In the summer of 2012 Minah Oh and I ran an REU where we worked on a problem in combinatorial linear algebra known as Rota's Basis Conjecture.

The students we worked with were:  Stephanie Bittner (Virginia Wesleyan College), Michael Cheung (Elizabethtown College), Xuyi Guo (Stanford University), and Adam Zweber (Carleton College).  Everyone worked extremely hard and after eight weeks we were able to come up with some nice results.

Left to right:  Minah, Adam, Stephanie, Mike, Xuyi, Josh

Before describing the students' work, here is one statement of Rota's Basis Conjecture.  Suppose you are given n bases of an n-dimensional vector space.  Additionally suppose that each basis is assigned a particular color:  say the first basis is red, the second blue, etc.  Then Rota's Basis Conjecture asserts that one can always repartition the multiset union of these bases into n "rainbow" bases--that is, each new basis will contain exactly one vector of each color.  This innocent-looking conjecture has been open for over twenty years.  

Both mathematics REU groups out for lunch (the other group was supervised by Brant Jones and Edwin O'Shea).

Adam, Stephanie, and Xuyi defined and studied the "incidence matrix of disjoint transversals."  Specifically, suppose you are given n disjoint sets each of size n.  Form a matrix with rows and columns indexed by the collection of transversals of these n sets.  Put a one in the (i,j)-entry if transversal i is disjoint from transversal j, otherwise put a zero.  Aside from being an interesting combinatorial object in its own right, this zero-one matrix is very closely related to the n-dimensional case of Rota's Basis Conjecture.  The students managed to understand and describe several numerical invariants of this matrix, in particular its eigenvalues and Smith normal form.  See their full report here.  [Update:  see our final paper.]

Mike took a different approach to the problem, and spent most of the summer working on a computer program that can prove Rota's Basis Conjecture for any dimension.  Actually, Rota's Basis Conjecture can be generalized to an analagous statement about n bases in a rank n matroid, and Mike works in this full generality.  The generalized Rota's Basis Conjecture was previously known to be true only for the case n=3, and this summer Mike managed to prove the case n=4.  See his full report here.

The students presented their work to the Department of Mathematics and Statistics at JMU, see the videos/slides below:

Part 1 - Stephanie
Part 2 - Mike
Part 3 - Questions

Part 4 - Adam and Xuyi

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