Mathematics Winter School 

26/02/2021, 15:00 — 16:00 — Online
Catarina Carvalho, Instituto Superior Técnico

The index of an equation

If we are given a linear system of equations on finite dimensional spaces, we know from Linear Algebra how many solutions there are and how to find them. On the other hand, most mathematical modelling involves the analysis of some system of differential equations describing our model; here the solutions are functions, living in an infinite dimensional world. Finding the actual solutions may be very hard in practice, but if we are lucky, we have a powerful tool: there is an integer, called the index, that gives us information about the number of solutions without computing them. In the right setting, this number is very stable and behaves well under many natural transformations.

In this talk, we place this invariant in the setting of Operator theory, that is, the study of linear maps between normed vector spaces, let it be Euclidean space, spaces of sequences or function spaces. We make a tour through some of the main concepts and examples, and consider the class of Fredholm operators, the ones that lead to a well-defined index. We see what stability means and explain how the strong properties of the index lead in some cases to remarkable index formulas, depending on the shape of the space our model takes place, thus establishing a strong link between analysis and topology.