Quivers are another name for directed graphs. In this talk we will define what a quiver representation is and explain some basic results, constructions, and examples of them. One major result is Gabriel's theorem, which says that a quiver has finitely many indecomposable representations if and only if it is a simply laced Dynkin diagram. We will explain how these indecomposable representations are in bijection with the positive roots of the corresponding simple lie algebra. Finally we will construct quiver varieties as a way to describe the space of representations of a quiver and show how the Springer resolution in type A appears as an example.

The bi-orderability of link groups has become a fascinating research topic. In this talk, we will survey some useful tools that have been developed to investigate the bi-orderabily of link complements. In particular, we will apply these techniques to the double twist link groups.

Random maps are natural models of random discrete surfaces. Random planar maps have been known for fifteen years to converge in the Benjamini--Schramm sense to infinite objects with fractal properties such as the UIPT. We will treat the case of random maps with genus proportional to the size, where limiting objects exhibit hyperbolic properties. Based on joint works with Baptiste Louf.

A big surface S means one whose pi1 is not finitely generated. For example, S could have infinitely many punctures, or infinitely many handles, or contain a disk minus Cantor set. In any of these cases, the mapping class group of S turns out not to satisfy what is called the "Tits Alternative". (And these cases cover almost all big surfaces.)

We will give an overview of big surfaces and the Tits Alternative. This "alternative" is one of the standard properties that a group might or might not have. We will also introduce a famous group found by Grigorchuk, that plays a central role in the proof of the theorem.

Weil reciprocity is a classical result in the algebraic geometry of curves. It defines a bilinear pairing, known as the tame symbol, which is related to Poincaré duality and geometric local class field theory. In this talk we will discuss Deligne's generalization of the tame symbol to open Riemann surfaces. We will see that there is a complex analytic version of Weil Reciprocity, and that this circle of ideas also extends to the rigid analytic world.

The classical results of Modica and Mortola show that the minimizers of the Allen-Cahn energy converge to minimizers of the area functional, under certain assumptions. In this talk, we will start exploring how this result can be extended for the case of the classical isoperimetric problem. In the first part of the talk, we will discuss briefly the properties of the solutions of this functional isoperimetric problem: existence, uniqueness, regularity and convergence. In the second part, we will present global and local stability estimates for these solutions (estimates for the distance of almost minimizer to the minimizer in terms of their difference in energy) , recovering the classical results for the isoperimetric problem as the limiting case.