Category Archives: Open discussion

The Polynomial Hirsch Conjecture: Discussion Thread, Continued

Posted on October 6, 2009 by Gil Kalai

Here is a  link for the just-posted paper Diameter of Polyhedra: The Limits of Abstraction by Freidrich Eisenbrand, Nicolai Hahnle,  Sasha Razborov, and Thomas Rothvoss. And here is a link to the paper  by Sandeep Koranne and Anand Kulkarni “The d-step Conjecture is Almost true”  – … Continue reading →

Posted in Convex polytopes, Open discussion, Open problems | Tagged , | 16 Comments

The Polynomial Hirsch Conjecture: Discussion Thread

Posted on August 9, 2009 by Gil Kalai

This post is devoted to the polymath-proposal about the polynomial Hirsch conjecture. My intention is to start here a discussion thread on the problem and related problems. (Perhaps identifying further interesting related problems and research directions.) Earlier posts are: The polynomial Hirsch … Continue reading →

Posted in Convex polytopes, Open discussion, Open problems | Tagged , | 115 Comments

The Polynomial Hirsch Conjecture – How to Improve the Upper Bounds.

Posted on July 30, 2009 by Gil Kalai

I can see three main avenues toward making progress on the Polynomial Hirsch conjecture. One direction is trying to improve the upper bounds, for example,  by looking at the current proof and trying to see if it is wasteful and if so where … Continue reading →

Posted in Convex polytopes, Open discussion, Open problems | Tagged , | 14 Comments

The Polynomial Hirsch Conjecture, a Proposal for Polymath3 (Cont.)

Posted on July 28, 2009 by Gil Kalai

The Abstract Polynomial Hirsch Conjecture A convex polytope is the convex hull of a finite set of points in a real vector space. A polytope can be described as the intersection of a finite number of closed halfspaces. Polytopes have … Continue reading →

Posted in Convex polytopes, Open discussion, Open problems | Tagged , | 5 Comments

The Polynomial Hirsch Conjecture: A proposal for Polymath3

Posted on July 17, 2009 by Gil Kalai

This post is continued here.  Eddie Kim and Francisco Santos have just uploaded a survey article on the Hirsch Conjecture. The Hirsch conjecture: The graph of a d-polytope with n vertices  facets has diameter at most n-d. We devoted several … Continue reading →

Posted in Convex polytopes, Open discussion, Open problems | Tagged , , , | 38 Comments

An Open Discussion and Polls: Around Roth’s Theorem

Posted on March 25, 2009 by Gil Kalai

Suppose that  is a subset of of maximum cardinality not containing an arithmetic progression of length 3. Let . How does behave? We do not really know. Will it help talking about it? Can we somehow look beyond the horizon and try to guess what … Continue reading →

Posted in Combinatorics, Open discussion, Open problems | Tagged , , , | 29 Comments

Is Mathematics a Science?

Posted on May 25, 2008 by Gil Kalai

Many people do not regard mathematics as a science since it does not directly probe our physical reality; some mathematicians even like to think about mathematics as being closer to art, music or literature. But is there really a big … Continue reading →

Posted in Open discussion, Philosophy, What is Mathematics | Tagged , | 7 Comments