Publications and Preprints

Correlation bounds for fields and matroids Correlation bounds for fields and matroids [arXiv:1806.02675]
Abstract
Let G be a finite connected graph, and let T be a spanning tree of G chosen uniformly at random. The work of Kirchhoff on electrical networks can be used to show that the events e1∈T and e2∈T are negatively correlated for any distinct edges e1 and e2. What can be said for such events when the underlying matroid is not necessarily graphic? We use Hodge theory for matroids to bound the correlation between the events e∈B, where B is a randomly chosen basis of a matroid. As an application, we prove Mason's conjecture that the number of k-element independent sets of a matroid forms an ultra-log-concave sequence in k.
with June Huh and Botong Wang
The correlation constant of a field The correlation constant of a field [arXiv:1804.03071]
Abstract
We study the correlation of edges, vectors or elements to be in a randomly chosen spanning tree or a basis, respectively. Here we follow the guideline of Huh and Wang and introduce as a measure an invariant that is called the correlation constant of a graph, vector configuration, matroid or field. It follows from one of their results that these correlation constants are numbers between 0 and 2. Here, we show that the correlation constant of every field is at least 8/7. In our proof we explicitly construct vector configurations and matroids with positively correlated elements.
Multi-splits and tropical linear spaces from nested matroids Multi-splits and tropical linear spaces from nested matroids [arXiv:1707.02814]
Abstract
In this paper we present an explicit combinatorial description of a special class of facets of the secondary polytopes of hypersimplices. These facets correspond to polytopal subdivisions called multi-splits. We show a relation between the cells in a multi-split of the hypersimplex and nested matroids. Moreover, we get a description of all multi-splits of a product of simplices. Additionally, we present a computational result to derive explicit lower bounds on the number of facets of secondary polytopes of hypersimplices.
Algorithms for Tight Spans and Tropical Linear Spaces Algorithms for Tight Spans and Tropical Linear Spaces [arXiv:1612.03592]
Abstract
We describe a new method for computing tropical linear spaces and more general duals of polyhedral subdivisions. It is based on Ganter's algorithm (1984) for finite closure systems.
with Simon Hampe and Michael Joswig
Matroids from hypersimplex splits Matroids from hypersimplex splits
Abstract
A class of matroids is introduced which is very large as it strictly contains all paving matroids as special cases. As their key feature these split matroids can be studied via techniques from polyhedral geometry. It turns out that the structural properties of the split matroids can be exploited to obtain new results in tropical geometry, especially on the rays of the tropical Grassmannians.
with Michael Joswig
The degree of a tropical basis The degree of a tropical basis
Abstract
We give an explicit upper bound for the degree of a tropical basis of a homogeneous polynomial ideal. As an application f-vectors of tropical varieties are discussed. Various examples illustrate differences between Gröbner and tropical bases.
with Michael Joswig
Linear Programs and Convex Hulls Over Fields of Puiseux Fractions Linear Programs and Convex Hulls Over Fields of Puiseux Fractions
Abstract
We describe the implementation of a subfield of the field of formal Puiseux series in polymake. This is employed for solving linear programs and computing convex hulls depending on a real parameter. Moreover, this approach is also useful for computations in tropical geometry.
with Michael Joswig, Georg Loho and Benjamin Lorenz

Thesis

Matroidal subdivisions, Dressians and tropical Grassmannians Matroidal subdivisions, Dressians and tropical Grassmannians
In this thesis we study various aspects of tropical linear spaces and their moduli spaces, the tropical Grassmannians and Dressians. Tropical linear spaces are dual to matroid subdivisions. Motivated by the concept of splits, the simplest case of a subdivision, a new class of matroids is introduced, which can be studied via techniques from polyhedral geometry. This class is very large as it strictly contains all paving matroids. The structural properties of these split matroids can be exploited to obtain new results in tropical geometry, especially on the rays of the tropical Grassmannians and the dimension of the Dressian. In particular, a relation between matroid realizability and certain tropical linear spaces is elaborated. The rays of a Dressian correspond to facets of the secondary polytope of a hypersimplex. A special class of facets is obtained by a generalization of splits, called multi-splits or originally, in Herrmann’s work, k-splits. We give an explicit combinatorial description of all multi-splits of the hypersimplex. These are in correspondence to nested matroids and, via the tropical Stiefel map, also to multi-splits of products of simplices. Hence, we derive a description for all multi-splits of a product of simplices. Moreover, a computational result leads to explicit lower bounds on the total number of facets of secondary polytopes of hypersimplices. Other computational aspects are also part of our research: A new method for computing tropical linear spaces and more general duals of polyhedral subdivisions is developed and implemented in the software polymake. This is based on Ganter’s algorithm (1984) for finite closure systems. Additionally, we describe the implementation of a subfield of the field of formal Puiseux series. This is employed for solving linear programs and computing convex hulls depending on a real parameter. Moreover, this approach is useful for computations in convex and algebraic tropical geometry. Tropical varieties, as for example tropical linear spaces or tropical Grassmannians, are intersections of finitely many tropical hypersurfaces. The set of corresponding polynomials is a tropical basis. We give an explicit upper bound for the degree of a general tropical basis of a homogeneous polynomial ideal. As an application f-vectors of tropical varieties are discussed. Various examples illustrate differences between Gröbner bases and tropical bases.
and the slides of my doctoral defense are available here

Poster and Others

Dressians and tropical Grassmannians Dressians and tropical Grassmannians [Poster for MEGA2015]
Poster Dressians and tropical Grassmannians
presented at MEGA2015
Tropical Grassmannian TropGr(2,6) Tropical Grassmannian TropGr(2,6) [Model]
The tropical Grassmannian TropGr(2,6)
interactive model for the DGD Gallery with Michael Joswig