LMS & LAG colloquium:

Abstract:The abelian sandpile model was expected to be scale-invariant since
its invention. Indeed, this property is the crucial part of the
self-organised criticality paradigm whose illustration was the purpose
to introduce the sandpile model. However, no precise renormalization
procedure was known until very recently. In our joint work with Moritz
Lang we suggest one: it goes through the extended sandpile group -- a
tropical abelian variety defined over Z constructed from a discrete
domain which is functorial under étale coverings. In this framework,
the classical sandpile group is seen as a subgroup of integer points
of the extended one and the renormalization is the truncation of the
homomorphism associated to the domain inclusion.

Abstract: The purpose of these two lectures is to report on recent 
results about Hochschild-Kostant-Rosenberg theorems over any base ring. For this, I'll 
start by presenting an object called the "filtered circle" as well as its relations to Witt vectors.
I will use it in order to produce filtered loop spaces and extract from them the 
Hochschild-Kostant-Rosenberg theorem. In a second part of the lectures I will apply this theorem to the 
definition and the construction of shifted symplectic structures in non-zero
characteristic situations. If time permit I'll also explain how to define singular supports for
bounded coherent complexes over arbitrary local complete interections schemes.
(joint work with T. Moulinos and M. Robalo).