The space of totally nonnegative real matrices, namely the real n by n matrices with all minors nonnegative, intersected with the ``unipotent radical'' of upper triangular matrices with 1's on the diagonal carries important information related to Lusztig's theory of canonical bases in representation theory.   This space of matrices (and generalizations of it beyond type A) is naturally stratified according to which minors are positive and which are 0, with the resulting stratified space described combinatorially by a well known partially ordered set called the Bruhat order.   I will tell the story of these spaces and in particular of a map from a simplex to these spaces that has recently been used to better understand them.  The fibers of this map encode exactly the nonnegative real relations amongst exponentiated Chevalley generators of a Lie algebra.   This talk will especially focus on recent joint work with Jim Davis and Ezra Miller uncovering overall combinatorial and topological structure governing these fibers.  Plenty of background, examples, and pictures will be provided along the way.