Totally positive spaces : topology and applications

Abstract

This thesis studies topological spaces arising in total positivity. Examples include the totally nonnegative Grassmannian Gr[subscripts >_0](k, n), Lusztig's totally nonnegative part (G/P)[subscripts >_0] of a partial flag variety, Lam's compactification of the space of electrical networks, and the space of (boundary correlation matrices of) planar Ising networks. We show that all these spaces are homeomorphic to closed balls. In addition, we confirm conjectures of Postnikov and Williams that the CW complexes Gr[subscripts >_0](k, n) and (G/P)[subscripts >_0] are regular. This implies that the closure of each positroid cell inside Gr[subscripts >_0](k, n) is homeomorphic to a closed ball. We discuss the close relationship between the above spaces and the physics of scattering amplitudes, which has served as a motivation for most of our results. In the second part of the thesis, we investigate the space of planar Ising networks. We give a simple stratification-preserving homeomorphism between this space and the totally nonnegative orthogonal Grassmannian, describing boundary correlation matrices of the planar Ising model by inequalities. Under our correspondence, Kramers-Wannier's high/low temperature duality transforms into the cyclic symmetry of Gr[subscripts >_0](k, n).