The other reason to create a new post is to take stock of where we are and perhaps suggest some ways to go forward. Let me emphasize that the list below is not comprehensive, but is meant only to summarize the comments so far and to throw in a few ideas of my own. Assuming this project continues to gather steam, the plan is to populate the associated Polymath Wiki page with a more comprehensive list of references and statements of partial results. If you have an idea that does not seem to fit into any of the categories below, please consider that to be an invitation to leave a comment about your idea, not an indication that it is not of interest!

I want to start with an idea that I mentioned in my MathOverflow post but not in my previous Polymath Blog post. I think it is very promising, and I don’t think many people have looked at it. Geelen and Humphries proved that Rota’s Basis Conjecture is true for paving matroids. In the case of vector spaces, what this means is that they proved the conjecture in the case where every (*n* – 1)-element subset of the given set of *n*^{2} vectors is linearly independent. It is natural to ask if *n* – 1 can be reduced to *n* – 2. I have not digested the Geelen–Humphries paper so I do not know how easy or hard this might be, but it certainly could not hurt to have more people study this paper and make an attempt to extend its results. If an oracle were to tell me that Rota’s Basis Conjecture has a 10-page proof and were to ask me what I thought the method was, then at this point in time I would guess that the proof proceeds by induction on the size of the smallest circuit. Even if I am totally wrong, I think we will definitely learn something by understanding exactly why this approach cannot be extended.

Let me now review the progress on the three ideas I mentioned in my first blog post. In Idea 1, I asked if the *n*^{2} vectors could be partitioned into at most 2*n* – 2 independent partial transversals. A nice proof that the answer is yes was given by domotorp. Eli Berger then made a comment that suggested that the topological methods of Aharoni and Berger could push this bound lower, but there was either an error in his suggestion or we misunderstood it. It would be good to get this point clarified. I should also mention that Aharoni mentioned to me offline that he unfortunately could not participate actively in Polymath but that he did have an answer to my question about their topological methods, which is that the topological concepts they were using were intrinsically not strong enough to bring the bound down to *n* + 1, let alone *n*. It might nevertheless be valuable to understand exactly how far we can go by thinking about independent partial transversals. Ron Aharoni and Jonathan Farley both had interesting ideas along these lines; rather than reproduce them here, let me just say that you can find Aharoni’s comment (under the previous blog post) by searching for “Vizing” and Farley’s comment by searching for “Mirsky.”

Idea 2 was to look for additional obstructions to natural strengthenings of Rota’s Basis Conjecture, by computationally searching for counterexamples that arise if the number of columns is smaller than the number of rows. Luke Pebody started such a search but reported a bug. I still believe that this computational search is worth doing, because I suspect that any proof that Rota’s Basis Conjecture holds for all matroids is going to have to come to grips with these counterexamples.

Note that if we are interested just in vector spaces, we could do some Gröbner basis calculations. I am not sure that this would be any less computationally intensive than exhausting over all small matroids, but it might reveal additional structure that is peculiar to the vector space case.

There has been minimal progress in this (admittedly vague) direction. I will quote Ellenberg’s initial thoughts: “If you were going to degenerate, what you would need to do is say: is there any version of this question that makes sense when the basic object is, instead of a basis of an *n*-dimensional vector space *V*, a 0-dimensional subscheme of *V* of degree *n* which is not contained in any hyperplane? For instance, in 2-space you could have something which was totally supported at the point (0,1) but which was “fat” in the horizontal direction of degree 2. This is the scheme *S* such that what it means for a curve *C* to contain *S* is that *S* passes through (0,1) and has a horizontal tangent there.”

Let me also mention that Jan Draisma sent me email recently with the following remarks: “A possible idea would be to consider a counterexample as lying in some suitable equivariant Hilbert scheme in which being a counterexample is a closed condition, then degenerate to a counterexample stable under a Borel subgroup of *GL _{n}*, and come to a contradiction. ‘Equivariant’ should reflect the action of

Now let me move on to other ideas suggested in the comments. There were several thoughts about the Alon–Tarsi Conjecture that the Alon–Tarsi constant *L _{n}*

Σ_{A} (–1)^{σ(A)} det(*A*)^{n},

where the sum is over all zero-one matrices and *σ*(*A*) is the number of zero entries in *A*. This formula suggests various possibilities. For example one could try to prove that *L _{n}*

Another possibility, suggested by Gil Kalai, is to consider a Gaussian analogue. Instead of random zero-one matrices, consider random Gaussian matrices and try to understand the Hermite expansion of det^{n}, in particular showing that the coefficient corresponding to all ones is nonzero. This might be easier and might give some insight.

Note also that in the comments to my MathOverflow post, Abdelmalek Abdesselam proposed an analogue of the Alon–Tarsi conjecture for odd *n*. I do not think that many people have looked at this.

Some generalizations and special cases of the conjecture were mentioned in the comments. Proving the conjecture for graphic matroids or binary matroids would be an enormous advance. There is a generalization due to Jeff Kahn, in which we have *n*^{2} bases *B _{ij}* and we have to pick

Juan Sebastian Lozano suggested the following reformulation: Does there exist a group *G* such that *V* is a representation of *G* and there exists *g _{i}* ∈

span{*g*^{0}*b*, …, *g*^{n – 1}*b*} = *V*

where *g ^{i}* =

Fedor Petrov mentioned a theorem by him and Roman Karasev that looks potentially relevant (or at least the method of proof might be useful). Let *p* be an odd prime, and let *V* be the **F**_{p}-vector space of dimension *k*. Denote *V*^{*} = *V* \ {0} and put *m* = |*V*^{*}|/2 = (*p ^{k}* – 1)/2. Suppose we are given

(*v*_{11}, …, *v*_{1k}), (*v*_{21}, …, *v*_{2k}), …, (*v*_{m1}, …, *v*_{mk}).

Then there exist pairwise distinct *x*_{1}, …, *x _{m}*,

Gil Kalai notes that the Alon–Tarsi conjecture is related to the coloring polynomial of a graph and asks if we can learn anything by considering more general polynomials such as

Π {(*x _{i}* –

where the *λ _{e}* are weights associated to the edges

Filed under: polymath proposals ]]>

As you may have noticed, the layout of this blog has changed. I was trying to address a request by one of the commenters here to try to enable the links to recent comments to change colour if they were clicked on; unfortunately I was not able to do so, and in the course of doing so managed to change the theme in such a manner that the original theme (“Rubric”, which has been retired by wordpress) is no longer recoverable. I hope the new theme is not too jarring in design (it is the closest I could find to the original layout, which tried to maximise the width of the main posts in order to facilitate detailed comments). If there are any experts in CSS, wordpress, and/or design who can help improve the layout, please feel free to add suggestions in the comments of this post. (In particular, if there is a way to widen the main portion of the blog further, please let me know.)

Filed under: planning ]]>

I proposed Rota’s basis conjecture as a possible Polymath project on MathOverflow last year. If you have not read my proposal, I strongly recommend that you read it now, because in it, I sketched some reasons why I thought this would make a good Polymath project, as well as some known partial results and potential avenues for progress.

Recently, I emailed several likely participants, and a number of them responded enthusiastically, enough in my opinion to warrant an attempt to start a Polymath project. I have discussed the possibility with the polymath blog admins and since I do not have a blog of my own, they have generously agreed to host the project here on the polymath blog itself. This means that you should comment freely in the comments section below.

Rota’s basis conjecture states that if *B*_{1}, *B*_{2}, …, *B _{n}* are

1. the *n* vectors in row *i* are the members of the *i*th basis *B _{i}* (in some order), and

2. in each column of the matrix, the *n* vectors in that column form a basis of *V*.

If this project gets enough momentum to be formally declared “Polymath 12” then it will be important to give a thorough summary of what is already known, and to lay out in some detail all the promising directions. However, at this early stage, I think that it is important to have some “quick wins” to get things moving, so I would like to present a couple of new ideas that I think could lead to some new partial results quickly, and also invite others to present their own ideas.

The first idea is to extend an old result of Aharoni and Berger that I think has not received too much attention from others. Suppose we have two matroids on the same ground set *E*. By definition, a *common independent set* is a subset of *E* that is independent in both matroids. We can try to partition *E* into a disjoint union of common independent sets, and can ask the question, what is the smallest number *β* of common independent sets that we need?

Here is the relation to Rota’s basis conjecture. The ground set *E* has *n*^{2} elements, and one of the matroids is defined by the given set of *n*^{2} vectors (here, if the same vector appears in more than one basis, we treat the different occurrences as being distinct). The second matroid is the so-called *transversal matroid* whose independent sets are precisely those subsets of *E* that contain at most one element from each *B _{i}*. From this point of view, Rota’s basis conjecture says that

Aharoni and Berger have proved a general theorem about matroids that implies, in the specific case of Rota’s basis conjecture, that *β* ≤ 2*n*. They also have a very general conjecture on matroids that would imply that *β* ≤ *n* + 1 for Rota’s basis conjecture.

Let me now make the simple observation that it is easy to prove directly that *β* ≤ 2*n* – 1 for Rota’s basis conjecture. We begin with a lemma. Suppose we have a matroid and suppose that *I*_{1}, …, *I _{n}* are independent sets with |

Returning to the setup for Rota’s basis conjecture, we can write out the *n*^{2} given vectors in a grid with the elements of *B _{i}* in row

So the first question, which I don’t think has been looked at much and which hopefully should not be too hard, is:

**Can we show that β ≤ 2n – 2?**

In one of my papers I introduced the idea of looking for certain kinds of obstructions to an inductive proof of the conjecture. Specifically, suppose that instead of *n* bases, we are given *n* independent sets *I*_{1}, …, *I _{n}*, each of size

The last idea I want to present here is very vague. It is inspired by a paper by Ellenberg and Erman that I recently learned about. The result of the paper itself isn’t relevant, but I thought that the method might be. Roughly speaking, they reduce a certain combinatorial problem involving points and lines in a vector space to a “degenerate” case that is more tractable. Since various “degenerate” cases of Rota’s basis conjecture are known, perhaps the same idea could be applied to extend those degenerate cases to more cases.

As an example of a known degenerate case, let us first generalize Rota’s basis conjecture slightly as follows. Let us allow the vector space *V* to have some dimension *d* > *n*, and instead of *n* bases, let us take any *n* independent sets *I*_{1}, …, *I _{n}*, each of size

Let me end this initial blog post here, with just one further comment that a couple of people that I have communicated with recently have some other concrete ideas that we can sink our teeth into immediately. I am going to invite them to explain those ideas in the comments to this blog post.

Filed under: polymath proposals ]]>

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 with there is an

0-1 matrix matrix whose determinant equals . 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 . (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

Filed under: polymath proposals ]]>

Two great Peter Frankls

Tim Gowers launched polymath 11 aimed at Peter Frankl’s conjecture asserting that for every union-closed family there is an element that belongs to at lease half the sets in the family. Here are links to Post number 0 and Post number 1. (Meanwhile polymath10 continues to run on “Combinatorics and More.”)

Filed under: news ]]>

Here is the link to a mathoverflow question asking for polymath proposals. There are some very interesting proposals. I am quite curious to see some proposals in applied mathematics, and various areas of geometry, algebra, analysis and logic.

Filed under: news, polymath proposals ]]>

**Dinesh Thakur David Speyer**

A beautiful polymath proposal by Dinesh Thakur was posted by Terry Tao on the this blog. The task was to explain some remarkable, numerically observed, identities involving the irreducible polynomials in the polynomial ring over the finite field of characteristic two. David Speyer managed to prove Thakur’s observed identities! Here is the draft of the paper. Congratulations to Dinesh and David!

Filed under: news ]]>

Filed under: polymath proposals ]]>

Let be the ring of polynomials over the finite field of two elements, and let

be the set of irreducible polynomials in this ring. Then infinite series such as

and

can be expanded as formal infinite power series in the variable .

It was numerically observed in http://arxiv.org/abs/1512.02685 that one appears to have the remarkable cancellation

and

For instance, one has

and all other terms in are of order or higher, so this shows that has -valuation at least 3. Similarly, if one expands the first sum for all primes of degree (in ) up to 37, one obtains (the calculation took about a month on one computer), implying that the -valuation of the infinite sum is at least 38; in fact a bit of theory can improve this to 42. (But we do not know whether this 42 is the answer to everything!).

For the second sum, calculation for degrees up to 28 shows that the difference between the two sides has -valuation at least 88.

The polymath proposal is to investigate this phenomenon further (perhaps by more extensive numerical calculations) and supply a theoretical explanation for it.

Background links:

- The paper http://arxiv.org/abs/1512.02685 where these (and many more guesses of this type) are given with some background on zeta deformation etc, and
- http://www.math.rochester.edu/people/faculty/dthakur2/primesymmetryrev.pdf where the updated version is and will be maintained.

Below the fold is some more technical information regarding the above calculations.

To show how complicated the cancellations are we record triples with degree , then -power for the first sum for degree primes followed by the -valuation

for the first sum for degrees up to .

Since the computation was done to precision , we only know that for degree the sum had -valuation at least , but do not know precise value. (I would like to know!).

Remarks: It might be best for the first sum to calculate to precision about (ignoring and more) and second about . (The valuations are even, so e.g., or will be waste.). Rather than making a full list (huge) and then summing, it is better to calculate both sums term by term (simultaneously) as the P’s are produced.

Probably different range of polynomials can be checked for irreducibility and sum calculated for it on different computers and then combined at the end. Simplest calculation will be degree 38 calculation for the first sum to precision (since the earlier calculation had this precision). (To go beyond , I suggest , as you cannot improve it later). There are many more sums for different ‘s and powers ‘s. I have written up some answers in the links above, any of which can be checked independently.

Filed under: planning, polymath proposals ]]>

Polymath10 has started on my blog. The aim is to prove the Erdos-Rado sunflower conjecture (also known as the delta-system conjecture). Here is the wikipage.

Filed under: news ]]>