Michael Joswig
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View article: Faster Algebraic Shifting
Faster Algebraic Shifting Open
Improved algorithms for computing (partial and full) exterior algebraic shifts of hypergraphs and simplicial complexes are presented. The main benefit is in positive characteristic. Experiments with an implementation in OSCAR are reported.
View article: The polyhedral geometry of truthful auctions
The polyhedral geometry of truthful auctions Open
The difference set of an outcome in an auction is the set of types that the auction mechanism maps to the outcome. We give a complete characterization of the geometry of the difference sets that can appear for a dominant strategy incentive…
View article: Wronski Pairs of Honeycomb Curves
Wronski Pairs of Honeycomb Curves Open
We study certain generic systems of real polynomial equations associated with triangulations of convex polytopes and investigate their number of real solutions. Our main focus is set on pairs of plane algebraic curves which form a so-calle…
View article: Partial Algebraic Shifting
Partial Algebraic Shifting Open
We study algebraic shifting of uniform hypergraphs and finite simplicial complexes in the exterior algebra with respect to matrices which are not necessarily generic. Several questions raised by Kalai (2002) are addressed. For instance, it…
View article: Rays of the secondary fan of the Hypersimplex(2,7)
Rays of the secondary fan of the Hypersimplex(2,7) Open
This dataset contains four files describing the rays of secondary fan of the hypersimplex(2,7). hypersimplex_2_7_idmap.dat.xz maps IDs to the rays of the secondary fan, such that we may refer to the rays by that ID. hypersimplex_2_7_orbit_…
View article: The mrdi File Format Specification
The mrdi File Format Specification Open
JSON schema describing the mrdi file format. The mrdi file format can be used for storing and sharing results in computer algebra. Loading and saving data in this format is has been implemented in the computer algebra system OSCAR since ve…
View article: Developments in tropical convexity
Developments in tropical convexity Open
The term "tropical convexity" was coined by Develin and Sturmfels who published a landmark paper with that title in 2004. However, the topic has much older roots and is deeply connected to linear and combinatorial optimization and other ar…
View article: Confirmable Workflows in OSCAR
Confirmable Workflows in OSCAR Open
We discuss what is special about the reproducibility of workflows in computer algebra. It is emphasized how the programming language Julia and the new computer algebra system OSCAR support such a reproducibility, and how users can benefit …
View article: Quantum automorphisms of matroids
Quantum automorphisms of matroids Open
Motivated by the vast literature of quantum automorphism groups of graphs, we define and study quantum automorphism groups of matroids. A key feature of quantum groups is that there are many quantizations of a classical group, and this phe…
View article: Master regulators of biological systems in higher dimensions
Master regulators of biological systems in higher dimensions Open
A longstanding goal of biology is to identify the key genes and species that critically impact evolution, ecology, and health. Network analysis has revealed keystone species that regulate ecosystems and master regulators that regulate cell…
View article: Asymmetric tropical distances and power diagrams
Asymmetric tropical distances and power diagrams Open
We investigate Voronoi diagrams with respect to an asymmetric tropical distance function, in particular for infinite point sets. These Voronoi diagrams turn out to be much better behaved than those arising from the standard tropical distan…
View article: Some thoughts and experiments on Bergman’s compact amalgamation problem
Some thoughts and experiments on Bergman’s compact amalgamation problem Open
We study the question whether copies of $$S^{1}$$ in $$\textrm{SU}(3)$$ can be amalgamated in a compact group. This is the simplest instance of a fundamental open problem in the theory of compact groups raised by George Bergman i…
View article: Order and chain polytopes of maximal ranked posets
Order and chain polytopes of maximal ranked posets Open
The order and chain polytopes, introduced by Richard P. Stanley, form a pair of Ehrhart equivalent polytopes associated to a given finite poset. A conjecture by Takayuki Hibi and Nan Li states that the $f$-vector of the chain polytope domi…
View article: Polyhedral Geometry in OSCAR
Polyhedral Geometry in OSCAR Open
OSCAR is an innovative new computer algebra system which combines and extends the power of its four cornerstone systems - GAP (group theory), Singular (algebra and algebraic geometry), Polymake (polyhedral geometry), and Antic (number theo…
View article: Tropical medians by transportation
Tropical medians by transportation Open
Fermat–Weber points with respect to an asymmetric tropical distance function are studied. It turns out that they correspond to the optimal solutions of a transportation problem. The results are applied to obtain a new method for computing …
View article: Some thoughts and experiments on Bergman's compact amalgamation problem
Some thoughts and experiments on Bergman's compact amalgamation problem Open
We study the question whether copies of $S^1$ in $\mathrm{SU}(3)$ can be amalgamated in a compact group. This is the simplest instance of a fundamental open problem in the theory of compact groups raised by George Bergman in 1987. Consider…
View article: Convergent Hahn series and tropical geometry of higher rank
Convergent Hahn series and tropical geometry of higher rank Open
We propose to study the tropical geometry specifically arising from convergent Hahn series in multiple indeterminates. One application is a new view on stable intersections of tropical hypersurfaces. Another one is perturbations of rank on…
View article: Generalized Permutahedra and Positive Flag Dressians
Generalized Permutahedra and Positive Flag Dressians Open
We study valuated matroids, their tropical incidence relations, flag matroids, and total positivity. This leads to a characterization of permutahedral subdivisions, namely subdivisions of regular permutahedra into generalized permutahedra.…
View article: Parametric Fermat-Weber and tropical supertrees
Parametric Fermat-Weber and tropical supertrees Open
We study a parametric version of the Fermat-Weber problem with respect to an asymmetric distance function, which occurs naturally in tropical geometry. Our results yield a method for constructing phylogenetic supertrees.
View article: The Polyhedral Geometry of Truthful Auctions
The Polyhedral Geometry of Truthful Auctions Open
The difference set of an outcome in an auction is the set of types that the auction mechanism maps to the outcome. We give a complete characterization of the geometry of the difference sets that can appear for a dominant strategy incentive…
View article: Asymmetric tropical distances and power diagrams
Asymmetric tropical distances and power diagrams Open
We investigate the Voronoi diagrams with respect to an asymmetric tropical distance function, also for infinite point sets. These turn out to be much better behaved than the tropical Voronoi diagrams arising from the standard tropical dist…
View article: Frontiers of sphere recognition in practice
Frontiers of sphere recognition in practice Open
Sphere recognition is known to be undecidable in dimensions five and beyond, and no polynomial time method is known in dimensions three and four. Here we report on positive and negative computational results with the goal to explore the li…
View article: Algebraic Degrees of 3-Dimensional Polytopes
Algebraic Degrees of 3-Dimensional Polytopes Open
Results of Koebe (Ber. Sächs. Akad. Wiss. Leipzig, Math.-phys. Kl. 88 , 141–164, 1936), Schramm (Invent. Math. 107 (3), 543560, 1992), and Springborn (Math. Z. 249 , 513–517, 2005) yield realizations of 3-polytopes with edges tangent to th…
View article: Tropical medians by transportation
Tropical medians by transportation Open
Fermat-Weber points with respect to an asymmetric tropical distance function are studied. It turns out that they correspond to the optimal solutions of a transportation problem. The results are applied to obtain a new method for computing …