L3

We revisit small-noise expansions in the spirit of Benarous, Baudoin-Ouyang, Deuschel-Friz-Jacquier-Violante for bivariate diffusions driven by fractional Brownian motions with different Hurst exponents. A particular focus is devoted to rough stochastic volatility models which have recently attracted considerable attention.
We derive suitable expansions (small-time, energy, tails) in these fractional stochastic volatility models and infer corresponding expansions for implied volatility. This sheds light (i) on the influence of the Hurst parameter in the time-decay of the smile and (ii) on the asymptotic behaviour of the tail of the smile, including higher orders.

SPDEs with Lévy noise can be used to model chemical, physical or biological phenomena which contain uncertainties. When discretising these SPDEs in order to solve them numerically the problem might be of large order. The goal is to save computational time by replacing large scale systems by systems of low order capturing the main information of the full model. In this talk, we therefore discuss balancing related MOR techniques. We summarise already existing results and discuss recent achievements.

The Regularity Structures introduced by Martin Hairer allow us to describe the solution of a singular SPDEs by a Taylor expansion with new monomials.  We present the two Hopf Algebras used in this theory for defining the structure group and the renormalisation group. We will point out the importance of recursive formulae with twisted antipodes.

Signature of a path provides a top down summary of the path as a driving signal. There have been substantial recent progress in reconstructing paths from its signature, (Lyons-Xu 2016, Geng 2016). In this talk, we focus on obtaining certain quantitative features of paths from their signatures. Hambly-Lyons' showed that the normalized limit of signature gives the length of a C^3 path. The result was recently extended by Lyons-Xu to C^1 paths. The extension of this result to bounded variation paths remains open. We will discuss this open problem.

TBC

Atomistic Molecular Dynamics is a well established biomolecular modelling tool that uses the wealth of information available in the Protein Data Bank (PDB). However, biophysical techniques that provide structural information at the mesoscale, such as cryo-electron microscopy and 3D tomography, are now sufficiently mature that they merit their own online repository called the EMDataBank (EMDB). We have developed a continuum mechanics description of proteins which uses this new experimental data as input to the simulations, and which we are developing into a software tool for use by the biomolecular science community. The model is a Finite Element algorithm which we have generalised to include the thermal fluctuations that drive protein conformational changes, and which is therefore known as Fluctuating Finite Element Analysis (FFEA) [1].

We will explain the physical principles underlying FFEA and provide a practical overview of how a typical FFEA simulation is set up and executed. We will then demonstrate how FFEA can be used to model flexible biomolecular complexes from EM and other structural data using our simulations of the molecular motors and protein self-assembly as illustrative examples. We then speculate how FFEA might be integrated with atomistic models to provide a multi-scale description of biomolecular structure and dynamics.

1. Oliver R., Read D. J., Harlen O. G. & Harris S. A. “A Stochastic finite element model for the dynamics of globular macromolecules”, (2013) J. Comp. Phys. 239, 147-165.