Computation and Physics in Algebraic Geometry

Physics provides new, tantalizing problems that we solve by developing and implementing innovative and effective geometric tools in nonlinear algebra. The techniques we employ also rely on numerical and symbolic computations performed with computer algebra. First, we study solutions to the Kadomtsev-Petviashvili equation that arise from singular curves. The Kadomtsev-Petviashvili equation is a partial differential equation describing nonlinear wave motion whose solutions can be built from an algebraic curve. Such a surprising connection established by Krichever and Shiota also led to an entirely new point of view on a classical problem in algebraic geometry known as the Schottky problem. To explore the connection with curves with at worst nodal singularities, we define the Hirota variety, which parameterizes KP solutions arising from such curves. Studying the geometry of the Hirota variety provides a new approach to the Schottky problem. We investigate it for irreducible rational nodal curves, giving a partial solution to the weak Schottky problem in this case. Second, we formulate questions from scattering amplitudes in a broader context using very affine varieties and D-module theory. The interplay between geometry and combinatorics in particle physics indeed suggests an underlying, coherent mathematical structure behind the study of particle interactions. In this thesis, we gain a better understanding of mathematical objects, such as moduli spaces of point configurations and generalized Euler integrals, for which particle physics provides concrete, non-trivial examples, and we prove some conjectures stated in the physics literature. Finally, we study linear spaces of symmetric matrices, addressing questions motivated by algebraic statistics, optimization, and enumerative geometry. This includes giving explicit formulas for the maximum likelihood degree and studying tangency problems for quadric surfaces in projective space from the point of view of real algebraic geometry

Grassmannians, measure partitions and waists of spheres

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

The Templates for Some Classes of Quaternary Matroids

Subject to hypotheses based on the matroid structure theory of Geelen, Gerards, and Whittle, we completely characterize the highly connected members of the class of golden-mean matroids and several other closely related classes of quaternary matroids. This leads to a determination of the eventual extremal functions for these classes. One of the main tools for obtaining these results is the notion of a frame template. Consequently, we also study frame templates in significant depth.Comment: 83 pages; minor corrections in Version 4; accepted for publication by Journal of Combinatorial Theory, Series