14 research outputs found
New light on Bergman complexes by decomposing matroid types
Bergman complexes are polyhedral complexes associated to matroids. Faces of
these complexes are certain matroids, called matroid types, too. In order to
understand the structure of these faces we decompose matroid types into direct
summands. Ardila/Klivans proved that the Bergman Complex of a matroid can be
subdivided into the order complex of the proper part of its lattice of flats.
Beyond that Feichtner/Sturmfels showed that the Bergman complex can even be
subdivided to the even coarser nested set complex. We will give a much shorter
and more general proof of this fact. Generalizing formulas proposed by
Ardila/Klivans and Feichtner/Sturmfels for special cases, we present a
decomposition into direct sums working for faces of any of these complexes.
Additionally we show that it is the finest possible decomposition for faces of
the Bergman complex.Comment: 18 pages, 1 figure, based on my diploma thesi
Geometric, Algebraic, and Topological Combinatorics
The 2019 Oberwolfach meeting "Geometric, Algebraic and Topological Combinatorics"
was organized by Gil Kalai (Jerusalem), Isabella Novik (Seattle),
Francisco Santos (Santander), and Volkmar Welker (Marburg). It covered
a wide variety of aspects of Discrete Geometry, Algebraic Combinatorics
with geometric flavor, and Topological Combinatorics. Some of the
highlights of the conference included (1) Karim Adiprasito presented his
very recent proof of the -conjecture for spheres (as a talk and as a "Q\&A"
evening session) (2) Federico Ardila gave an overview on "The geometry of matroids",
including his recent extension with Denham and Huh of previous work of Adiprasito, Huh and Katz
Matroidal Cayley-Bacharach and independence/dependence of geometric properties of matroids
We consider the relationship between a matroidal analogue of the degree
Cayley-Bacharach property (finite sets of points failing to impose independent
conditions on degree hypersurfaces) and geometric properties of matroids.
If the matroid polytopes in question are nestohedra, we show that the minimal
degree matroidal Cayley-Bacharach property denoted is determined by
the structure of the building sets used to construct them. This analysis also
applies for other degrees . Also, it does not seem to affect the
combinatorial equivalence class of the matroid polytope.
However, there are close connections to minimal nontrivial degrees and
the geometry of the matroids in question for paving matroids (which are
conjecturally generic among matroids of a given rank) and matroids constructed
out of supersolvable hyperplane arrangements. The case of paving matroids is
still related to with properties of building sets since it is closely connected
to (Hilbert series of) Chow rings of matroids, which are combinatorial models
of the cohomology of wonderful compactifications. Finally, our analysis of
supersolvable line and hyperplane arrangements give a family of matroids which
are natrually related to independence conditions imposed by points one plane
curves or can be analyzed recursively.Comment: 16 pages; Comments welcome
Supersolvability of built lattices and Koszulness of generalized Chow rings
We give an explicit quadratic Grobner basis for generalized Chow rings of
supersolvable built lattices, with the help of the operadic structure on
geometric lattices introduced in a previous article. This shows that the
generalized Chow rings associated to minimal building sets of supersolvable
lattices are Koszul. As another consequence, we get that the cohomology
algebras of the components of the extended modular operad in genus 0 are
Koszul.Comment: Second version. Cleaned up a few proofs. Comments are welcom
LIPIcs, Volume 251, ITCS 2023, Complete Volume
LIPIcs, Volume 251, ITCS 2023, Complete Volum
IST Austria Thesis
We describe arrangements of three-dimensional spheres from a geometrical and topological point of view. Real data (fitting this setup) often consist of soft spheres which show certain degree of deformation while strongly packing against each other. In this context, we answer the following questions: If we model a soft packing of spheres by hard spheres that are allowed to overlap, can we measure the volume in the overlapped areas? Can we be more specific about the overlap volume, i.e. quantify how much volume is there covered exactly twice, three times, or k times? What would be a good optimization criteria that rule the arrangement of soft spheres while making a good use of the available space? Fixing a particular criterion, what would be the optimal sphere configuration? The first result of this thesis are short formulas for the computation of volumes covered by at least k of the balls. The formulas exploit information contained in the order-k Voronoi diagrams and its closely related Level-k complex. The used complexes lead to a natural generalization into poset diagrams, a theoretical formalism that contains the order-k and degree-k diagrams as special cases. In parallel, we define different criteria to determine what could be considered an optimal arrangement from a geometrical point of view. Fixing a criterion, we find optimal soft packing configurations in 2D and 3D where the ball centers lie on a lattice. As a last step, we use tools from computational topology on real physical data, to show the potentials of higher-order diagrams in the description of melting crystals. The results of the experiments leaves us with an open window to apply the theories developed in this thesis in real applications