In this Fridays@4 event – for this week renamed Fridays@1 (with lunchtime pizza) – Torkel Loman from the Mathematics Institute and Alastair McCullough from the Department of Computer Science will present their talks.

Torkel Loman
The behaviours of noisy feedback loops and where (in parameter space) to find them

Alastair McCullough 
Tech, Coffee, and the Regulation of Truth: An Enterprise Barista's Story

Torkel's abstract

Mixed positive/negative feedback loops (networks where a single component both activates and deactivates its own productions) are common across biological systems, and also the subject of this talk. Here (inspired by systems for e.g. bacterial antibiotics resistance), we create a minimal mathematical model of such a feedback loop. Our model (a stochastic delay differential equation) depends on only 6, biologically interpretable, parameters. We describe 10 distinct behaviours that such feedback loops can produce, and map their occurrence across 6-dimensional parameter space.

This talk is about a classical problem in differential geometry and global analysis: the isometric immersions of Riemannian manifolds into Euclidean spaces. We focus on the PDE approach to isometric immersions, i.e., the analysis of Gauss--Codazzi--Ricci equations, especially in the regime of low Sobolev regularity. Such equations are not purely elliptic, parabolic, or hyperbolic in general, hence calling for analytical tools for PDEs of mixed types. We discuss various recent contributions -- in line with the pioneering works by G.-Q. Chen, M. Slemrod, and D. Wang [Proc. Amer. Math. Soc. (2010); Comm. Math. Phys. (2010)] -- on the weak continuity of Gauss--Codazzi--Ricci equations, the weak stability of isometric immersions, and the fundamental theorem of submanifold theory with low regularity. Two mixed-type PDE techniques are emphasised throughout these developments: the method of compensated compactness and the theory of Coulomb--Uhlenbeck gauges.