University of Oxford

The Hanna Neumann Conjecture provides a bound on the rank of the intersection of finitely generated subgroups of a free group. We will follow Mineyev's recent elementary and beautiful proof of this longstanding conjecture.

Several stochastic simulation algorithms (SSAs) have been recently

proposed for modelling reaction-diffusion processes in cellular and molecular biology. In this talk, two commonly used SSAs will  be studied. The first SSA is an on-lattice model described by the  reaction-diffusion master equation. The second SSA is an off-lattice model based on the simulation of Brownian motion of individual  molecules and their reactive collisions. The connections between SSAs  and the deterministic models (based on reaction- diffusion PDEs) will  be presented. I will consider chemical reactions both at a surface  and in the bulk. I will show how the "microscopic" parameters should  be chosen to achieve the correct "macroscopic" reaction rate. This  choice is found to depend on which SSA is used. I will also present  multiscale algorithms which use models with a different level of  detail in different parts of the computational domain

We treat the first n terms of general orthogonal series evolving with n as the partial sum process, and proved that under Menshov-Rademacher condition, the partial sum process can be enhanced into a geometric 2-rough process. For Fourier series, the condition can be improved, with an equivalent condition on limit function identified.

 In the recent series of papers Kleban, Simmons, and Ziff gave a non-rigorous computation  (base on Conformal Field Theory) of probabilities of several connectivity events for critical percolation. In particular they showed that the probability that there is a percolation cluster connecting two points on the boundary and a point inside the domain can be factorized in therms of pairwise connection probabilities. We are going to use SLE techniques to rigorously compute probabilities of several connectivity events and prove the factorization formula.

Boundary layers are often studied with no pressure gradient

or with an imposed pressure gradient. Either of these assumptions

can lead to difficulty in obtaining solutions. A major advance in fluid

dynamics last century (1969) was the development of a triple deck

formulation for boundary layers where the pressure is not

specified but emerges through an interaction between

boundary layer and the inviscid outer flow. This has given rise to

new computational problems and computations have in turn

fed ideas back into theoretical developments. In this survey talk

based on my new book, I will look at three problems:

flow past a plate, flow separation and flow in channels

and discuss the interaction between theory and computation

in advancing boundary layer theory.

The general importance of preconditioning in combination with an

appropriate iterative technique for solving large scale linear(ised)

systems is widely appreciated. For definite problems (where the

eigenvalues lie in a half-plane) there are a number of preconditioning

techniques with a range of applicability, though there remain many

difficult problems. For indefinite systems (where there are eigenvalues

in both half-planes), techniques are generally not so well developed.

Constraints arise in many physical and mathematical problems and

invariably give rise to indefinite linear(ised) systems: the incompressible

Navier-Stokes equations describe conservation of momentum in the

presence of viscous dissipation subject to the constraint of

conservation of mass, for transmission problems the solution on an

interior domain is often solved subject to a boundary integral which

imposes the exterior field, in optimisation the appearance of

constraints is ubiquitous...

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We will describe two approaches to preconditioning such constrained

systems and will present analysis and numerical results for each. In

particular, we will describe the applicability of these techniques to

approximations of incompressible Navier-Stokes problems using mixed

finite element approximation.