Toric Ideals, Polytopes, and Convex Neural Codes

How does the brain encode the spatial structure of the external world? A partial answer comes through place cells, hippocampal neurons which become associated to approximately convex regions of the world known as their place fields. When an organism is in the place field of some place cell, that cell will fire at an increased rate. A neural code describes the set of firing patterns observed in a set of neurons in terms of which subsets fire together and which do not. If the neurons the code describes are place cells, then the neural code gives some information about the relationships between the place fields–for instance, two place fields intersect if and only if their associated place cells fire together. Since place fields are convex, we are interested in determining which neural codes can be realized with convex sets and in finding convex sets which generate a given neural code when taken as place fields. To this end, we study algebraic invariants associated to neural codes, such as neural ideals and toric ideals. We work with a special class of convex codes, known as inductively pierced codes, and seek to identify these codes through the Gröbner bases of their toric ideals

Tropical Geometry in Singular

Das Ziel dieser Dissertation ist die Entwicklung und Implementation eines Algorithmus zur Berechnung von tropischen Varietäten über allgemeine bewertete Körper. Die Berechnung von tropischen Varietäten über Körper mit trivialer Bewertung ist ein hinreichend gelöstes Problem. Hierfür kombinieren die Autoren Bogart, Jensen, Speyer, Sturmfels und Thomas eindrucksvoll klassische Techniken der Computeralgebra mit konstruktiven Methoden der konvexer Geometrie. Haben wir allerdings einen Grundkörper mit nicht-trivialer Bewertung, wie zum Beispiel den Körper der $p$-adischen Zahlen $\mathbb{Q}_p$, dann stößt die konventionelle Gröbnerbasentheorie scheinbar an ihre Grenzen. Die zugrundeliegenden Monomordnungen sind nicht geeignet um Problemstellungen zu untersuchen, die von einer nicht-trivialen Bewertung auf den Koeffizienten abhängig sind. Dies führte zu einer Reihe von Arbeiten, welche die gängige Gröbnerbasentheorie modifizieren um die Bewertung des Grundkörpers einzubeziehen.$$\phantom{newline}$$ In dieser Arbeit präsentieren wir einen alternativen Ansatz und zeigen, wie sich die Bewertung mittels einer speziell eingeführten Variable emulieren lässt, so dass eine Modifikation der klassischen Werkzeuge nicht notwendig ist. Im Rahmen dessen wird Theorie der Standardbasen auf Potenzreihen über einen Koeffizientenring verallgemeinert. Hierbei wird besonders Wert darauf gelegt, dass alle Algorithmen bei polynomialen Eingabedaten mit ihren klassischen Pendants übereinstimmen, sodass für praktische Zwecke auf bereits etablierte Softwaresysteme zurückgegriffen werden kann. Darüber hinaus wird die Konstruktion des Gröbnerfächers sowie die Technik des Gröbnerwalks für leicht inhomogene Ideale eingeführt. Dies ist notwendig, da bei der Einführung der neuen Variable die Homogenität des Ausgangsideal gebrochen wird.$$\phantom{newline}$$ Alle Algorithmen wurden in Singular implementiert und sind als Teil der offiziellen Distribution erhältlich. Es ist die erste Implementation, welches in der Lage ist tropische Varietäten mit $p$-adischer Bewertung auszurechnen. Im Rahmen der Arbeit entstand ebenfalls ein Singular Paket für konvexe Geometrie, sowie eine Schnittstelle zu Polymake

Exploiting Torus Actions: Immaculate Line Bundles on Toric Varieties and Parametrizations of Gröbner Cells

This dissertation contains two chapters on the use of torus actions in algebraic geometry. In chapter 2 we study ”immaculate line bundles” on projective toric varieties. The cohomology groups of those line bundles vanish in all degrees, including the 0-th degree. Immaculate line bundles can be seen as building blocks of full exceptional sequences of line bundles of the variety. All the immaculate line bundles of a toric variety X = TV(Σ) can be identified in two steps. First identify those subsets of the rays Σ(1) whose geometric realization is not k-acyclic, they will be called tempting. Those subsets of the rays give ”maculate sets/regions” in the class group of the variety. A line bundle is immaculate, if it is not in any of those maculate sets. So the first step in finding immaculate line bundles is to find all tempting subsets. When X is projective, the main result for this is that primitive collections – subsets of the rays that do not span a cone, but each proper subset spans a cone – are always tempting. And a subset of rays can only be tempting if it is the union of primitive collections. The same has to hold for the complement, too. We give descriptions of the immaculate line bundles for different examples. In particular, we describe the immaculate locus for projective toric varieties of Picard rank 3. Most of the results have been published in [ABKW20]. In chapter 3 we study the Hilbert scheme of n points in affine plane. It describes all ideals in the polynomial ring of two variables whose quotient is an n-dimensional vector space. The Hilbert scheme can be decomposed into so called Gröbner cells. They consist of those ideals that have a prescribed leading term ideal with respect to a given term order. The Gröbner cells for the lexicographic and the degree-lexicographic order are parametrized in [CV08] and [Con11], respectively, by canonical Hilbert-Burch matrices. A Hilbert-Burch matrix of an ideal is a matrix generating the syzygies of the ideal. Its maximal minors also generate the ideal. These results are generalized in two directions. Firstly, we consider the ring of formal power series. Here we give a parametrization of the cells that respects the Hilbert function stratification of the punctual Hilbert scheme. In particular, this cellular decomposition restricts to a cellular decomposition of the subscheme consisting of ideals with a prescribed Hilbert function. We use the parametrization to describe subsets of the Gröbner cells associated to lex-segment ideals with a given minimal number of generators. These subsets are quasi-affine varieties inside the cell. Most of these results have been published in [HW21] and [HW23]. The second way of changing the setting is to consider a general term order on the polynomial ring. We give a surjection to the Gröbner cell with respect to this ordering and parametrizations of subsets of the cell, as well as a conjecture how the parametrization of the whole cell should look like. We also study intersections of Gröbner cells with respect to different term orders.Die vorliegende Dissertation besteht aus zwei Kapiteln zu zwei unterschiedlichen Anwendungen von Toruswirkungen in der algebraischen Geometrie. Die wichtigsten Objekte des Kapitels 2 sind unbefleckte Geradenbündel auf projektiven torischen Varietäten X = TV(Σ), Geradenbündel, deren Kohomologiegruppen alle verschwinden. Unbefleckte Geradenbündel können als Bausteine für exzeptionelle Sequenzen aus Geradenbündeln dienen und somit die derivierte Kategorie der Varietät beschreiben. Die Bestimmung von unbefleckten Geradenbündeln lässt sich in zwei Schritte aufteilen. Es lassen sich Teilmengen der Strahlen Σ(1) des die torische Varietät beschreibenden Fächers Σ identifizieren, deren geometrische Realisierungen nicht k-azyklisch sind. Diese verlockenden Teilmengen der Strahlen definieren befleckte Teilmengen der Klassengruppe Cl(X). Ein Geradenbündel ist genau dann unbefleckt, wenn es in keiner befleckten Teilmenge von Cl(X) liegt. Die Bestimmung aller unbefleckten Geradenbündel lässt sich also in zwei Schritte aufteilen. Das Bestimmen der verlockenden Teilmengen der Strahlen und das Bestimmen der zugehörigen befleckten Regionen. Primitive Kollektionen – Teilmengen der Strahlen, die selbst keinen Kegel des Fächers aufspannen, aber jede ihrer Teilmenge spannt einen Kegel des Fächers auf – sind verlockend und außerdem ist eine Teilmenge nur dann verlockend, wenn sie eine Vereinigung von primitiven Kollektionen ist. Dies muss auch für das Komplement gelten. Wir geben die Beschreibung für die unbefleckten Geradenbündel für verschiedene Beispielklassen von projektiven torischen Varietäten. Insbesondere beschreiben wir die unbefleckten Geradenbündel für projektive torische Varietäten von Picardrang 3. Die meisten dieser Ergebnisse sind in [ABKW20] erschienen. In Kapitel 3 geht es um das Hilbertschema von n Punkten in der affinen Ebene. Seine Punkte sind Ideale im Polynomenring k[x, y], deren Quotient ein n-dimensionaler k-Vektorraum ist. Das Hilbertschema kann in sogenannte Gröbnerzellen unterteilt werden. Sie umfassen Ideale, die bezüglich einer Termordnung τ ein festgelegtes Leitideal haben. In [CV08] und [Con11] werden für die lexikographische und gradlexikographische Termordnung Parametrisierung der Gröbnerzellen durch kanonische Hilbert-Burch Matrizen angegeben. Hilbert-Burch Matrizen beschreiben die Syzygien des Ideals und ihre maximalen Minoren erzeugen das Ideal. Die Ergebnisse werden in zwei Richtungen verallgemeinert. Zunächst betrachten wir Ideale im Ring der formalen Potenzreihen. Wir geben eine Parametrisierung der Zellen, bei der die lokale Struktur der Ideale berücksichtigt wird. Insbesondere lässt sich diese zelluläre Unterteilung des lokalen Hilbertschemas auf eine zelluläre Unterteilung des Unterschemas einschränken, das nur Ideale mit einer gegebenen Hilbertfunktion beinhaltet. Durch diese Parametrisierung lassen sich für Ideale in diesen Zellen kanonische Hilbert-Burch Matrizen definieren. Diese benutzen wir um Teilmengen der Gröbnerzellen mit einer vorgegebenen minimalen Anzahl von Erzeugern zu beschreiben. Diese Teilmengen sind quasi-affine Varietäten in der Gröbnerzelle. Die meisten der Resultate sind in [HW21] und [HW23] erschienen. Die zweite Möglichkeit das Setting zu ändern, ist beliebige Termordnungen auf dem Polynomenring zu betrachten. Im zweiten Teil von Kapitel 3 geben wir eine Surjektion auf diese Gröbnerzellen, sowie Parametrisierungen von Teilmengen und geben eine Vermutung, wie eine Parametrisierung der ganzen Zelle aussieht. Außerdem untersuchen wir Schnitte von Gröbnerzellen bezüglich verschiedener Termordnungen

Tropical totally positive cluster varieties

We study the relation between the integer tropical points of a cluster variety (satisfying the full Fock-Goncharov conjecture) and the totally positive part of the tropicalization of an ideal presenting the corresponding cluster algebra. Suppose we are given a presentation of the cluster algebra by a Khovanskii basis for a collection of ${\bf g}$-vector valuations associated with several seeds related by mutations. In presence of a full rank fully extended exchange matrix we construct the rays of a subfan of the totally positive part of the tropicalization of the ideal that coincides combinatorially with the subgraph of the exchange graph of the cluster algebra corresponding to the collection of seeds. Moreover, geometric information about Gross-Hacking-Keel-Kontsevich's toric degenerations associated with seeds gets identified with the Gr\"obner toric degenerations obtained from maximal cones in the tropicalization. As application we prove a conjecture about the relation between Rietsch-Williams' valuations for Grassmannians arising from plabic graphs \cite{RW17} to Kaveh-Manon's work on valuations from the tropicalization of an ideal \cite{KM16}. In a second application we give a partial answer to the question if the Feigin-Fourier-Littelmann-Vinberg degeneration of the full flag variety in type $\mathtt A$ is isomorphic to a degeneration obtained from the cluster structure.Comment: Comments are very welcom

Gauge Backgrounds and Zero-Mode Counting in F-Theory

Computing the exact spectrum of charged massless matter is a crucial step towards understanding the effective field theory describing F-theory vacua in four dimensions. In this work we further develop a coherent framework to determine the charged massless matter in F-theory compactified on elliptic fourfolds, and demonstrate its application in a concrete example. The gauge background is represented, via duality with M-theory, by algebraic cycles modulo rational equivalence. Intersection theory within the Chow ring allows us to extract coherent sheaves on the base of the elliptic fibration whose cohomology groups encode the charged zero-mode spectrum. The dimensions of these cohomology groups are computed with the help of modern techniques from algebraic geometry, which we implement in the software gap. We exemplify this approach in models with an Abelian and non-Abelian gauge group and observe jumps in the exact massless spectrum as the complex structure moduli are varied. An extended mathematical appendix gives a self-contained introduction to the algebro-geometric concepts underlying our framework.Comment: 41 pages + extended appendice

Computing topological zeta functions of groups, algebras, and modules, II

Building on our previous work (arXiv:1405.5711), we develop the first practical algorithm for computing topological zeta functions of nilpotent groups, non-associative algebras, and modules. While we previously depended upon non-degeneracy assumptions, the theory developed here allows us to overcome these restrictions in various interesting cases.Comment: 33 pages; sequel to arXiv:1405.571

Parametric shortest-path algorithms via tropical geometry

We study parameterized versions of classical algorithms for computing shortest-path trees. This is most easily expressed in terms of tropical geometry. Applications include shortest paths in traffic networks with variable link travel times.Comment: 24 pages and 8 figure

Toric Varieties and Numerical Algorithms for Solving Polynomial Systems

This work utilizes toric varieties for solving systems of equations. In particular, it includes two numerical homotopy continuation algorithms for numerically solving systems of equations. The first algorithm, the Cox homotopy, solves a system of equations on a compact toric variety. The Cox homotopy tracks points in the total coordinate space of the toric variety and can be viewed as a homogeneous version of the polyhedral homotopy of Huber and Sturmfels. The second algorithm, the Khovanskii homotopy, solves a system of equations on a variety in the presence of a finite Khovanskii basis. This homotopy takes advantage of Anderson’s flat degeneration to a toric variety. The Khovanskii homotopy utilizes the Newton-Okounkov body of the system, whose normalized volume gives a bound on the number of solutions to the system. Both homotopy algorithms provide the computational advantage of tracking paths in a compact space while also minimizing the total number of paths tracked. The Khovanskii homotopy is optimal with respect to the number of paths tracked, and the Cox homotopy is optimal when the system is Bernstein-general

Tropical Positivity and Semialgebraic Sets from Polytopes

This dissertation presents recent contributions in tropical geometry with a view towards positivity, and on certain semialgebraic sets which are constructed from polytopes. Tropical geometry is an emerging field in mathematics, combining elements of algebraic geometry and polyhedral geometry. A key in establishing this bridge is the concept of tropicalization, which is often described as mapping an algebraic variety to its 'combinatorial shadow'. This shadow is a polyhedral complex and thus allows to study the algebraic variety by combinatorial means. Recently, the positive part, i.e. the intersection of the variety with the positive orthant, has enjoyed rising attention. A driving question in recent years is: Can we characterize the tropicalization of the positive part? In this thesis we introduce the novel notion of positive-tropical generators, a concept which may serve as a tool for studying positive parts in tropical geometry in a combinatorial fashion. We initiate the study of these as positive analogues of tropical bases, and extend our theory to the notion of signed-tropical generators for more general signed tropicalizations. Applying this to the tropicalization of determinantal varieties, we develop criteria for characterizing their positive part. Motivated by questions from optimization, we focus on the study of low-rank matrices, in particular matrices of rank 2 and 3. We show that in rank 2 the minors form a set of positive-tropical generators, which fully classifies the positive part. In rank 3 we develop the starship criterion, a geometric criterion which certifies non-positivity. Moreover, in the case of square-matrices of corank 1, we fully classify the signed tropicalization of the determinantal variety, even beyond the positive part. Afterwards, we turn to the study of polytropes, which are those polytopes that are both tropically and classically convex. In the literature they are also established as alcoved polytopes of type A. We describe methods from toric geometry for computing multivariate versions of volume, Ehrhart and h^*-polynomials of lattice polytropes. These algorithms are applied to all polytropes of dimensions 2,3 and 4, yielding a large class of integer polynomials. We give a complete combinatorial description of the coefficients of volume polynomials of 3-dimensional polytropes in terms of regular central subdivisions of the fundamental polytope, which is the root polytope of type A. Finally, we provide a partial characterization of the analogous coefficients in dimension 4. In the second half of the thesis, we shift the focus to study semialgebraic sets by combinatorial means. Intersection bodies are objects arising in geometric tomography and are known not to be semialgebraic in general. We study intersection bodies of polytopes and show that such an intersection body is always a semialgebraic set. Computing the irreducible components of the algebraic boundary, we provide an upper bound for the degree of these components. Furthermore, we give a full classification for the convexity of intersection bodies of polytopes in the plane. Towards the end of this thesis, we move to the study of a problem from game theory, considering the correlated equilibrium polytope $P_G$ of a game G from a combinatorial point of view. We introduce the region of full-dimensionality for this class of polytopes, and prove that it is a semialgebraic set for any game. Through the use of oriented matroid strata, we propose a structured method for classifying the possible combinatorial types of $P_G$, and show that for (2 x n)-games, the algebraic boundary of each stratum is a union of coordinate hyperplanes and binomial hypersurfaces. Finally, we provide a computational proof that there exists a unique combinatorial type of maximal dimension for (2 x 3)-games.:Introduction 1. Background 2. Tropical Positivity and Determinantal Varieties 3. Multivariate Volume, Ehrhart, and h^*-Polynomials of Polytropes 4. Combinatorics of Correlated Equilibri