L3

Description:In his seminal work from the 1950s, William Feller classified all one-dimensional diffusions in terms of their ability to access the boundary (Feller's test for explosions) and to enter the interior from the boundary. Feller's technique is restricted to diffusion processes as the corresponding differential generators allow explicit computations and the use of Hille-Yosida theory. In the present article we study exit and entrance from infinity for jump diffusions driven by a stable process.Many results have been proved for jump diffusions, employing a variety of techniques developed after Feller's work but exit and entrance from infinite boundaries has long remained open. We show that the these processes have features not observes in the diffusion setting. We derive necessary and sufficient conditions on σ so that (i) non-exploding solutions exist and (ii) the corresponding transition semigroup extends to an entrance point at `infinity'. Our proofs are based on very recent developments for path transformations of stable processes via the Lamperti-Kiu representation and new Wiener-Hopf factorisations for Lévy processes that lie therein. The arguments draw together original and intricate applications of results using the Riesz-Bogdan--Żak transformation, entrance laws for self-similar Markov processes, perpetual integrals of Lévy processes and fluctuation theory, which have not been used before in the SDE setting, thereby allowing us to employ classical theory such as Hunt-Nagasawa duality and Getoor's characterisation of transience and recurrence.

Backward Stochastic Differential Equations (BSDEs) have been successfully applied  to represent the value of optimal control problems for controlled

stochastic differential equations. Since in the classical framework several restrictions on the scope of applicability of this method remained, in recent times several approaches have been devised to obtain the desired probabilistic representation in more general situations. We will review the so called  randomization method, originally introduced by B. Bouchard in the framework of optimal switching problems, which consists in introducing an auxiliary,`randomized'' problem with the same value as the original one, where the control process is replaced by an exogenous random point process,and optimization is performed over a family of equivalent probability measures. The value of the randomized problem is then represented

by means of a special class of BSDEs with a constraint on one of the unknown processes.This methodology will be applied in the framework of controlled evolution equations (with immediate applications to controlled SPDEs), a case for which very few results are known so far.

Your brain has its own waterscape: whether you are reading or sleeping, fluid flows around or through the brain tissue and clears waste in the process. These physiological processes are crucial for the well-being of the brain. In spite of their importance we understand them but little. Mathematics and numerics could play a crucial role in gaining new insight. Indeed, medical doctors express an urgent need for modeling of water transport through the brain, to overcome limitations in traditional techniques. Surprisingly little attention has been paid to the numerics of the brain’s waterscape however, and fundamental knowledge is missing. In this talk, I will discuss mathematical models and numerical methods for the brain's waterscape across scales - from viewing the brain as a poroelastic medium at the macroscale and zooming in to studying electrical, chemical and mechanical interactions between brain cells at the microscale.
 

The challenge is to produce a reduced order model which predicts the maximum temperature rise of a thermally conducting object subjected to a power deposition profile supplied by an external code. The target conducting object is basically cuboidal but with one or more shaped faces and may have complex internal cooling structures, the deposition profile may be time dependent and exhibit hot spots and sharp edged shadows among other features. An additional feature is the importance of radiation which makes the problem nonlinear, and investigation of control strategies is also of interest. Overall there appears to be a sequence of problems of degree of difficulty sufficient to tax the most gifted student, starting with a line profile on a cuboid (quasi-2D) with linearised radiation term, and moving towards increased difficulty.

We present analysis and computations of a non-local thin film model developed by Kalogirou et al (2016) for a perturbed two-layer Couette flow when the thickness of the more viscous fluid layer next to the stationary wall is small compared to the thickness of the less viscous fluid. Travelling wave solutions and their stability are determined numerically, and secondary bifurcation points identified in the process. We also determine regions in parameter space where bistability is observed with two branches being linearly stable at the same time. The travelling wave solutions are mathematically justified through a quasi-solution analysis in a neighbourhood of an empirically constructed approximate solution. This relies in part on precise asymptotics of integrals of Airy functions for large wave numbers. The primary bifurcation about the trivial state is shown rigorously to be supercritical, and the dependence of bifurcation points, as a function of Reynolds number R and the primary wavelength 2πν−1/2 of the disturbance, is determined analytically. We also present recent results on time periodic solutions arising from Hoof-Bifurcation of the primary solution branch.

(This work is in collaboration with D. Papageorgiou & E. Oliveira ) 
 

Optical super-resolution microscopy enables the observations of individual bio-molecules. The arrangement and dynamic behaviour of such molecules is studied to get insights into cellular processes which in turn lead to various application such as treatments for cancer diseases. STFC's Central Laser Facility provides (among other) public access to super-resolution microscope techniques via research grants. The access includes sample preparation, imaging facilities and data analysis support. Data analysis includes single molecule tracking algorithms that produce molecule traces or tracks from time series of molecule observations. While current algorithms are gradually getting away from "connecting the dots" and using probabilistic methods, they often fail to quantify the uncertainties in the results. We have developed a method that samples a probability distribution of tracking solutions using the Metropolis-Hastings algorithm. Such a method can produce likely alternative solutions together with uncertainties in the results. While the method works well for smaller data sets, it is still inefficient for the amount of data that is commonly collected with microscopes. Given the observations of the molecules, tracking solutions are discrete, which gives the proposal distribution of the sampler a peculiar form. In order for the sampler to work efficiently, the proposal density needs to be well designed. We will discuss the properties of tracking solutions and the problems of the proposal function design from the point of view of discrete mathematics, specifically in terms of graphs. Can mathematical theory help to design a efficient proposal function?

Here is the scientific program.

Keynote speakers:

Roger Kamm, Cecil and Ida Green Distinguished Professor of Biological and Mechanical Engineering, MIT

Alicia El Haj,  Interdisciplinary Chair of Cell Engineering, Healthcare Technology Institute, University of Birmingham

Invited speakers (confirmed to date):

Davide Ambrosi, Politecnico di Torino, Italy

Anthony Callanan, University of Edinburgh, UK

Ruth Cameron, University of Cambridge, UK

Sonia Contera, University of Oxford, UK

Linda Cummings, New Jersey Institute of Technology, USA

Mohit Dalwadi, University of Oxford, UK

John Dunlop, University of Salzburg, Austria

John King, Nottingham, UK

Nati Korin, Technion, Israel

Catriona Lally, Trinity College Dublin, Ireland

Sandra Loerakker, TU Eindhoven, Netherlands

Ivan Martin, University of Basel, Switzerland

Scott McCue, Queensland University of Technology, Australia

Pierre-Alexis Mouthuy, University of Oxford, UK

Tom Mullin,  University of Oxford, UK

Ramin Nasehi, Politecnico di Milano, Italy

Reuben O'Dea, University of Nottingham, UK

James Oliver, University of Oxford, UK

Ioannis Papantoniou, KU Leuven, Belgium

Ansgar Petersen, Julius Wolf Institute Berlin, Germany

Luigi Preziosi, Politecnico di Torino, Italy

Rebecca Shipley, University College London, UK

Barbara Wagner, Weierstrass Institute for Applied Analysis and Stochastics, Berlin

Cathy Ye, Oxford University, UK

Feihu Zhao, TU Eindhoven, Netherlands