A constant theme in the research of Victor Klee and Branko Grunbaum has been the discovery and development of surprising and powerful connections between combinatorics and discrete geometry and other branches of mathematics. This conference is aimed to be both a retrospective on these developments and an educational opportunity for the younger generation who will be exposed to a survey of the wide-spread contributions of the honorees. A second important component of the conference will be three 1-2 hour long problem sessions spread through the three days that will discuss open problems arising from the work of Klee and Grunbaum.
Victor Klee and Branko Grunbaum founded several branches of modern combinatorics and discrete geometry and spent their long careers (a total of 100 years between them) at the University of Washington in Seattle. Their contributions has also made an impact on several areas connected to mathematics such as optimization, computer science, statistics, and the life sciences. This conference is intended to be a world class celebration of the illustrious careers of these mathematical giants.
The grant money was used to support speakers and graduate students attending the conference "The Mathematics of Klee and Grünbaum: 100 years in Seattle" that took place in Seattle, July 28-30, 2010. This conference in honor of Victor Klee and Branko Grunbaum included twenty 45-minutes talks (on a wide range of topics in discrete geometry and combinatorics) and three 30-minutes problem sessions. In addition, Branko Grunbaum has edited and amended Vic Klee's unpublished collection (from 1960) of open problems entitled "Unsolved problems in intuitive geometry." This collection was made available to all the participants. The list of speakers, the titles and abstracts of their talks, videos of some of the talks, as well as the above mentioned collection of problems is available on the web-page of the conference: https://sites.google.com/a/alaska.edu/kleegrunbaum/ One of the highlights of the conference was a talk by Francisco Santos’ describing his recent counterexample to the Hirsch conjecture. This 53-year-old conjecture posits that the diameter of a convex polytope with n facets in d dimensions is at most n − d. Aside from its mathematical interest, this conjecture is important because the simplex method for solving a linear program walks along a path on the surface of a polytope (often in dimensions as large as d = 10000). Thus the diameter of the polytope provides a lower bound on its worst-case complexity, and establishing an upper bound on the diameter raises that complexity lower bound. Where Santos' counterexample leaves us was nicely described in Gil Kalai’s talk at the conference on the "polynomial Hirsch conjecture," which states that the diameter is bounded by some polynomial in d and n. The polynomial Hirsch conjecture is wide open at present. More than 120 people attended the conference, among them many graduate students and post-docs, as well as a few secondary school math teachers. The talks were of excellent quality. It is our hope that the talks and the open problems presented at the conference will stimulate a lot of research in the next few years.
