Past

After a short introduction to homogeneous relational structures (structures such that all local symmetries are global), I will discuss some different topics relating homogeneity to homomorphisms: a family of notions of 'homomorphism-homogeneity' that generalise homogeneity; generic endomorphisms of homogeneous structures; and constraint satisfaction problems.

Using the spectral multiplicities of the standard torus, we
endow the Laplace eigenspaces with Gaussian probability measures.
This induces a notion of random Gaussian eigenfunctions
on the torus ("arithmetic random waves''.)  We study the
distribution of the nodal length of random Laplace eigenfunctions for high
eigenvalues,and our primary result is that the asymptotics for the variance is
non-universal, and is intimately related to the arithmetic of
lattice points lying on a circle with radius corresponding to the
energy. This work is joint with Manjunath Krishnapur and Par Kurlberg

Three-wave interactions form the basis of our understanding of many

nonlinear pattern forming systems because they encapsulate the most basic

nonlinear interactions. In problems with two comparable length scales, such

as the Faraday wave experiment with multi-frequency forcing, consideration

of three-wave interactions can explain the presence of the spatio-temporal

chaos found in some experiments, enabling some previously unexplained

results to be interpreted in a new light. The predictions are illustrated

with numerical simulations of a model partial differential equation.

When modelling biochemical reactions within cells, it is vitally important to take into account the effect of intrinsic noise in the system, due to the small copy numbers of some of the chemical species. Deterministic systems can give vastly different types of behaviour for the same parameter sets of reaction rates as their stochastic analogues, giving us an incorrect view of the bifurcation behaviour.

The stochastic description of this problem gives rise to a multi-dimensional Markov jump process, which can be approximated by a system of stochastic differential equations. Long-time behaviour of the process can be better understood by looking at the steady-state solution of the corresponding Fokker-Planck equation.

In this talk we consider a new finite element method which uses simulated trajectories of the Markov-jump process to inform the choice of mesh in order to approximate this invariant distribution. The method has been implemented for systems in 3 dimensions, but we shall also consider systems of higher dimension.

We will describe the space of Bridgeland stability conditions

of the derived category of some CY3 algebras of quivers drawn on the

Riemann sphere. We give a biholomorphic map from the upper-half plane to

the space of stability conditions lifting the period map of a meromorphic

differential on a 1-dimensional family of elliptic curves. The map is

equivariant with respect to the actions of a subgroup of $\mathrm{PSL}(2,\mathbb Z)$ on the

left by monodromy of the rational elliptic surface and on the right by

autoequivalences of the derived category.

The complement of a divisor in the rational elliptic surface can be

identified with Hitchin's moduli space of connections on the projective

line with prescribed poles of a certain order at marked points. This is

the space of initial conditions of one of the Painleve equations whose

solutions describe isomonodromic deformations of these connections.

This is joint work with Uri Onn. We use motivic integration to get the growth rate of the sequence consisting of the number of conjugacy classes in quotients of G(O) by congruence subgroups, where $G$ is suitable algebraic group over the rationals and $O$ the ring of integers of a number field.

The proof uses tools from the work of Nir Avni on representation growth of arithmetic groups and results of Cluckers and Loeser on motivic rationality and motivic specialization.

Successful navigation through a complicated and evolving environment is a fundamental task carried out by an enormous range of organisms, with migration paths staggering in their length and intricacy. Selecting a path requires the detection, processing and integration of a myriad of cues drawn from the surrounding environment and in many instances it is the intrinsic orientation of the environment that provides a valuable navigational aid.

In this talk I will describe the use of transport models to describe migration in oriented environments, and demonstrate the scaling approaches that allow us to derive macroscopic models for movement.

I will illustrate the methods through a number of apposite examples, including the migration of cells in the extracellular matrix, the macroscopic growth of brain tumours and the movement of wolves in boreal forest.

Motivated by the study of micro-vascular disease, we have been investigating the relationship between the structure of capillary networks and the resulting blood perfusion through the muscular walls of the heart. In order to derive equations describing effective fluid transport, we employ an averaging technique called homogenisation, based on a separation of length scales. We find that the tissue-scale flow is governed by Darcy's Law, whose coefficients we are able to explicitly calculate by averaging the solution of the microscopic capillary-scale equations. By sampling from available data acquired via high-resolution imaging of the coronary capillaries, we automatically construct physiologically-realistic vessel networks on which we then numerically solve our capillary-scale equations. By validating against the explicit solution of Poiseuille flow in a discrete network of vessels, we show that our homogenisation method is indeed able to efficiently capture the averaged flow properties.

We prove existence of a global semigroup of conservative solutions of the nonlinear variational wave equation $u_{tt}-c(u) (c(u)u_x)_x=0$. The equation was derived by Saxton as a model for liquid crystals. This equation shares many of the peculiarities of the Hunter–Saxton and the Camassa–Holm equations. In particular, the equation possesses two distinct classes of solutions denoted conservative and dissipative. In order to solve the Cauchy problem uniquely it is necessary to augment the equation properly. In this talk we describe how this is done for conservative solutions. The talk is based on joint work with X. Raynaud.

The theory of modular forms owes in many ways lots of its results to the existence of the Hecke operators and their nice properties. However, even acting on modular forms of level 1, lots of basic questions remain unresolved. We will describe and prove some known properties of the Hecke operators, and state Maeda's conjecture. This conjecture, if true, has many deep consequences in the theory. In particular, we will indicate how it implies the nonvanishing of certain L-functions.

The notion of an E-infinity ring spectrum arose about thirty years ago,

and was studied in depth by Peter May et al, then later reinterpreted

in the framework of EKMM as equivalent to that of a commutative S-algebra.

A great deal of work on the existence of E-infinity structures using

various obstruction theories has led to a considerable enlargement of

the body of known examples. Despite this, there are some gaps in our

knowledge. The question that is a major motivation for this talk is

`Does the Brown-Peterson spectrum BP for a prime p admit an E-infinity

ring structure?'. This has been an important outstanding problem for

almost four decades, despite various attempts to answer it.

I will explain what BP is and give a brief history of the above problem.

Then I will discuss a construction that gives a new E-infinity ring spectrum

which agrees with BP if the latter has an E-infinity structure. However,

I do not know how to prove this without assuming such a structure!