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.