Past

In this talk we discusss some notions of small sets in Polish groups. We give some examples and applications of these notions in analysis and group theory. Moreover, we introduce a new notion of smallness which we call Haar meager sets. This notion coincides with the meager sets in locally compact groups. However, it is strictly stronger in the setting of nonlocally compact groups. We argue that this is the right topological analogue of Christian's Haar null sets. The speaker gratefully acknowledges the support of the LMS under a Scheme 2 Grant.

OxMOS Team Meetings are held regularly during term and are open to all. Two members of OxMOS will give a short update on their recent research.

In this talk I'll explore the meaning of the Hankel determinant formula for the general solutions of the Painleve' equations both from the analytic and the geometric point of view. I'll start with the simple example of PII and I'll show how the generating function for the Hankel determinant satisfies two Riccati equations. These linearize into the Jimbo-Miwa-Ueno isomonodromic deformation problem. Indeed this occurs for all the Painleve' equations PII,..,PVI and it is due to the link between their solutions and the infinite Toda lattice equation. I'll then explore the geometric meaning of the Hankel determinants by looking at the (suitably defined) spectral curve of the Toda lattice equation.

Chakravarty and Ablowitz (PRL vol.76 p.857, 1996). showed that a fifth-order equation arizing from the Bianchi IX system can be described asmonodromy evolving (non-preserving) deformations. In my talk, we will show that general Halphen-type systems, which comes from generic DH-IX systems, can be represented as monodromy evolving deformations.

  In this talk we will discuss some recent developments in the study of nonlinear partial differential equations of mixed type, including the mixed parabolic-hyperbolic type and mixed elliptic-hyperbolic type. Examples include nonlinear degenerate diffusion-convection equations and transonic flow equations in fluid mechanics, as well as nonlinear equations of mixed type in a fluid mechanical formulation for isometric embedding problems in differential geometry. Further ideas, trends, and open problems in this direction will be also addressed.  

We present a continuous-time equilibrium-based model for large economic agent, where she trades with market makers at their utility indifference prices. The presentation is based on a joint project with Peter Bank.

There will be three discussion meetings based on aspects of the

programme open to all internal project members. Others interested in

attending should contact Carlos Mora-Corral.

Team meetings, held roughly every four weeks, are open to anyone who is

interested. OxMOS post docs and Dphil students will give updates on the

research.

 

If new particles are produced at the LHC, it is vital that we can extract as much information as possible from them about the underlying theory.  I will discuss some recent work on extracting spin information from invariant mass distributions of new particles.  I will then introduce the Kullback-Leibler method of quantifying our ability to distinguish different scenarios.  

Whether the 3D incompressible Euler or Navier-Stokes equations

can develop a finite time singularity from smooth initial data has been

an outstanding open problem. Here we review some existing computational

and theoretical work on possible finite blow-up of the 3D Euler equations.

We show that the geometric regularity of vortex filaments, even in an

extremely localized region, can lead to dynamic depletion of vortex

stretching, thus avoid finite time blowup of the 3D Euler equations.

Further, we perform large scale computations of the 3D Euler equations

to re-examine the two slightly perturbed anti-parallel vortex tubes which

is considered as one of the most attractive candidates for a finite time

blowup of the 3D Euler equations. We found that there is tremendous dynamic

depletion of vortex stretching and the maximum vorticity does not grow faster

than double exponential in time. Finally, we present a new class of solutions

for the 3D Euler and Navier-Stokes equations, which exhibit very interesting

dynamic growth property. By exploiting the special nonlinear structure of the

equations, we prove nonlinear stability and the global regularity of this class of solutions.

Self-interacting diffusions are solutions to SDEs with a drift term depending

on the process and its normalized occupation measure $\mu_t$ (via an interaction

potential and a confinement potential): $$\mathrm{d}X_t = \mathrm{d}B_t -\left(

\nabla V(X_t)+ \nabla W*{\mu_t}(X_t) \right) \mathrm{d}t ; \mathrm{d}\mu_t = (\delta_{X_t}

- \mu_t)\frac{\mathrm{d}t}{r+t}; X_0 = x,\,\ \mu_0=\mu$$ where $(\mu_t)$ is the

process defined by $$\mu_t := \frac{r\mu + \int_0^t \delta_{X_s}\mathrm{d}s}{r+t}.$$

We establish a relation between the asymptotic behaviour of $\mu_t$ and the

asymptotic behaviour of a deterministic dynamical flow (defined on the space of

the Borel probability measures). We will also give some sufficient conditions

for the convergence of $\mu_t$. Finally, we will illustrate our study with an

example in the case $d=2$.