Yuri Safarov (King's College London)
ysafarov@mth.kcl.ac.uk
A (possibly infinite) matrix is said to be doubly stochastic if all its entries are non-negative and the sum of entries in every row and every column is equal to one. According to Birkhoff's theorem,
(1) the extreme points of the convex set of doubly stochastic matrices are permutation matrices and
(2) the set of doubly stochastic matrices coincides with the closed convex hull of the set of permutation matrices.
The main aim of the talk is to show that, under certain conditions, Birkhoff's theorem can be extended to weighted graphs (or countable families of discrete probability spaces which have nonempty intersections). We shall also discuss some applications to spectral theory.