# The polymath blog

## August 13, 2016

### MO Polymath question: Summary of Proposals

Filed under: polymath proposals — Gil Kalai @ 7:23 pm

### Summary of proposals (updated: August 10, 2016)

1) The LogRank conjecture. Proposed by Arul.

2) The circulant Hadamard matrix conjecture. Proposed by Richard Stanley.

3) Finding combinatorial models for the Kronecker coefficients. Proposed by Per Alexandersson.

4) Eight lonely runners. Proposed by Mark Lewko.

5) A problem by Ruzsa:
Finding the slowest possible exponential growth rate of a mapping from N to Z that is not a polynomial and yet shares with (integer) polynomials the congruence-preserving property n−m∣f(n)−f(m). Proposed by Vesselin Dimitrov.

6) Finding the Matrix Multiplication Exponent ω. (Running time of best algorithm for matrix multiplication.) Proposed by Ryan O’Donnell.

7) The Moser Worm problem and Bellman’s Lost in a forest problem. Proposed by Philip Gibbs.

8) Rational Simplex Conjecture ( by Cheeger and Simons). Proposed by Sasha Kolpakov.

9) determinants for 0-1 matrices Proving that for every integer $m$ with $|m|\le c(\sqrt{n}/2)^n$ there is an $n \times n$
0-1 matrix matrix whose determinant equals $m$.  Proposed by Gerhard Paseman.

10) Proving or disproving that the Euler’s constant is irrational. Proposed by Sylvain JULIEN.

11) The Greedy Superstring Conjecture. Proposed by Laszlo Kozma.

12) Understanding the behavior and structure of covering arrays. Proposed by Ryan.

13) The group isomorphism problem, proposed by Arul based on an early proposal by Lipton.

14) Frankl’s union closed set conjecture (Proposed by Dominic van der Zypen; Also one of the proposals by Gowers in this post). (Launched)

15) Komlos’s conjecture in Discrepancy Theory. Proposed by Arul.

16) Rota’s Basis Conjecture. Proposed by Timothy Chow.

17)+18) I contributed two proposals. One in ANT is to A problem in ANT show that
$latex 2^n+5$ is  composite for almost all positive integers $n$. (Might be too hard.) Another is to prove a remarkable combinatorial identity on certain Permanents.

19) Real world applications of large cardinals Proposed by Joseph van Name. There were a few more proposals in comments.

20) A project around a cluster of tiling problems. In particular: Is the Heech number bounded for polygonal monotiles? Is it decidable to determine if a single given polygonal tile can tile the plane monohedrally? Even for a single polyomino? Proposed by Joseph O’Rourke

## 2 Comments »

1. Dear Gil, Ron Peled and I were able to prove the identity from your proposal (no. 18). The main tool is Borchardt’s identity, which reduces the permanent calculation to the evaluation of two determinants, which in our case are circulant. Here’s a link to the draft of the short proof: dropbox.com/s/p4hdl2qb7pgppn7/identity.pdf?dl=0 .

Comment by Ofir Gorodetsky — August 26, 2016 @ 6:54 am

2. […] 截至到8月10日，Polymath项目在MathOverflow上公开征集到了20个适宜作为项目提案的组合问题。某些读者或许会对这份清单感兴趣。 […]

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