In this series of three lectures, we will discuss two important and relatively new methods in combinatorics. Firstly, the probabilistic method developed by Erdős and which has now seen numerous applications in various areas of mathematics such as number theory, linear algebra, additive combinatorics, real analysis, as well as in computer science. Secondly, we will give an introduction to the polynomial method and present few surprising applications of linear algebra type-arguments to problems in combinatorics and geometry.

The speaker will be presenting some of the results from his DPhil thesis (2017, University of Oxford). In particular, the talk will be centered on the problem of determining point-to-point reachability for discrete linear time-invariant dynamical systems, when the set of controls is either a convex polyhedron or a finite union of convex polyhedra. The speaker will present a proof that the latter case is undecidable, by encoding Hilbert's Tenth Problem; time permitting, a proof of hardness of the former case will also be presented.

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.