Seminars in December 2010

• HwanChul Yoo (유환철), Triangulations of Product of Simplices and Tropical Oriented Matroid

  Triangulations of Product of Simplices and Tropical Oriented Matroid

  HwanChul Yoo (유환철)
  Department of Mathematics, MIT

  2010/12/22 Wed 4:30PM-5:30PM (Room 3433)

  In 2006 at MSRI, nine tropical geometers and combinatorialists met and announced the list of ten key open problems in (algebraic and combinatorial side of) tropical geometry. Axiomatization of tropical oriented matroids was one of them. After the work of Develin and Sturmfels, tropical oriented matroids were conjectured to be in bijection with subdivisions of product of simplices as well as with tropical pseudohyperplane arrangements. Ardila and Develin defined tropical oriented matroid, and showed one direction that tropical oriented matroids encode subdivision of product of simplices. Recently, in joint work with Oh, we proved that every triangulation of product of simplices encodes a tropical oriented matroid.

  In this talk, I will give a survey on this topic, and discuss this well known conjecture. I will also suggest a new class of combinatorial objects that may describe all subdivisions of a bigger class of polytopes.

  Tags:

  • 유환철

• (CS Colloquium) David Eppstein, Graph algorithms in computational geometry

  Graph algorithms in computational geometry

  David Eppstein
  University of California, Irvine, CA, USA

  2010/12/13 Mon 4:00PM-5:30PM (CS Bldg., Room 1501)</em>

  TBA. This is a colloquium of department of computer science.

  Tags:

  • DavidEppstein

• Xavier Goaoc, Helly numbers and nerve theorems

  Helly numbers and nerve theorems

  Xavier Goaoc
  LORIA, INRIA Nancy – Grand Est, Villers-Lès-Nancy cedex, France.

  2010/12/10 Fri 4PM-5PM

  The Helly number of a collection of sets is the size of its largest inclusionwise minimal subfamily with empty intersection. The precise conditions that lead to bounded Helly numbers have been studied since the 1920’s, when Helly showed that the Helly number of any collection of compact convex sets in R^d has Helly number at most d+1.

  I will discuss a proof that any collection of subsets of R^d where the intersection of any subfamily consists of at most r connected components, each of which is contractible, has Helly number at most r(d+1). I will show how this implies, in a unified manner, quantitative bounds for several Helly-type theorems in geometric transversal theory.

  Our main ingredients are a new variant of the nerve, a “homological nerve theorem” for this structure and an extension of a projection theorem of Kalai and Meshulam.

  This is joint work with Eric Colin de Verdiere and Gregory Ginot.

  Tags:

  • XavierGoaoc