We propose a new geometric method of IR factorization in sector decomposition. The problem is converted into a set of problems in convex geometry. The latter problems are solved using algorithms in combinatorial geometry. This method provides a deterministic algorithm and never falls into an infinite loop. The number of resulting sectors depends on the algorithm of triangulation. Our test imple...

This workshop intends to celebrate and encourage international research collaboration in the area of discrete and computational geometry, particularly between Canada and Japan. The workshop features talks on theoretical and applied aspects of discrete geometry, with an emphasis on computational questions, and computational approaches to mathematics. Topics of particular interest include geometr...

In this paper, we explain the relations between combinatorial optimization and real algebraic geometry with a special focus to the quadratic assignment problem. We demonstrate how to write a quadratic optimization problem over discrete feasible set as a linear optimization problem over the cone of completely positive matrices. The latter formulation enables a hierarchy of approximations which r...

Mixed volumes of lattice polytopes play a central role in numerical and tropical algebraic geometry. We present an implementation of a new algorithm for their computation based on tropical homotopy continuation, which is a combinatorial procedure using ideas from numerical algebraic geometry. While the mathematical aspects of the algorithm are presented elsewhere, here we mainly address technic...

In this report, we discuss the geometric properties of spatial additively weighted Voronoi (AWV-) cells. First, we give a short survey of previously known properties of the geometry of these cells. Then, we give a detailed account on the geometry of the edges of AWV-cells. Finally, and this is our main contribution, we prove a new and tight lower bound of Θ(n2) on the worst-case combinatorial c...

Let Σ be a subset of the set of prime numbers which is either equal to the entire set of prime numbers or of cardinality one. In the present paper, we continue our study of the pro-Σ fundamental groups of hyperbolic curves and their associated configuration spaces over algebraically closed fields in which the primes of Σ are invertible. The starting point of the theory of the present paper is a...

Extremal Combinatorics is one of the most active topics in Discrete Mathematics, dealing with problems that are often motivated by questions in other areas, including Theoretical Computer Science, Geometry and Game Theory. This paper contains a collection of problems and results in the area, including solutions or partial solutions to open problems suggested by various researchers. The topics c...

This report discusses the problem of mismatch between the geometry modelers and the end-applications that need to use the geometry data. The mismatch is seen at various levels: data formats, geometric artifacts, quantization errors, combinatorial complexity and lack of desired structure. To make the geometry usable, it has to undergo extensive preprocessing which is also referred to as “healing...

We show that when n is greater than 3, the number of points in a combinatorial geometry (or simple matroid) G of rank n containing no minor isomorphic to the 8-point line is at most 1 4 (5n − 1). This bound is sharp and is attained if and only if the geometry G is the projective geometry PG(n − 1, 5) over the field GF(5). This paper appeared in: Mathematical Essays in Honor of Gian-Carlo Rota, ...

of the Dissertation On the Algebra and Geometry of a Manifold’s Chains and Cochains by Scott Owen Wilson Doctor of Philosophy in Mathematics Stony Brook University 2005 This dissertation consists of two parts, each of which describes new algebraic and geometric structures defined on chain complexes associated to a manifold. In the first part we define, on the simplicial cochains of a triangulat...