Categorification and quantisation of cluster type structures

The project fits into the area of representation theory which is
governed by symmetries. Representation theory seeks for
symmetries, in order to shed light on the observed objects, and also
to collect them into abstract entities, which aid to distinguish and
compare different objects. The main type of entities investigated in
this project are those modelled by a Lie algebra, whose prominent
role in physics is represented by infinitesimal symmetries of physical
systems.
We aim to look for different incarnations of this fundamental object.
We propose to unify these incarnations on a higher level. In
particular, we would like to understand how multiplication and
addition can be regarded as different phenomena of the same
underlying structure.
More precisely, we will mainly study cluster algebras, i.e. algebras
with a combinatorial flavour which encode information about a basis
of an ("extended") Lie algebra, associated to Lie algebras and we
intend to relate their additive and multiplicative categorifications, by
applying and developing (higher) geometric methods to
representation theory. This will also naturally bring us to ``tau-tilting
theory", which might be in turn viewed as yet another categorification
of cluster algebras. Hereby we study the behaviour of the tau-tilting
theory of an algebra after deformation. This is a method to build new
objects which share many characteristics but possibly with the
exception of a few key features.