Manchester

I shall give a non-technical survey of Pure Inductive Logic, a branch of Carnap's Inductive Logic which was

anticipated early on in that subject but has only recently begun to be developed as an area of Mathematical Logic. My intention

is to cover its origins and aims, and to pick out some of the key concepts which have emerged in the last decade or so.

Masser recently proved a bound on the number of rational points of bounded height on the graph of the zeta function restricted to the interval [2,3]. Masser's bound substantially improves on bounds obtained by Bombieri-Pila-Wilkie. I'll discuss some results obtained in joint work with Gareth Boxall in which we prove bounds only slightly weaker than Masser's for several more natural analytic functions.

 We present a new, more conceptual proof of our result that, when a finite group acts on a polynomial ring, the regularity of the ring of invariants is at most zero, and hence one can write down bounds on the degrees of the generators and relations. This new proof considers the action of the group on the Cech complex and looks at when it splits over the group algebra. It also applies to a more general class of rings than just polynomial ones.

Motivated by industrial and biological applications, the Waves

Group at Manchester has in recent years been interested in

developing methods for obtaining the effective properties of

complex composite materials. As time allows we shall discuss a

number of issues, such as differences between composites

with periodic and aperiodic distributions of inclusions, and

modelling of nonlinear composites.

Feature Selection is a ubiquitous problem in across data mining,

bioinformatics, and pattern recognition, known variously as variable

selection, dimensionality reduction, and others. Methods based on

information theory have tremendously popular over the past decade, with

dozens of 'novel' algorithms, and hundreds of applications published in

domains across the spectrum of science/engineering. In this work, we

asked the question 'what are the implicit underlying statistical

assumptions of feature selection methods based on mutual information?'

The main result I will present is a unifying probabilistic framework for

information theoretic feature selection, bringing almost two decades of

research on heuristic methods under a single theoretical interpretation.

The displacement of a liquid by an air finger is a generic two-phase flow that

underpins applications as diverse as microfluidics, thin-film coating, enhanced

oil recovery, and biomechanics of the lungs. I will present two intriguing

examples of such flows where, firstly, oscillations in the shape of propagating

bubbles are induced by a simple change in tube geometry, and secondly, flexible

vessel boundaries suppress viscous fingering instability.

1) A simple change in pore geometry can radically alter the behaviour of a

fluid displacing air finger, indicating that models based on idealized pore

geometries fail to capture key features of complex practical flows. In

particular, partial occlusion of a rectangular cross-section can force a

transition from a steadily-propagating centred finger to a state that exhibits

spatial oscillations via periodic sideways motion of the interface at a fixed

location behind the finger tip. We characterize the dynamics of the

oscillations and show that they arise from a global homoclinic connection

between the stable and unstable manifolds of a steady, symmetry-broken

solution.

2) Growth of complex dendritic fingers at the interface of air and a viscous

fluid in the narrow gap between two parallel plates is an archetypical problem

of pattern formation. We find a surprisingly effective means of suppressing

this instability by replacing one of the plates with an elastic membrane. The

resulting fluid-structure interaction fundamentally alters the interfacial

patterns that develop and considerably delays the onset of fingering. We

analyse the dependence of the instability on the parameters of the system and

present scaling arguments to explain the experimentally observed behaviour.

One of the cores in modern probability theory is the stochastic integral introduced by K.

Ito in the 1940s. Due to the randomness and the irregularity of typical stochastic

integrators (such as the Wiener process) one can not follow a classical approach as in

calculus to define the stochastic integral.

For Hilbert spaces Ito's theory of stochastic integration in finite

dimensions can be generalised. There are several even quite early attempts to tackle

stochastic integration in more general spaces such as Banach spaces but none of them

provides the generality and powerful tool as the theory in Hilbert spaces.

In this talk, we begin with introducing the stochastic integral in Hilbert spaces based

on the classical theory and with explaining the restriction of this approach to Hilbert

spaces. We tackle the problem of stochastic integration in Banach spaces by introducing

a stochastic version of a Pettis integral. In the case of a Wiener process as an integrator,

the stochastic Pettis integrability of a function is related to the extensively studied class of

$\gamma$-radonifying operators. Surprisingly, it turns out that for more general integrators

which are non-Gaussian and discontinuous (Levy processes) such a relation can still be

established but with another subclass of radonifying operators.

The talks will discuss relations between two major conjectures in the theory of groups of finite Morley rank, a modern chapter of model theoretic algebra. One conjecture, the famous the Cherlin-Zilber Algebraicity Conjecture formulated in 1970-s states that infinite simple groups of finite Morley rank are isomorphic to simple algebraic groups over algebraically closed fields. The other conjecture, due to Hrushovski and more recent, states that a generic automorphism of a simple group of finite Morley rank has pseudofinite group of fixed points.

Hrushovski showed that the Cherlin-Zilber Conjecture implies his conjecture. Proving Hrushovski's Conjecture and reversing the implication would provide a new efficient approach to proof of Cherlin-Zilber Conjecture.

Meanwhile, the machinery that is already available for the work at pseudofinite/finite Morley rank interface already yields an interesting

result: an alternative proof of the Larsen-Pink Theorem (the latter says, roughly speaking, that "large" finite simple groups of matrices are Chevalley groups over finite fields).

I'll give a brief survey of what is known about the density of rational points on definable sets in o-minimal expansions of the real field, then discuss improving these results in certain cases.