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We study how the supporting hyperplanes produced by the projection process can complement the method of alternating projections and its variants for the convex set intersection problem. For the problem of finding the closest point in the…
The family of graphs that can be constructed from isolated vertices by disjoint union and graph join operations are called cographs. These graphs can be represented in a tree-like representation termed parse tree or cotree. In this paper,…
Beyond-planarity focuses on combinatorial properties of classes of non-planar graphs that allow for representations satisfying certain local geometric or topological constraints on their edge crossings. Beside the study of a specific graph…
The purpose of this thesis is to study classical combinatorial objects, such as polytopes, polytopal complexes, and subspace arrangements, using tools that have been developed in combinatorial topology, especially those tools developed in…
We describe a singular variety associated to the smallest degree Pinchuk map and calculate its intersection homology. The result describes the geometry at infinity of the Pinchuk's map.
We characterize in terms of the Goldman Lie algebra which conjugacy classes in the fundamental group of a surface with non empty boundary are represented by simple closed curves. We prove the following: A non power conjugacy class X…
Bisztriczky defines a multiplex as a generalization of a simplex, and an ordinary polytope as a generalization of a cyclic polytope. This paper presents results concerning the combinatorics of multiplexes and ordinary polytopes. The flag…
We consider a general concept of composition and decomposition of objects, and discuss a few natural properties one may expect from a reasonable choice thereof. It will be demonstrated how this leads to multiplication and co- multiplication…
We study the structure of the set of all possible affine hyperplane sections of a convex polytope. We present two different cell decompositions of this set, induced by hyperplane arrangements. Using our decomposition, we bound the number of…
Motivated by geometry processing for surfaces with non-trivial topology, we study discrete harmonic maps between closed surfaces of genus at least two. Harmonic maps provide a natural framework for comparing surfaces by minimizing…
Associated varieties are geometric objects appearing in infinite-dimensional representations of semisimple Lie algebras (groups). By applying Fourier transformations to the natural orthogonal oscillator representations of special linear Lie…
In this note, we initiate a study of the finite-dimensional representation theory of a class of algebras that correspond to noncommutative deformations of compact surfaces of arbitrary genus. Low dimensional representations are investigated…
Cooperative driving at signal-free intersections, which aims to improve driving safety and efficiency for connected and automated vehicles, has attracted increasing interest in recent years. However, existing cooperative driving strategies…
A discrete harmonic surface is a trivalent graph which satisfies the balancing condition in the 3-dimensional Euclidean space and achieves energy minimizing under local deformations. Given a topological trivalent graph, a holomorphic…
We consider the relations between different measures of complexity for free homotopy classes of curves on a surface $\Sigma$, including the minimum number of self-intersections, the minimum length of the words representing them in a…
The toric residue is a map depending on n+1 semi-ample divisors on a complete toric variety of dimension n. It appears in a variety of contexts such as sparse polynomial systems, mirror symmetry, and GKZ hypergeometric functions. In this…
We investigate the intersection body of a convex polytope using tools from combinatorics and real algebraic geometry. In particular, we show that the intersection body of a polytope is always a semialgebraic set and provide an algorithm for…
In this paper, we present a new framework that exploits combinatorial optimization for efficiently generating a large variety of combinatorial objects based on graphs, matroids, posets and polytopes. Our method relies on a simple and…
We study a general smallest intersecting ball problem and its soft-margin variant in high-dimensional Euclidean spaces for input objects that are compact and convex. These two problems link and unify a series of fundamental problems in…
We consider the problem of cooperative intersection management. It arises in automated transportation systems for people or goods but also in multi-robots environment. Therefore many solutions have been proposed to avoid collisions. The…