A commutative-by-finite Hopf algebra is an extension of a commutative Hopf algebra by a finite dimensional Hopf algebra. Examples include many large classes of algebras - commutative Hopf algebras and finite Hopf algebras of course,  group algebras of abelian-by-finite groups, enveloping algebras of Lie algebras in positive characteristic, quantum groups at a root of unity,.... I'll review aspects of their structure and their representation theory, and mention a number of open questions. I'll aim to make the talk accessible without previous knowledge of Hopf algebras.

After a short history of this problem we will first give some examples of matrices that are always product of idempotent matrices. We will then present the case of matrices over a division rings, local rings, quasi-euclidean rings. In the second part of the talk we will consider the case of singular nonnegative matrices (over the real numbers) and examine when they are product of nonnegative idempotent matrices. We will end this talk mentioning nice results obtained by Hannah and O'Meara about decomposing elements of a regular rings into idempotent.

I will introduce Serre's uniformity problem and try to convey part of its importance by showing how a solution to it may be useful when dealing with some diophantine equations. In the end, I will present some of my contribution to the study of this question.

Ramsey theory is a branch of combinatorics whose aim is to find some "order" inside chaos, more precisely one is usually concerned in finding large monochromatic substructures in any finite colouring of a ‘rich’ structure.

A right R-module M is called a Utumi Module (U-module) if, whenever A and B are isomorphic submodules of M with A ∩ B = 0, there exist two summands K and T of M such that A is an essential submodule of K, B is an essential submodule of T and K ⊕ T is a direct summand of M . The class of U -modules is a simultaneous and strict generalization of three fundamental classes of modules; namely the quasi-continuous, the square-free and the automorphism-invariant modules.