In this talk, I am going to report on some on-going research at the interface between Rough Paths Theory and Schramm-Loewner evolutions (SLE). In this project, we try to adapt techniques from Rough Differential Equations to the study of the Loewner Differential Equation. The main ideas concern the restart of the backward Loewner differential equation from the singularity in the upper half plane. I am going to describe some general tools that we developed in the last months that lead to a better understanding of the dynamics in the closed upper half plane under the backward Loewner flow.

Joint work with Prof. Dmitry Belyaev and Prof. Terry Lyons

# Past Forthcoming Seminars

Associated to a finite graph without loops is the Kac-Moody Lie algebra for the Cartan matrix whose off diagonal entries are (minus) the adjacency matrix for the graph. Two famous conjectures of Kac, proved by Hausel, Letellier and Villegas, hint that there may be some larger cohomologically graded algebra associated to the graph (even if there are loops), providing "higher" Kac moody Lie algebras, or at least their positive halves. Using work with Sven Meinhardt, I will give a geometric construction of the (full) Kac-Moody algebra for a general finite graph, using cohomological DT theory. Along the way we'll see a proof of the positivity conjecture for the modified Kac polynomials of Bozec, Schiffmann and Vasserot counting various types of representations of quivers.

The existence of planetary and stellar magnetic fields is attributed to the dynamo instability, the mechanism by which a background turbulent flow spontaneously generates a magnetic field by the constructive refolding of magnetic field lines. Many efforts have been made by several experimental groups to reproduce the dynamo instability in the laboratory using liquid metals. However, so far, unconstrained dynamos driven by turbulent flows have not been achieved in the intrinsically low magnetic Prandtl number $P_m$ (i.e. $Pm = Rm/Re << 1$) laboratory experiments. In this seminar I will demonstrate that the critical magnetic Reynolds number $Rm_c$ for turbulent non-helical dynamos in the low $P_m$ limit can be significantly reduced if the flow is submitted to global rotation. Even for moderate rotation rates the required energy injection rate can be reduced by a factor more than 1000. Our finding thus points into a new paradigm for the design of new liquid metal dynamo experiments.

Image use continues to increase in both biomedical sciences and clinical practice. State of the art acquisition techniques allow characterisation from subcellular to whole organ scale, providing quantitative information of structure and function. In the heart, for example, images acquired from a single modality (cardiac MRI) can characterise micro- and macrostructure, describe mechanical function and measure blood flow. In the lungs, new contrast agents can be used to visualise the flow of gas in free breathing subjects. This provides rich new sources of information as well as new challenges to extract data in a way that is useful to clinicians as well as computer modellers.

I will describe efforts in my group to use the latest advances in machine learning to analyse images, and explain how we are applying these to the development of accurate computer models of the heart.

We discuss pathwise pricing-hedging dualities in continuous time and on a frictionless market consisting of finitely many risky assets with continuous price trajectories.

The costs to Vodafone of calls terminating on other networks – especially fixed networks – are largely determined by the termination charges levied by other telecoms operators. We interconnect to several other telecoms operators, who charge differently; within one interconnect operator, costs vary depending on which of their switching centres we deliver calls to, and what the terminating phone number is. So, while these termination costs depend partly on factors that we cannot control (such as the number called, the call duration and the time of day), they are also influenced by some factors that we can control. In particular, we can route calls within our network before handing them over from our network to the other telecoms operator; where this “handover” occurs has an impact on termination cost.

Vodafone would like to develop a repeatable capability to determine call delivery cost efficiency and identify where network routing changes can be made to improve matters, and determine traffic growth forecasts.

In this talk I will discuss the problem of finding Einstein metrics in the homogeneous and cohomogeneity one setting.

In particular, I will describe a recent result concerning existence of solutions to the Dirichlet problem for cohomogeneity one Einstein metrics.

We formulate and solve a class of Backward Stochastic Differential Equations (BSDEs) driven by the compensated random measure associated to a given marked point process on a general state space. We present basic well-posedness results in L 2 and in L 1 . We show that in the setting of point processes it is possible to solve the equation recursively, by replacing the BSDE by an ordinary differential equation in between jumps. Finally we address applications to optimal control of marked point processes, where the solution of a suitable BSDE allows to identify the value function and the optimal control. The talk is based on joint works with Marco Fuhrman and Jean Jacod.