Positive Configuration Space

We define and study the totally nonnegative part of the Chow quotient of the Grassmannian, or more simply the nonnegative configuration space. This space has a natural stratification by positive Chow cells, and we show that nonnegative configuration space is homeomorphic to a polytope as a stratified space. We establish bijections between positive Chow cells and the following sets: (a) regular subdivisions of the hypersimplex into positroid polytopes, (b) the set of cones in the positive tropical Grassmannian, and (c) the set of cones in the positive Dressian. Our work is motivated by connections to super Yang–Mills scattering amplitudes, which will be discussed in a sequel.

WHEN ARE THERE ENOUGH PROJECTIVE PERVERSE SHEAVES?

Let X be a topologically stratified space, p be any perversity on X and k be a field. We show that the category of p-perverse sheaves on X, constructible with respect to the stratification and with coefficients in k, is equivalent to the category of finite-dimensional modules over a finite-dimensional algebra if and only if X has finitely many strata and the same holds for the category of local systems on each of these. The main component in the proof is a construction of projective covers for simple perverse sheaves.

Reconstructing linearly embedded graphs: A first step to stratified space learning

<p style='text-indent:20px;'>In this paper, we consider the simplest class of stratified spaces – linearly embedded graphs. We present an algorithm that learns the abstract structure of an embedded graph and models the specific embedding from a point cloud sampled from it. We use tools and inspiration from computational geometry, algebraic topology, and topological data analysis and prove the correctness of the identified abstract structure under assumptions on the embedding. The algorithm is implemented in the Julia package Skyler, which we used for the numerical simulations in this paper.</p>

THE STRATIFIED STRUCTURE OF SPACES OF SMOOTH ORBIFOLD MAPPINGS

We consider four notions of maps between smooth C∞ orbifolds [Formula: see text], [Formula: see text] with [Formula: see text] compact (without boundary). We show that one of these notions is natural and necessary in order to uniquely define the notion of orbibundle pullback. For the notion of complete orbifold map, we show that the corresponding set of Cr maps between [Formula: see text] and [Formula: see text] with the Cr topology carries the structure of a smooth C∞ Banach (r finite)/Fréchet (r = ∞) manifold. For the notion of complete reduced orbifold map, the corresponding set of Cr maps between [Formula: see text] and [Formula: see text] with the Cr topology carries the structure of a smooth C∞ Banach (r finite)/Fréchet (r = ∞) orbifold. The remaining two notions carry a stratified structure: The Cr orbifold maps between [Formula: see text] and [Formula: see text] is locally a stratified space with strata modeled on smooth C∞ Banach (r finite)/Fréchet (r = ∞) manifolds while the set of Cr reduced orbifold maps between [Formula: see text] and [Formula: see text] locally has the structure of a stratified space with strata modeled on smooth C∞ Banach (r finite)/Fréchet (r = ∞) orbifolds. Furthermore, we give the explicit relationship between these notions of orbifold map. Applying our results to the special case of orbifold diffeomorphism groups, we show that they inherit the structure of C∞ Banach (r finite)/Fréchet (r = ∞) manifolds. In fact, for r finite they are topological groups, and for r = ∞ they are convenient Fréchet Lie groups.