Category Archives: Open problems

Stefan Steinerberger: The Ulam Sequence

This post is authored by Stefan Steinerberger. The Ulam sequence is defined by starting with 1,2 and then repeatedly adding the smallest integer that is (1) larger than the last element and (2) can be written as the sum of two … Continue reading

Posted in Guest blogger, Open problems | Tagged , | 8 Comments

Polymath 10 Post 3: How are we doing?

The main purpose of this post is to start a new research thread for Polymath 10  dealing with the Erdos-Rado Sunflower problem.  (Here are links to post 2 and post 1.) Here is a  very quick review of where we … Continue reading

Posted in Combinatorics, Mathematics over the Internet, Open problems, Polymath10 | Tagged , | 103 Comments

More Reasons for Small Influence

Readers of the big-league ToC blogs have already heard about the breakthrough paper An average-case depth hierarchy theorem for Boolean circuits by Benjamin Rossman, Rocco Servedio, and Li-Yang Tan. Here are blog reports on Computational complexity, on the Shtetl Optimized, and of Godel … Continue reading

Posted in Combinatorics, Computer Science and Optimization, Open problems, Probability | 12 Comments

Coloring Simple Polytopes and Triangulations

Coloring Edge-coloring of simple polytopes One of the equivalent formulation of the four-color theorem asserts that: Theorem (4CT) : Every cubic bridgeless planar graph is 3-edge colorable So we can color the edges by three colors such that every two … Continue reading

Posted in Combinatorics, Open problems | Tagged , | 10 Comments

Jim Geelen, Bert Gerards, and Geoff Whittle Solved Rota’s Conjecture on Matroids

Gian Carlo Rota Rota’s conjecture I just saw in the Notices of the AMS a paper by Geelen, Gerards, and Whittle where they announce and give a high level description of their recent proof of Rota’s conjecture. The 1970 conjecture asserts … Continue reading

Posted in Combinatorics, Open problems, Updates | Tagged , , , , , , | 7 Comments

My Mathematical Dialogue with Jürgen Eckhoff

Jürgen Eckhoff, Ascona 1999 Jürgen Eckhoff is a German mathematician working in the areas of convexity and combinatorics. Our mathematical paths have met a remarkable number of times. We also met quite a few times in person since our first … Continue reading

Posted in Combinatorics, Convex polytopes, Open problems | Tagged , , , , , | 1 Comment

Navier-Stokes Fluid Computers

Smart fluid Terry Tao posted a very intriguing post on the Navier-Stokes equation, based on a recently uploaded paper Finite time blowup for an averaged three-dimensional Navier-Stokes equation. The paper proved a remarkable negative answer for the regularity conjecture for a certain … Continue reading

Posted in Analysis, Applied mathematics, Computer Science and Optimization, Open problems | Tagged , , , , , | 11 Comments

Amazing: Peter Keevash Constructed General Steiner Systems and Designs

Here is one of the central and oldest problems in combinatorics: Problem: Can you find a collection S of q-subsets from an n-element set X set so that every r-subset of X is included in precisely λ sets in the collection? … Continue reading

Posted in Combinatorics, Open problems | Tagged , , | 13 Comments

Many triangulated three-spheres!

The news Eran Nevo and Stedman Wilson have constructed triangulations with n vertices of the 3-dimensional sphere! This settled an old problem which stood open for several decades. Here is a link to their paper How many n-vertex triangulations does the 3 … Continue reading

Posted in Combinatorics, Convex polytopes, Geometry, Open problems | Tagged , | Leave a comment

Polymath 8 – a Success!

Yitang Zhang Update (July 22, ’14). The polymath8b paper “Variants of the Selberg sieve, and bounded intervals containing many primes“, is now on the arXiv. See also this post on Terry Tao’s blog. Since the last update, we also had here … Continue reading

Posted in Mathematics over the Internet, Number theory, Open problems | Tagged , , , , | 10 Comments