Point-free topology can often seem like an algebraic almost-topology, 
> not quite the same but still interesting to those with an interest in 
> it. There is also a tradition of it in computer science, traceable back 
> to Scott's topological model of the untyped lambda-calculus, and 
> developing through Abramsky's 1987 thesis. There the point-free approach 
> can be seen as giving new insights (from a logic of observations), 
> albeit in a context where it is equivalent to point-set topology. It was 
> in that tradition that I wrote my own book "Topology via Logic".
> 
> Absent from my book, however, was a rather deeper connection with logic 
> that was already known from topos theory: if one respects certain 
> logical constraints (of geometric logic), then the maps one constructs 
> are automatically continuous. Given a generic point x of X, if one 
> geometrically constructs a point of Y, then one has constructed a 
> continuous map from X to Y. This is in fact a point-free result, even 
> though it unashamedly uses points.
> 
> This "continuity via logic", continuity as geometricity, takes one 
> rather further than simple continuity of maps. Sheaves and bundles can 
> be understood as continuous set-valued or space-valued maps, and topos 
> theory makes this meaningful - with the proviso that, to make it run 
> cleanly, all spaces have to be point-free. In the resulting fibrewise 
> topology via logic, every geometric construction of spaces (example: 
> point-free hyperspaces, or powerlocales) leads automatically to a 
> fibrewise construction on bundles.
> 
> I shall present an overview of this framework, as well as touching on 
> recent work using Joyal's Arithmetic Universes. This bears on the logic 
> itself, and aims to replace the geometric logic, with its infinitary 
> disjunctions, by a finitary "arithmetic type theory" that still has the 
> intrinsic continuity, and is strong enough to encompass significant 
> amounts of real analysis.

Last year, G. Kings and D. Rossler related the degree zero part of the polylogarithm
on abelian schemes pol^0 with another object previously defined by V. Maillot and D.
Rossler. More precisely, they proved that the canonical class of currents constructed
by Maillot and Rossler provides us with the realization of pol^0 in analytic Deligne
cohomology.
I will show that, adding some properness conditions, it is possible to give a
refinement of Kings and Rossler’s result involving Deligne-Beilinson cohomology
instead of analytic Deligne cohomology.

The wall-crossing behaviour of Donaldson-Thomas invariants in CY3 categories is controlled by a beautiful formula involving the group of automorphisms of a symplectic algebraic torus. This formula invites one to solve a certain Riemann-Hilbert problem. I will start by explaining how to solve this problem in the simplest possible case (this is undergraduate stuff!). I will then talk about a more general class of examples of the wall-crossing formula involving moduli spaces of quadratic differentials.

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.