Barcelona Introduction to Mathematical Research 2023 Summer Program

What algebraic topology can do for you

In this minicourse, the goal is to introduce the basic idea underlying the research area of algebraic topology. Roughly speaking, algebraic topology provides algebraic tools to distinguish/detect  properties of topological spaces up to continuous deformations. The notion of continuous deformation is formalized in the concept of homotopy which leads to the the area of homotopy theory. First I will introduce the notion of homotopy theory and how it can be used to define well-behaved algebraic structures associated to spaces (homotopy groups, homology) which allow to translate ‘equations’ in topology to ‘equations’ in algebraic settings where more structure can be used. I will finish with a couple of relevant examples on how these tools where crucial to solve questions which apparently are not related to algebraic topology.

Linear representations of finite groups with some classical applications

One of the main tools for the study of groups is representation and character theory. In this course, we will learn the rudiments of the theory of representations of finite groups over the complex numbers. In the first part, we will cover the basic material: the complete reducibility theorem, the orthogonality relations, the group algebra, the integrality properties of characters, and the induction of characters. In the second part, we will apply this theory to provide beautiful and elegant proofs of some classical results about finite groups (the theorems of Burnside, Minkowski, and Frobenius). We will not assume the audience has previous knowledge of Galois theory or algebraic number theory: instead we will briefly introduce along the way the notions from these areas that become necessary for our discussion.

Some pearls around Roth’s theorem
Roth’s theorem is one of the cornerstones in combinatorial number theory. It states that it is not possible to have dense sets of integers avoiding (non-trivial) 3-arithmetic progressions. This theorem, which solves the first non-trivial case of an important conjecture due to Erdős and Turán, has been very influential in the development of modern combinatorics, and several  Field medallists and Abel prize awardees had worked on that particular problem (Roth himself Bourgain, Gowers,Tao and Szemerédi, among others).

In this short course we will see a couple of proofs of this result (combinatorial and analytic) and exploit what in combinatorics we know as the dichotomy “structureness-randomness”. Limitations of the methods in order to deal with longer arithmetic progressions will be discussed as well.