Modular curves are moduli spaces of central importance in arithmetic geometry. In this talk, I will introduce these geometric objects and present some number theoretic results whose proofs used them in an essential way.

A gendo-symmetric algebra is the endomorphism ring of a generator over a symmetric algebra. After discussing various characterisations and examples, I will explain a comultiplication on these algebras.

This is joint work with Ming Fang.

The representation dimension of an algebra was introduced by Maurice Auslander in the 70's of the last century with the aim of measuring how far an algebra is to be representation-finite. Recall that an algebra A is representation finite provided there are only finitely many non-isomorphic indecomposable finitely generated A-modules.

Ernest Kummer's work on Fermat's last theorem (FLT) basically forms the starting point of algebraic number theory. Although FLT was finally settled by very different means, the classical approaches are still interesting to look at. We will look at the approaches to the first case of FLT, by Kummer, Mirimanoff and others.

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.