Géométrie tropicale et algèbres amassées – GTAA

The aim of this project is to investigate interactions between tropical geometry and cluster theory, to develop their techniques, and to apply the acquired tools to study fundamental topics belonging to these intertwined domains. Among these topics, one can mention : tropicalization of Gromov-Witten, Welschinger and Solomon invariants, which should lead to a unified theory of these invariants; quantization and compactification of tropical and cluster varieties, which in particular should open a way to the use of cluster algebras in constructions of complex and real algebraic varieties; applications to complex and real enumerative geometries, including new recursive formulas and asymptotic study of enumerative invariants; exploration of the geometry of tropical and cluster varieties, which in particular should provide applications of theory of clusters in geometry of plane algebraic curves and vice versa; construction of distinguished bases in cluster algebras, which should yield in particular new ways of constructing canonical bases and, via duality, shed new light on tropical and cluster varieties; categorification of cluster algebras and varieties, which is the most promising tool to the solution of many tantalizing combinatorial conjectures on these objects.