If the abundances of the constituent molecules of a biochemical reaction system  are sufficiently high then their concentrations are typically modelled by a coupled set of ordinary differential equations (ODEs).  If, however, the abundances are low then the standard deterministic models do not provide a good representation of the behaviour of the system and stochastic models are used.  In this talk, I will first introduce both the stochastic and deterministic models.  I will then provide theorems that allow us to determine the qualitative behaviour of the underlying mathematical models from easily checked properties of the associated reaction network.  I will present results pertaining to so-called ``complex-balanced'' models and those satisfying ``absolute concentration robustness'' (ACR).  In particular, I will show how  ACR models, which are stable when modelled deterministically, necessarily undergo an extinction event in the stochastic setting.  I will then characterise the behaviour of these models prior to extinction.

I will describe a construction of the Weinstein symplectic category of Lagrangian correspondences in the context of shifted symplectic geometry. I will then explain how one can linearize this category starting from a "quantization" of  (-1)-shifted symplectic derived stacks: we assign a perverse sheaf to each (-1)-shifted symplectic derived stack (already done by Joyce and his collaborators) and a map of perverse sheaves to each (-1)-shifted Lagrangian correspondence (still conjectural).

In dimension three, convex surface theory implies that every tight contact structure on a connected sum M # N can be constructed as a connected sum of tight contact structures on M and N. I will explain some examples showing that this is not true in any dimension greater than three.  The proof is based on a recent higher-dimensional version of a classic result of Eliashberg about the symplectic fillings of contact manifolds obtained by subcritical surgery. This is joint work with Paolo Ghiggini and Klaus Niederkrüger.

Adaptive cubic regularization methods have recently emerged as a credible alternative to line search and trust-region for smooth nonconvex optimization, with optimal complexity amongst second-order methods. Here we consider a general class of adaptive regularization methods, that use first- or higher-order local Taylor models of the objective regularized by a(ny) power of the step size. We investigate the worst-case complexity/global rate of convergence of these algorithms, in the presence of varying (unknown) smoothness of the objective. We find that some methods automatically adapt their complexity to the degree of smoothness of the objective; while others take advantage of the power of the regularization step to satisfy increasingly better bounds with the order of the models. This work is joint with Nick Gould (RAL) and Philippe Toint (Namur).