abstract: Tropical geometry has important applications to both complex and real enumerative geometry. Mikhalkin correspondence theorems provide a way to translate some classical enumerative problems to tropical ones. I will state basic definitions and properties of tropical variety focusing on the case of curves. I will then explain how to tropicalize (complex and real) enumerative problems of the following type: What is the number of plane curves of degree d and genus g passing through 3d-1+g points? Finally, I plan to give a nice application of this correspondence to a real enumerative problem with tangency conditions.

abstract: According to Mikhalkin's Correspondance Theorem (cf Benoit Bertrand's lectures), classical enumerative problems (i.e. the art of counting curves) can be solved tropically. Moreover, the tropical appoach allows one to count real curves as well, what is in generally not possible with previous technics. The goal of these lectures is to explain why tropical enumerative geometry is easier than the classical one. First I will define some enumerative invariants in complex (Gromov Witten invariants) and real (Welschinger invariants) geometry, and I will explain how to get formulas to compute them via tropical geometry. These lectures are based on previous works by I. Itenberg, A. Gathmann, V. Kharlamov, H. Markwig, G. Mikhalkin, E. Shustin and myself.

abstract: I will first treat some basic theorems in tropical geometry, e.g., give various characterisations of the tropicalisation of a variety, and some insight into the Bieri-Groves theorem. Then I will zoom in on a nice application by myself and Karin Baur to secant varieties. Finally, I plan to treat a few interesting tropical constructions, e.g., Speyer-Sturmfels's tropical Grassmannian of
lines.

abstract: Cluster algebras were invented in 2000 by S. Fomin and A. Zelevinsky. The original motivations came from Lusztig's work on canonical bases in quantum groups and total positivity in algebraic groups. It has turned out that cluster algebras are related to a surprisingly large number of subjects, notably higher Teichmüller theory and quiver representation theory. In these lectures, we will introduce cluster algebras by a series of examples and explore their links with quiver representations following mainly work by Caldero-Chapoton, Buan-Marsh-Reiten, and, if time permits, Geiss-Leclerc-Schröer.

abstract: TBA