Dennis Stanton and I co-mentored REU students in Summer 2005, 2006, 2007.
We will co-mentor an REU again in Summer 2008,
with the application deadline of February 21, 2008.
Here is the application info.
Starting in the summer of 2000, I've been involved in mentoring summer REU projects in the School of Mathematics at the Univ. of Minnesota. Many of the projects involved spanning trees of graphs, Kirchhoff's Matrix-Tree theorem and its variants, graph Laplacians, chip-firing games and critical groups of graphs.
In particular, the critical group of a graph is an isomorphism invariant in the form of a finite abelian group. Its order is the number of spanning trees in the graph. Although there are many classes of graphs for which the spanning tree number is known, often through calculation of Laplacian eigenvalues, the structure of the critical group had been computed explicitly for very few examples prior to some of these REU's.
Some published papers that resulted from these REU's may be found toward the bottom of my papers web page
Here's a rough summary of what happened in the REU's.
We performed experiments in Maple to guess the structure of the critical group for threshold graphs. A conjecture was formed in the "generic" case, and proven in some very special cases. See the REU report on the Math REU page.
Mulvaney produced software for visualizing algebraic curves in the real affine plane using MATLAB. In particular, one can use it to animate one-parameter families of such curves. See the REU report on the Math REU page.
Christianson proved the conjecture made by Bendich and Bogart during Summer 2000. See the REU report on the Math REU page, and "The critical group of a threshold graph" on my papers page (appeared in Lin. Alg. Appl.)
Jacobson and Niedemaier calculated the critical group structure for complete multipartite graphs, and calculated (almost all of) the structure for Cartesian products of complete graphs. See the REU report on the Math REU page, and "Critical groups for complete multipartite graphs and Cartesian products of complete graphs" on my papers page (to appear in J. Graph Theory).
Hirschman gave a simple sign-reversing involution proof for the recent Pfaffian Matrix-Tree Theorem of Masbaum and Vaintrob. See the REU report on the Math REU page, and "Note on the Pfaffian Matrix-Tree Theorem" on my papers page (to appear in Graphs and Combinatorics).
Kim looked at enumerating spanning trees in Cartesian products of complete graphs; see the REU report on the Math REU page. Pursuing this project further eventually led to the paper (joint with Jeremy Martin) "Factorizations of some weighted spanning tree enumerators" on my papers page (to appear in J. Combin. Theory Ser. A.).
Calderer and Plautz gave a simple deletion-contraction proof on the relation between the recently defined G-parking functions and an evaluation of a Tutte polynomial; their report is here.
Berget investigated the critical group structure for some regular line graphs (in particular, for some line graphs of regular graphs). It was known that there is a simple relation between the number of spanning trees for a regular graph and for its line graph. Berget's results suggest there may be many regular graphs where there is a also a similar relation between their critical groups. His report is here.
Abuzzahab, Korson, Meyer and Li investigated a cyclic (and even dihedral) sieving phenomena occurring in the context of MacMahon's formula generating function for plane partitions in a box. They formulated some fascinating conjectures, and managed to prove one of these, along with special cases of the others. Their report is here.
Miller found a beautiful conjecture about the Smith normal form of A^T A, where A is the incidence matrix between two adjacent ranks in Young's lattice. His report is here.
Early on, Abuzzahab and Meyer did some preliminary investigations into a possible cyclic sieving phenomenon occuring the context of the hook-formula counting linear extensions of posets which are forests of rooted trees. This didn't look so promising, and so they were eventually re-deployed. But their draft report on what they did find is here.
The whole group re-examined and generalized a result of M. Bona stating that the number of partitions of an n-set with exactly 1-crossing is (2n-5 choose n-4). The more general result also lends itself to a nice instance of the cyclic sieving phenomenon. The report is here. The draft of a short note based on this is here.
Alex Miller made great progress on proving his conjecture from Summer 2006, and generalized it to r-differential posets, even proving it for the Fibonacci differential posets. His report is here.
Alex Cloninger and Noah Stephens-Davidowitz worked on questions about various interesting symmetries on alternating sign matrices (including Wieland's "gyration") and accompanying cyclic sieving phenomena. Their report is here.
Fraser Chiu also looked at ASM's from the point of view of fully-packed loops and their associated pair of noncrossing matchings. He examined several questions about which pairings can be achieved. His report is here.
Yuncheng Lin looked at a subclass of partitions inside an M-by-N rectangle with restrictions on their "hook differences", introduced by Bressoud in the context of the notorious Borwein "+ - -"-conjecture. He showed that setting q=-1 in the q-count for these partitions gives the number of such partitions for a rectangle of dimensions roughly M/2-by-N/2, suggesting a "q=-1 phenomenon" in this context. His report is here.
Evgeny Demekhin and Gabriel Kreindler looked at "hole" problems related to degree sequences of k-uniform hypergraphs, that is, which vectors can/cannot be achieved by as combinations of a fixed subset of nonegative integer vectors whose coordinates sum to k, where the coefficients might be the rationals, the integers, the positive integers, etc. These are closely related to commutative algebra questions about affine semigroup rings generated by monomials of degree k, and their normalizations. The answers for k=2 are well-known, but their report shows that many things go wrong for k=3, although a few things work well for all k. Their report is here.
Some related preprints
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