Adrián Mateo

Project Title: Magnitude homology

One of the main concerns of mathematics is to measure size. This can mean different things in different contexts: for finite sets we count the number of elements, for objects in space we measure their volume, for possible outcomes we determine their probability, etc. It turns out that these and other mathematical properties can all be seen as special cases of one single property defined in an extraordinarily general setting: magnitude.

The generality of magnitude means that it can be specialised in many different contexts, as is the case of metric spaces (sets where one can measure distances between points), where it has uncovered a previously unknown notion of size. Magnitude homology is an alternative, more algebraic perspective that often can be used to study magnitude using already existing algebraic tools.

The main obstacle to further developments in magnitude homology is that it only agrees with magnitude when the spaces in consideration are finite. The first aim of this project is to expand the definition of magnitude homology so that this is no longer the case. Once achieved, the next step is to put this new theory to use. Some results about magnitude, which have difficult proofs requiring sophisticated analytical techniques, can likely be rephrased in the language of magnitude homology and given ‘simpler’ algebraic proofs.