Warwick

This talk will present a new characterisation of rectifiable subsets of a complete metric space in terms of local approximation, with respect to the Gromov-Hausdorff distance, by finite dimensional Banach spaces. Time permitting, we will discuss recent joint work with Hyde and Schul that provides quantitative analogues of this statement.
 

The study of gravity currents has long been of interest due to their prevalence in industry and in nature, one such example being the spreading of viscoplastic (yield-stress) fluid films. When a viscoplastic fluid is extruded onto a flat plate, the resulting gravity current expands axisymmetrically when the surface is dry and rough. In this talk, I will discuss two instabilities that arise when (1) the no-slip surface is replaced by a free-slip surface; and (2) the flat plate is wet by a thin coating of water.

We prove that for Bernoulli bond percolation on $\mathbb{Z}^d$, $d\geq2$, the percolation density $\theta(p)$ (defined as the probability of the origin lying in an infinite cluster) is an analytic function of the parameter in the supercritical interval $(p_c,1]$. This answers a question of Kesten from 1981.

The proof involves a little bit of elementary complex analysis (Weierstrass M-test), a few well-known results from percolation theory (Aizenman-Barsky/Menshikov theorem), but above all combinatorial ideas. We used a new notion of contours, bounds on the number of partitions of an integer, and the inclusion-exclusion principle, to obtain a refinement of a classical argument of Peierls that settled the 2-dimensional case in 2018. More recently, we coupled these techniques with a renormalisation argument to handle all dimensions.

Joint work with Christoforos Panagiotis.

Given a tournament with $d{n \choose 3}$ cycles of length three, how many cycles of length four must there be? Linial and Morgenstern (2016) conjectured that the minimum is asymptotically attained by "blowing up" a transitive tournament and orienting the edges randomly within the parts. This is reminiscent of the tight examples for the famous Triangle and Clique Density Theorems of Razborov, Nikiforov and Reiher. We prove the conjecture for $d \geq \frac{1}{36}$ using spectral methods. We also show that the family of tight examples is more complex than expected and fully characterise it for $d \geq \frac{1}{16}$. Joint work with Timothy Chan, Andrzej Grzesik and Daniel Král'.

The AO-model describes crystalline solids in the presence of defects like dislocation lines. We demonstrate that the model supports low-energy structures like grains and determine for simple geometries the grain boundary energy density. At small misorientation angles we recover the well-known Read-Shockley law. Due to the atomistic nature of the model it is possible to consider the the Boltzmann-Gibbs distribution at non-zero temperature. Using ideas by Froehlich and Spencer we prove rigorously the presence of long-range order if the temperature is sufficiently small.
 

The space of complete collineations is an important and beautiful chapter of algebraic geometry, which has its origins in the classical works of Chasles, Schubert and many others, dating back to the 19th century. It provides a 'wonderful compactification' (i.e. smooth with normal crossings boundary) of the space of full-rank maps between two (fixed) vector spaces. More recently, the space of complete collineations has been studied intensively and has been used to derive groundbreaking results in diverse areas of mathematics. One such striking example is L. Lafforgue's compactification of the stack of Drinfeld's shtukas, which he subsequently used to prove the Langlands correspondence for the general linear group. 

In joint work with M. Kapranov, we look at these classical spaces from a modern perspective: a complete collineation is simply a spectral sequence of two-term complexes of vector spaces. We develop a theory involving more full-fledged (simply graded) spectral sequences with arbitrarily many terms. We prove that the set of such spectral sequences has the structure of a smooth projective variety, the 'variety of complete complexes', which provides a desingularization, with normal crossings boundary, of the 'Buchsbaum-Eisenbud variety of complexes', i.e. a 'wonderful compactification' of the union of its maximal strata.
 

It is a standard heuristic that sums of oscillating number theoretic functions, like the M\"obius function or Dirichlet characters, should exhibit squareroot cancellation. It is often very difficult to prove anything as strong as that, and we generally expect that if we could prove squareroot cancellation it would be the best possible bound. I will discuss recent results showing that, in fact, certain averages of multiplicative functions exhibit a bit more than squareroot cancellation.