The role of tissue stiffness in controlling cell behaviours ranging from proliferation to signalling and activation is by now well accepted. A key focus of experimental studies into mechanotransduction are focal adhesions, localised patches of strong adhesion, where cell signalling has been established to occur. However, these adhesion sites themselves alter the mechanical equilibrium of the system determining the force balance and work done. To explore this I have developed an active matter continuum description of cellular contractility and will discuss recent results on the specific role of spatial positioning of adhesions in mechanotransduction. I show using energy arguments why the experimentally observed arrangements of focal adhesions develop and the implications this has for stiffness sensing and cellular contractility control. I will also show how adhesions play distinct roles in single cells and tissue layers respectively drawing on recent experimental work with Dr JR Davis (Manchester University) and Dr Nic Tapon (Crick Institute) with applications to epithelial layers and organoids.

I will discuss aspects of some work in progress with Tingxiang Zou, in which we continue the investigation of pseudofinite sets coarsely respecting structures of algebraic geometry, focusing on algebraic group actions. Using a version of Balog-Szemerédi-Gowers-Tao for group actions, we find quite weak hypotheses which rule out non-abelian group actions, and we are applying this to obtain new Elekes-Szabó results in which the general position hypothesis is fully weakened in one co-ordinate.

Optimal transport (OT) proves to be a powerful tool for non-parametric calibration: it allows us to take a favourite (non-calibrated) model and project it onto the space of all calibrated (martingale) models. The dual side of the problem leads to an HJB equation and a numerical algorithm to solve the projection. However, in general, this process is costly and leads to spiky vol surfaces. We are interested in special cases where the projection can be obtained semi-analytically. This leads us to the martingale equivalent of the seminal fluid-dynamics interpretation of the optimal transport (OT) problem developed by Benamou and Brenier. Specifically, given marginals, we look for the martingale which is the closest to a given archetypical model. If our archetype is the arithmetic Brownian motion, this gives the stretched Brownian motion (or the Bass martingale), studied previously by Backhoff-Veraguas, Beiglbock, Huesmann and Kallblad (and many others). Here we consider the financially more pertinent case of Black-Scholes (geometric BM) reference and show it can also be solved explicitly. In both cases, fast numerical algorithms are available.

Based on joint works with Julio Backhoff, Benjamin Joseph and Gregoire Leoper.  

This talk reports a work in progress. It will be done on a board.