Landscapes in Mathematical Sciences seminar,
University of Bath,
25 October 2013, and
Université Catholique de Louvain,
31 October 2013.
Abstract The magnitude of a square matrix is the sum of all the entries of its inverse. This strange definition, suitably used, produces a family of invariants in different contexts across mathematics. All of them can be loosely understood as "size". For example, the magnitude of a convex set is an invariant from which one can conjecturally recover many important classical quantities: volume, surface area, perimeter, and so on. The magnitude of a graph is a new invariant sharing features with the Tutte polynomial. The magnitude of a category is very closely related to the Euler characteristic of a topological space. Magnitude also appears in the difficult problem of quantifying biological diversity: under certain circumstances, the greatest possible diversity of an ecosystem is exactly its magnitude. I will give an aerial view of this landscape.
Slides In this pdf file.