Category Archives: Open discussion

Why is Mathematics Possible: Tim Gowers’s Take on the Matter

In a previous post I mentioned the question of why is mathematics possible. Among the interesting comments to the post, here is a comment by Tim Gowers: “Maybe the following would be a way of rephrasing your question. We know … Continue reading

Posted in Open discussion, Philosophy, What is Mathematics | Tagged , , , | 23 Comments

Why is mathematics possible?

Spectacular advances in number theory Last weeks we heard about two spectacular results in number theory.  As announced in Nature, Yitang Zhang proved that there are infinitely many pairs of consecutive primes which are at most 70 million apart! This is a sensational achievement. … Continue reading

Posted in Computer Science and Optimization, Number theory, Open discussion, Philosophy, Updates, What is Mathematics | 15 Comments

Polynomial Hirsch Conjecture 5: Abstractions and Counterexamples.

This is the 5th research thread of polymath3 studying the polynomial Hirsch conjecture. As you may remember, we are mainly interested in an abstract form of the problem about families of sets. (And a related version about families of multisets.) The … Continue reading

Posted in Open discussion, Open problems, Polymath3 | Tagged , | 59 Comments

Polymath3: Polynomial Hirsch Conjecture 4

So where are we? I guess we are trying all sorts of things, and perhaps we should try even more things. I find it very difficult to choose the more promising ideas, directions and comments as Tim Gowers and Terry Tao did so … Continue reading

Posted in Combinatorics, Convex polytopes, Open discussion, Open problems, Polymath3 | Tagged , | 73 Comments

Polymath3 : Polynomial Hirsch Conjecture 3

Here is the third research thread for the polynomial Hirsch conjecture.  I hope that people will feel as comfortable as possible to offer ideas about the problem we discuss. Even more important, to think about the problem either in the directions suggested by … Continue reading

Posted in Combinatorics, Convex polytopes, Open discussion, Open problems, Polymath3 | Tagged | 102 Comments

Polymath 3: The Polynomial Hirsch Conjecture 2

Here we start the second research thread about the polynomial Hirsch conjecture.  I hope that people will feel as comfortable as possible to offer ideas about the problem. The combinatorial problem looks simple and also everything that we know about it is rather simple: … Continue reading

Posted in Convex polytopes, Open discussion, Open problems, Polymath3 | Tagged , | 104 Comments

Polymath 3: Polynomial Hirsch Conjecture

I would like to start here a research thread of the long-promised Polymath3 on the polynomial Hirsch conjecture. I propose to try to solve the following purely combinatorial problem. Consider t disjoint families of subsets of {1,2,…,n}, . Suppose that … Continue reading

Posted in Convex polytopes, Open discussion, Open problems, Polymath3 | Tagged , | 119 Comments

Polymath Reflections

Polymath is a collective open way of doing mathematics. It  started over Gowers’s blog with the polymath1 project that was devoted to the Density Hales Jewett problem. Since then we  had Polymath2 related to Tsirelson spaces in Banach space theory , an  intensive Polymath4 devoted … Continue reading

Posted in Mathematics over the Internet, Open discussion | 2 Comments

When It Rains It Pours

After our success in exploring the phrase “more or less” in many languages here is a task of a similar nature There is a saying in Hebrew:  “Troubles come in packages”  צרות באות בצרורות  “Tzarot Baot bitzrorot”.  I am curious about analogs in other … Continue reading

Posted in Open discussion | Tagged , | 49 Comments

The Polynomial Hirsch Conjecture: Discussion Thread, Continued

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