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r 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 comments 4 february 23 2017 rota s basis conjecture polymath 12 filed under polymath proposals tchow8 11 44 pm this post tentatively kicks off polymath 12 on rota s basis conjecture 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 n bases of an n dimensional vector space v not necessarily distinct or disjoint then there exists an n n grid of vectors v ij such that 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 idea 1 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 β n i e that e may be partitioned into n disjoint common independent sets each necessarily of size n 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 i i i for all i call this a triangular system then i claim that there exists a way of choosing a v i from each i i in such a way that j 1 v i is independent the proof is easy we start with the forced choice v 1 i 1 and then note that by the independent set axiom since i 2 2 there must exist some v 2 i 2 that can be added to v 1 to produce an independent set of size 2 similarly once v 1 and v 2 are chosen it follows directly from the independent set axiom that we can add some v 3 i 3 and so on this proves the lemma now once j 1 has been constructed we can imagine removing the elements of j 1 from the original triangular system to obtain a smaller triangular system we can then repeat the argument on this smaller system to form an independent set j 2 that contains exactly one element from each i i for i 2 3 n this shows that the original triangular system can be partitioned into at most n common independent sets where as before the second matroid is the natural transversal matroid 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 i not worrying about whether the columns are bases and draw a diagonal to split the bases into two disjoint triangular systems one of size n and one of size n 1 so we can partition the vectors into at most n n 1 2 n 1 common independent sets q e d 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 2 n 2 idea 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 k n suppose further that these nk vectors counted with multiplicity can be partitioned into k bases somehow but not necessarily bases that contain exactly one vector from each row then we can ask if there exists an n k grid whose i th row comprises the elements of i i and whose columns are all bases in general the answer will be no but it is not so easy to come up with counterexamples i came up with two counterexamples with k 2 but i think it would be worth doing a computational search for more examples even k 2 and n 5 has not been checked as far as i know if there are not many counterexamples then there is some hope that we could classify them all and i think that this would be a big step towards proving the full conjecture note one family of counterexamples has been identified by harvey kiraly and lau idea 3 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 n then we ask for the usual n n grid except now we only require the columns to be independent and not necessarily bases as far as i know there is no known counterexample to this stronger conjecture moreover this stronger conjecture is known to be true if we fix a single standard basis b of v and insist that every i i be a subset of b two proofs of this fact may be found in section 2 of this paper 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 comments 112 august 13 2016 mo polymath question summary of proposals filed under polymath proposals gil kalai 7 23 pm here are the math overflow polymath proposals given in response to the polymath question 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 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 comments 2 february 7 2016 polymath 11 is now open filed under news gil kalai 3 45 am 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 comments 7 polymath proposals on math overflow filed under news polymath proposals gil kalai 3 21 am 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 comments 3 explaining polynomials identities success filed under news gil kalai 3 03 am 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 leave a comment january 2 2016 crowdmath project for high school students opens on march 1 filed under polymath proposals terence tao 4 25 pm tags art of problem solving mit primes the mit primes program and the art of problem solving are planning to run a crowdmath project for high school students with advanced mathematical backgrounds based on the polymath approach to mathematical research the project which officially starts on march 1 will be devoted to original research on a mathematics problem to be specified at the time of the project but judging from the reference material provided it will probably involve the combinatorics of 0 1 matrices participation is open to all high school students though they will need an art of problem solving account comments 2 december 28 2015 polymath proposal explaining identities for irreducible polynomials filed under planning polymath proposals terence tao 7 05 pm i am posting this proposal on behalf of dinesh thakur 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 more comments 63 previous page next page recent comments polymath plus ai c on two polymath of a sort propo ten recent questions on two polymath of a sort propo anonymous on rota s basis conjecture anonymous on rota s basis conjecture anonymous on polymath projects 2021 richard montgomery a on rota s basis conjecture anonymous on crowdmath projec anonymous on rota s basis conjecture anonymous on a sort of polymath on a famous anonymous on a new polymath proposal relat anonymous on polymath projects 2021 anonymous on polymath proposal bounded gap anonymous on research thread v determinst anonymous on a new polymath proposal relat anonymous on polymath8 a success polymath wiki most recent changes an error has occurred the feed is probably down try again later blogroll gil kalai michael nielsen terence tao the polymath wiki tim gowers projects mini polymath1 imo q6 2009 polymath1 density hales jewett polymath2 embedded c_0 or l p in explicit banach spaces polymath4 finding primes deterministically polymath5 erdos discrepancy problem proposals boshernitzan s problem cap sets other proposals polynomial hirsch conjecture proposals on wordpress pages about general discussion general polymath rules how to use latex in comments categories discussion 9 finding primes 8 hot spots 7 improving roth bounds 1 mock up 2 news 9 planning 7 polymath proposals 33 polymath5 1 research 13 search 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