Oriented Matroids and Combinatorial Neural Codes

A combinatorial neural code $\mathscr C\subseteq 2^{[n]}$ is convex if it arises as the intersection pattern of convex open subsets of $\mathbb R^d$. We relate the emerging theory of convex neural codes to the established theory of oriented matroids, both categorically and with respect to geometry and computational complexity. On the categorical side, we show that the map taking an acyclic oriented matroid to the code of positive parts of its topes is a faithful functor. We adapt the oriented matroid ideal introduced by Novik, Postnikov, and Sturmfels into a functor from the category of oriented matroids to the category of rings; then, we show that the resulting ring maps naturally to the neural ring of the matroid's neural code. For geometry and computational complexity, we show that a code has a realization with convex polytopes if and only if it lies below the code of a representable oriented matroid in the partial order of codes introduced by Jeffs. We show that previously published examples of non-convex codes do not lie below any oriented matroids, and we construct examples of non-convex codes lying below non-representable oriented matroids. By way of this construction, we can apply Mn\"{e}v-Sturmfels universality to show that deciding whether a combinatorial code is convex is NP-hard

Finding branch-decompositions of matroids, hypergraphs, and more

Given $n$ subspaces of a finite-dimensional vector space over a fixed finite field $\mathcal F$, we wish to find a "branch-decomposition" of these subspaces of width at most $k$, that is a subcubic tree $T$ with $n$ leaves mapped bijectively to the subspaces such that for every edge $e$ of $T$, the sum of subspaces associated with leaves in one component of $T-e$ and the sum of subspaces associated with leaves in the other component have the intersection of dimension at most $k$. This problem includes the problems of computing branch-width of $\mathcal F$-represented matroids, rank-width of graphs, branch-width of hypergraphs, and carving-width of graphs. We present a fixed-parameter algorithm to construct such a branch-decomposition of width at most $k$, if it exists, for input subspaces of a finite-dimensional vector space over $\mathcal F$. Our algorithm is analogous to the algorithm of Bodlaender and Kloks (1996) on tree-width of graphs. To extend their framework to branch-decompositions of vector spaces, we developed highly generic tools for branch-decompositions on vector spaces. The only known previous fixed-parameter algorithm for branch-width of $\mathcal F$-represented matroids was due to Hlin\v{e}n\'y and Oum (2008) that runs in time $O(n^3)$ where $n$ is the number of elements of the input $\mathcal F$-represented matroid. But their method is highly indirect. Their algorithm uses the non-trivial fact by Geelen et al. (2003) that the number of forbidden minors is finite and uses the algorithm of Hlin\v{e}n\'y (2005) on checking monadic second-order formulas on $\mathcal F$-represented matroids of small branch-width. Our result does not depend on such a fact and is completely self-contained, and yet matches their asymptotic running time for each fixed $k$.Comment: 73 pages, 10 figure

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

Combinatorial geometry of neural codes, neural data analysis, and neural networks

This dissertation explores applications of discrete geometry in mathematical neuroscience. We begin with convex neural codes, which model the activity of hippocampal place cells and other neurons with convex receptive fields. In Chapter 4, we introduce order-forcing, a tool for constraining convex realizations of codes, and use it to construct new examples of non-convex codes with no local obstructions. In Chapter 5, we relate oriented matroids to convex neural codes, showing that a code has a realization with convex polytopes iff it is the image of a representable oriented matroid under a neural code morphism. We also show that determining whether a code is convex is at least as difficult as determining whether an oriented matroid is representable, implying that the problem of determining whether a code is convex is NP-hard. Next, we turn to the problem of the underlying rank of a matrix. This problem is motivated by the problem of determining the dimensionality of (neural) data which has been corrupted by an unknown monotone transformation. In Chapter 6, we introduce two tools for computing underlying rank, the minimal nodes and the Radon rank. We apply these to analyze calcium imaging data from a larval zebrafish. In Chapter 7, we explore the underlying rank in more detail, establish connections to oriented matroid theory, and show that computing underlying rank is also NP-hard. Finally, we study the dynamics of threshold-linear networks (TLNs), a simple model of the activity of neural circuits. In Chapter 9, we describe the nullcline arrangement of a threshold linear network, and show that a subset of its chambers are an attracting set. In Chapter 10, we focus on combinatorial threshold linear networks (CTLNs), which are TLNs defined from a directed graph. We prove that if the graph of a CTLN is a directed acyclic graph, then all trajectories of the CTLN approach a fixed point.Comment: 193 pages, 69 figure

Proceedings of the 26th International Symposium on Theoretical Aspects of Computer Science (STACS'09)

The Symposium on Theoretical Aspects of Computer Science (STACS) is held alternately in France and in Germany. The conference of February 26-28, 2009, held in Freiburg, is the 26th in this series. Previous meetings took place in Paris (1984), Saarbr¨ucken (1985), Orsay (1986), Passau (1987), Bordeaux (1988), Paderborn (1989), Rouen (1990), Hamburg (1991), Cachan (1992), W¨urzburg (1993), Caen (1994), M¨unchen (1995), Grenoble (1996), L¨ubeck (1997), Paris (1998), Trier (1999), Lille (2000), Dresden (2001), Antibes (2002), Berlin (2003), Montpellier (2004), Stuttgart (2005), Marseille (2006), Aachen (2007), and Bordeaux (2008). ..