Convex Polytopes, Algebraic Geometry, and Combinatorics

Applications of Convex and Algebraic Geometry to Graphs and Polytopes

Geometry, Complexity, and Combinatorics of Permutation Polytopes

Each group G of permutation matrices gives rise to a permutation polytope P(G) = cony(G) c Re×d, and for any x ~ W, an orbit polytope P(G, x) = conv(G, x). A broad subclass is formed by the Young permutation polytopes, which correspond bijectively to partitions 2 = (21, ..., 2k)~-n of positive integers, and arise from the Young representations of the symmetric group. Young polytopes provide a f...

Applications and Combinatorics in Algebraic Geometry

Algebraic Geometry is a deep and well-established field within pure mathematics that is increasingly finding applications outside of mathematics. These applications in turn are the source of new questions and challenges for the subject. Many applications flow from and contribute to the more combinatorial and computational parts of algebraic geometry, and this often involves real-number or posit...

Classification of Finite Metric Spaces and Combinatorics of Convex Polytopes

We describe the canonical correspondence between finite metric spaces and symmetric convex polytopes, and formulate the problem about classification of the metric spaces in terms of combinatorial structure of those polytopes.

Nonrational, Nonsimple Convex Polytopes in Symplectic Geometry

In this research announcement we associate to each convex polytope, possibly nonrational and nonsimple, a family of compact spaces that are stratified by quasifolds, i.e., the strata are locally modeled by Rk modulo the action of a discrete, possibly infinite, group. Each stratified space is endowed with a symplectic structure and a moment mapping having the property that its image gives the or...