom Hutchcroft.

The Schmidt subspace theorem is a far-reaching generalization of the Thue-Siegel-Roth theorem in diophantine approximation. In this talk I will give an interpretation of Schmidt's subspace theorem in terms of the dynamics of diagonal flows on homogeneous spaces and describe how the exceptional subspaces arise from certain rational Schubert varieties associated to the family of linear forms through the notion of Harder-Narasimhan filtration and an associated slope formalism. This geometric understanding opens the way to a natural generalization of Schmidt's theorem to the setting of diophantine approximation on submanifolds of $GL_d$, which is our main result. In turn this allows us to recover and generalize the main results of Kleinbock and Margulis regarding diophantine exponents of submanifolds. I will also mention an application to diophantine approximation on Lie groups. Joint work with Nicolas de Saxcé.

In long-range percolation on $\mathbb{Z}^d$, each potential edge $\{x,y\}$ is included independently at random with probability roughly $\beta\|x-y\|-d-\alpha$, where $\alpha > 0$ controls how long-range the model is and $\beta > 0$ is an intensity parameter. The smaller $\alpha$ is, the easier it is for very long edges to appear. We are normally interested in fixing $\alpha$ and studying the phase transition that occurs as $\beta$ is increased and an infinite cluster emerges. Perhaps surprisingly, the phase transition for long-range percolation is much better understood than that of nearest neighbour percolation, at least when $\alpha$ is small: It is a theorem of Noam Berger that if $\alpha < d$ then the phase transition is continuous, meaning that there are no infinite clusters at the critical value of $\beta$. (Proving the analogous result for nearest neighbour percolation is a notorious open problem!) In my talk I will describe a new, quantitative proof of Berger's theorem that yields power-law upper bounds on the distribution of the cluster of the origin at criticality.
    As a part of this proof, I will describe a new universal inequality stating that on any graph, the maximum size of a percolation cluster is of the same order as its median with high probability. This inequality can also be used to give streamlined new proofs of various classical results on e.g. Erdős-Rényi random graphs, which I will hopefully have time to talk a little bit about also.

Let $X$ be a random variable, taking values in $\{1,…,n\}$, with standard deviation $\sigma$ and let $f_X$ be its probability generating function. Pemantle conjectured that if $\sigma$ is large and $f_X$ has no roots close to 1 in the complex plane then $X$ must approximate a normal distribution. In this talk, I will discuss a complete resolution of Pemantle's conjecture. As an application, we resolve a conjecture of Ghosh, Liggett and Pemantle by proving a multivariate central limit theorem for, so called, strong Rayleigh distributions. I will also discuss how these sorts of results shed light on random variables that arise naturally in combinatorial settings. This talk is based on joint work with Marcus Michelen.

Given a log Calabi-Yau surface Y with maximal boundary D, I'll explain how to construct a mirror Landau-Ginzburg model, and sketch a proof of homological mirror symmetry for these pairs when (Y,D) is distinguished within its deformation class (this is mirror to an exact manifold). I'll explain how to relate this to the total space of the SYZ fibration predicted by Gross--Hacking--Keel, and, time permitting, explain ties with earlier work of Auroux--Katzarkov--Orlov and Abouzaid. Joint work with Paul Hacking.

For a family $A$ in $\{0,...,k\}^n$, its deletion shadow is the set obtained from $A$ by deleting from any of its vectors one coordinate. Given the size of $A$, how should we choose $A$ to minimise its deletion shadow? And what happens if instead we may delete only a coordinate that is zero? We discuss these problems, and give an exact solution to the second problem.

In 2013, Bhargava-Shankar proved that (in a suitable sense) the average rank of elliptic curves over Q is bounded above by 1.5, a landmark result which earned Bhargava the Fields medal. Later Bhargava-Gross proved similar results for hyperelliptic curves, and Poonen-Stoll deduced that most hyperelliptic curves of genus g>1 have very few rational points. The goal of my talk is to explain how simple curve singularities and simple Lie algebras come into the picture, via a modified Grothendieck-Brieskorn correspondence.

Moreover, I’ll explain how this viewpoint leads to new results on the arithmetic of curves in families, specifically for certain families of non-hyperelliptic genus 3 curves.

I will present joint work with A. Pillay on the theory of NIP formulas in arbitrary groups, which exhibit a local formulation of the notion of finitely satisfiable generics (as defined by Hrushovski, Peterzil, and Pillay). This setting generalizes ``local stable group theory" (i.e., the study of stable formulas in groups) and also the case of arbitrary NIP formulas in pseudofinite groups. Time permitting, I will mention an application of these results in additive combinatorics.

The homological monodromy of the universal family of cubic threefolds defines a representation of a certain Artin-type group into the symplectic group Sp(10;\Z). We use Thurston’s classification of surface automorphisms to prove this does not factor through the genus five mapping class group.  This gives a geometric group theory perspective on the well-known irrationality of cubic threefolds, as established by Clemens and Griffiths.