Manchester

An old conjecture of Graham asks whether there are infinitely many integers n such that \binom{2n}{n} is coprime to 105. This is equivalent to asking whether there are infinitely many integers which only have the digits 0,1 in base 3, 0,1,2 in base 5, and 0,1,2,3 in base 7. In general, one can ask whether there are infinitely many integers which only have 'small' digits in multiple bases simultaneously. For two bases this was established in 1975 by Erdos, Graham, Ruzsa, and Straus, but the case of three or more bases is much more mysterious. I will discuss recent joint work with Ernie Croot, in which we prove that (assuming the bases are sufficiently large) there are infinitely many integers such that almost all of the digits are small in all bases simultaneously. 

Habegger showed that a subvariety of a fibre power of the Legendre family of elliptic curves contains a Zariski-dense set of special points if and only if it is special. I'll explain this result, and discuss an effective version that Gal Binyamini, Harry Schmidt, Margaret Thomas and I proved.

The propagation of a deformable air finger or bubble into a fluid-filled channel with an imposed pressure gradient was first studied by Saffman and Taylor. Assuming large aspect ratio channels, the flow can be depth-averaged and the free-boundary problem for steady propagation solved by conformal mapping. Famously, at zero surface tension, fingers of any width may exist, but the inclusion of vanishingly small surface tension selects symmetric fingers of discrete finger widths. At finite surface tension, Vanden-Broeck later showed that other families of 'exotic' states exist, but these states are all linearly unstable.

In this talk, I will discuss the related problem of air bubble propagation into rigid channels with axially-uniform, but non-rectangular, cross-sections. By including a centred constriction in the channel, multiple modes of propagation can be stabilised, including symmetric, asymmetric and oscillatory states, with a correspondingly rich bifurcation structure. These phenomena can be predicted via depth-averaged modelling, and also observed in our experiments, with quantitative agreement between the two in appropriate parameter regimes. This agreement provides insight into the physical mechanisms underlying the observed behaviour. I will outline our efforts to understand how the system dynamics is affected by the presence of nearby unstable solution branches acting as edge states. Finally, I will discuss how feedback control and control-based continuation could be used for direct experimental observation of stable or unstable modes.

We discuss a nonlinear variant of Roth’s theorem on the existence of three-term progressions in dense sets of integers, focusing on an effective version of such a result. This is joint work with Sarah Peluse.
 

A famous theorem due to Erdős and Kac states that the number of prime divisors of an integer N behaves like a normal distribution. In this talk we consider analogues of this result in the setting of arithmetic geometry, and obtain probability distributions for questions related to local solubility of algebraic varieties. This is joint work with Efthymios Sofos.

Using some new results in probabilistic recognition of black box groups
(Joint work with Sukru Yalcinkaya), I will discuss some emerging
structures that beg for a model-theoretic analysis.

Motivated by the Dixmier-Moeglin equivalence, which belongs to the realm of algebra representations, we look at a differential version of this equivalence for algebraic D-groups, which belong to the realm of finite Morley rank groups in differentially closed fields. We will see how the proof of this equivalence reduces to a standard model-theoretic fact (on binding groups). Time permitting we will present an application to Hopf-Ore extensions. This is joint work with J. Bell and R. Moosa.

In joint work with Sylvy Anscombe we had found an abstract
valuation theoretic condition characterizing those fields F for which
the power series ring F[[t]] is existentially 0-definable in its
quotient field F((t)). In this talk I will report on recent joint work
with Sylvy Anscombe and Philip Dittmann in which the study of this
condition leads us to some beautiful results on the border of number
theory and model theory. In particular, I will suggest and apply a
p-adic analogue of Lagrange's Four Squares Theorem.

After giving some motivation, I will discuss work in progress with Harry Schmidt in which we give a pfaffian definition of Weierstrass elliptic functions, refining a result due to Macintyre. The complexity of our definition is bounded by an effective absolute constant. As an application we give an effective version of a result of Corvaja, Masser and Zannier on a sharpening of Manin-Mumford for non-split extensions of elliptic curves by the additive group. We also give a higher dimensional version of their result.

An abelian l-group G is essentially a partially ordered subgroup of functions from a set to a totally ordered abelian group such

that G is closed under taking finite infima and suprema. For example, G could be the continuous semi-linear functions defined on the open
unit square, or, G could be the continuous semi-algebraic functions defined in the plane with values in (0,\infty), where the group
operation is multiplication. I will show how G, under natural geometric assumptions, can be interpreted (in a weak sense) in its lattice of
zero sets. This will then be applied to the model theory of natural divisible abelian l-groups. For example we will see that the
aforementioned examples are elementary equivalent. (Parts of the results have been announced in a preliminary report from 1987 by F. Shen
and V. Weispfenning.)