Oxford University

In 1997 V. Bentkus and F. Götze introduced a technique for estimating $L^p$ norms of certain exponential sums without needing an explicit estimate for the exponential sum itself. One uses instead a kind of estimate I call a "moat lemma". I explain this term, and discuss the implications for several kinds of point-counting problem which we all know and love.

Thoughts on the Burnside problem

It is an open question whether a group with a finite classifying space is hyperbolic or contains a Baumslag Solitar Subgroup. An idea of Gromov was to use aperiodic tilings of the plane to try and disprove this conjecture. I will be looking at some of the attempts to find a counterexample.

Two-dimensional viscous fluid flow problems come about either because of a thin gap geometry (Hele-Shaw flow) or plane symmetry (Stokes flow). Such problems can also involve free boundaries between different fluids, and much has been achieved in this area, including by many at Oxford. In this seminar I will discuss some new results in this field.

Firstly I will talk about some of the results of my PhD on contracting inviscid bubbles in Hele-Shaw flow, in particular regarding the effects of surface tension and kinetic undercooling on the free boundary. When a bubble contracts to a point, these effects are dominant, and lead to a menagerie of possible extinction shapes. This limiting problem is a generalisation of the curve shortening flow equation from the study of geometric PDEs. We are currently exploring properties of this generalised flow rule.

Secondly I will discuss current work on applying a free boundary Stokes flow model to the evolution of subglacial water channels. These channels are maintained by the balance between inward creep of ice and melting due to the flow of water. While these channels are normally modelled as circular or semicircular in cross-section, the inward creep of a viscous fluid is unstable. We look at some simplistic viscous dissipation models and the effect they have on the stability of the channel shape. Ultimately, a more realistic turbulent flow model is needed to understand the morphology of the channel walls.

The essential information contained in most large data sets is

small when compared to the size of the data set. That is, the

data can be well approximated using relatively few terms in a

suitable transformation. Compressed sensing and matrix completion

show that this simplicity in the data can be exploited to reduce the

number of measurements. For instance, if a vector of length $N$

can be represented exactly using $k$ terms of a known basis

then $2k\log(N/k)$ measurements is typically sufficient to recover

the vector exactly. This can result in dramatic time savings when

k

Suppose we have a collection of blocks, which gradually split apart as time goes on. Each block waits an exponential amount
of time with parameter given by its size to some power alpha, independently of the other blocks. Every block then splits randomly,but according to the same distribution. In this talk, I will focus on the case where alpha is negative, which
means that smaller blocks split faster than larger ones. This gives rise to the phenomenon of loss of mass, whereby the smaller blocks split faster and faster until they are reduced to ``dust''. Indeed, it turns out that the whole state is reduced to dust in a finite time, almost surely (we call this the extinction time). A natural question is then: how do the block sizes behave as the process approaches its extinction time? The answer turns out to involve a somewhat unusual ``spine'' decomposition for the fragmentation, and Markov renewal theory.

This is joint work with Bénédicte Haas (Paris-Dauphine).