Matroid polytopes, nested sets and Bergman fans.
Minimality of toric arrangements
We prove that the complement of a toric arrangement has the homotopy type of a minimal CW-complex. As a corollary we deduce that the integer cohomology of these spaces is torsionfree. We apply discrete Morse theory to the toric Salvetti complex, providing a sequence of cellular collapses that leads to a minimal complex.
Movable configurations.
New lower bounds for Heilbronn numbers.
On seven points in the boundary of a plane convex body in large relative distances.
On the Complexity of a Single Cell in Certain Arrangements of Surfaces Related to Motion Planning.
On the diffeomorphic type of the complement to a line arrangement in a projective plane
We show that the diffeomorphic type of the complement to a line arrangement in a complex projective plane P 2 depends only on the graph of line intersections if no line in the arrangement contains more than two points in which at least two lines intersect. This result also holds for some special arrangements which do not satisfy this property. However it is not true in general, see [Rybnikov G., On the fundamental group of the complement of a complex hyperplane arrangement, Funct. Anal. Appl., 2011,...
On the Zone of a Surface in a Hyperplane Arrangement.
Oriented Matroids with Few Mutations.
Rational model of the subspace complement on atomic complexes.
Rational realization of the minimum ranks of nonnegative sign pattern matrices
A sign pattern matrix (or nonnegative sign pattern matrix) is a matrix whose entries are from the set (, respectively). The minimum rank (or rational minimum rank) of a sign pattern matrix is the minimum of the ranks of the matrices (rational matrices, respectively) whose entries have signs equal to the corresponding entries of . Using a correspondence between sign patterns with minimum rank and point-hyperplane configurations in and Steinitz’s theorem on the rational realizability of...
Sections of simplices.
Semimatroids and their Tutte polynomials.
Shelling Pseudopolyhedra.
Six little squares and how their numbers grow.
Some analogs of Zariski's Theorem on nodal line arrangements.
Stokes matrices of hypergeometric integrals
In this work we compute the Stokes matrices of the ordinary differential equation satisfied by the hypergeometric integrals associated to an arrangement of hyperplanes in generic position. This generalizes the computation done by J.-P. Ramis for confluent hypergeometric functions, which correspond to the arrangement of two points on the line. The proof is based on an explicit description of a base of canonical solutions as integrals on the cones of the arrangement, and combinatorial relations between...
Supersolvable orders and inductively free arrangements
In this paper, we define the supersolvable order of hyperplanes in a supersolvable arrangement, and obtain a class of inductively free arrangements according to this order. Our main results improve the conclusion that every supersolvable arrangement is inductively free. In addition, we assert that the inductively free arrangement with the required induction table is supersolvable.
Support properties of a family of connected compact sets
A problem of finding a system of proportionally located parallel supporting hyperplanes of a family of connected compact sets is analyzed. A special attention is paid to finding a common supporting halfspace. An existence theorem is proved and a method of solution is proposed.
Tetrahedra on deformed spheres and integral group cohomology.