In this thesis we apply methods from algebraic topology on questions from geometry, combinatorics and functional analysis. First we study amplituhedra – images of the totally nonnegative Grassmannians under projections that are induced by linear maps. They were introduced in Physics by Arkani-Hamed & Trnka (Journal of High Energy Physics, 2014) as model spaces that should provide a better understanding of the scattering amplitudes of quantum field theories. The topology of the amplituhedra has been known only in a few special cases, where they turned out to be homeomorphic to balls. Amplituhedra are special cases of Grassmann polytopes introduced by Lam (Current developments in mathematics 2014, Int. Press). We show that some further amplituhedra are homeomorphic to balls, and that some more Grassmann polytopes and amplituhedra are contractible. Next we study equipartitions of measures in a Euclidean space by certain families of convex sets. Our first result gives partitions of the ambient space into convex prisms – products of convex sets, that equipart a given set of measures, and our second result gives partitions of the Euclidean space by regular linear fans, that also equipart a given set of measures. The next result is a continuous analogue of the conjecture of Holmsen, Kynčl and Valculescu (Computational Geometry, 2017). For given a large enough family of positive finite absolutely continuous measures in the Euclidean space, we prove that there exists a partition of the ambient space, such that every set in the partition has positive measure with respect to at least c of the given measures, where we allow c to be greater than the dimension of the ambient Euclidean space. Additionally, we obtain an equipartition of one of the measures. The proof relies on a configuration space/test map scheme that translates this problem into a novel question from equivariant topology: We show non-existence of equivariant maps from the ordered configuration space into the union of an affine arrangement. Furthermore, we prove an extension of the Gromov’s theorem on the waists of spheres (Geometric and Functional Analysis, 2003). Gromov showed that for every n > k ≥ 1 and for every continuous map f : S n → R k from a sphere to a Euclidean space, there exists a point z ∈ R k , such that the volume of the tubular neighborhood of the inverse image f −1 (z) is at least as large as the volume of the tubular neighborhood of the (n − k)-dimensional equatorial sphere. We show that if the map f is Z p -equivariant for a prime p, and if the action of Z p on S n and R k satisfies certain properties, one can choose z in Gromov’s theorem to be the origin in R k . Finally, we study oriented matroid Grassmannians, also called MacPhersonians. An oriented matroid Grassmannian is the order complex of the set of all oriented matroids of a fixed rank and a fixed number of elements, ordered by weak maps. They were introduced by MacPherson (Topological Methods in Modern Mathematics, 1993), and firstly used by Gel’fand and MacPherson to give a combinatorial formula for Pontrjagin classes. For a given rank and a number of elements, the MacPhersonian is conjectured to be homotopy equivalent to the corresponding Grassmannian. We give some computational evidence in rank 3 and 4 that support the conjecture.
