Our 11th annual REU program in combinatorics will be held in the summer of 2013,at the University of Minnesota, Minneapolis.
We expect at least 6-8 participants to spend 8 weeks in residence sometime between May 27, 2013 and August 2, 2013.
The application deadline is Thursday, February 7, 2013.
Applications are submitted through the
MathPrograms listing, which lists the necessary application materials.
Some published papers that resulted from my early REUs are on my papers web page
Here are the reports from the REUs, in reverse chronological order.
Rediet Abebe and Joshua Pfeffer studied simplicial complex generalizations of two conjectures on the graph Laplacian eigenvalues. The first generalizes the Grone-Merris conjecture, which is known for graphs by work of H. Bai, but wide open in all higher-dimensions. Abebe and Pfeffer make progress toward the second inequality in this conjecture for 2-dimensional complexes. The second generalization is of a conjecture of A. Brouwer for graphs that is still open. They give several potential generalizations, including one which they verify both for shifted complexes and for simplicial trees in the sense of S. Faridi. Here is their report.
Eric Chen and Dennis Tseng proved the recent "Splitting subspace" conjecture of S. Ghorpade and S. Ram, answering a 15-year-old question of Niederreiter: in a degree mn extension of the finite field Fq when one chooses a primitive element σ, how many m-dimensional Fq-subspaces W have the property that the n different translates W,σW,σ2W,...,σn-1W "split" the extension as a direct sum? Chen and Tseng prove Ghorpade and Ram's conjecture that this number is [n]q^m qm(m-1)(n-1), independent of σ. Their method lets them compute a product formula for a far-reaching generalization of this enumeration problem. Here is their report, and their arXiv preprint.
Xin Chen and Jane Wang studied the so-called "super Catalan numbers" S(m,n) = (2m)!(2n)!⁄m!n!(m + n)! of I. Gessel. S(m,n) is known to be an integer, and has a combinatorial interpretation due to Gessel and Xin for m=2,3. Chen and Wang give simple lattice path interpretations for S(n,m) when n-m is at most 3, and a not-so-simple such interpretation for n-m=4. In addition, their methods give them expressions for the q-analogues of S(m,n) as polynomials in q with nonnegative coefficients for n-m at most 3. They also give some connections of S(m,n) with annular noncrossing partitions, and examine some other ratios of factorials that turn out to be integers, including conjectures about q-analogues having nonnegative coefficients as polynomials in q. Here is their report, and their arXiv preprint.
Horia Mania worked on Wilmes' Conjecture on the Betti numbers in the minimal free resolution of certain ideals related to abelian sandpiles. He used Hochster's formula to prove the conjecture for the first Betti number, and introduced ideas, such as boundary divisors, that may be helpful for a combinatorial proof for the higher Betti numbers.
Here is his report,
and his arXiv preprint.
In independent work appearing Fall 2012, two groups, Mohammadi and Shokrieh
(their arXiv preprint)
and Manjunath, Schreyer, Wilmes
(their arXiv preprint)
used alternative algebraic methods to prove the conjecture for all Betti numbers.
Dennis Tseng considered maps induced on critical groups by graph coverings. For n-sheeted coverings, the map on critical groups surjects, and splits at primes p not dividing n. For regular coverings one can identify its kernel as a naturally defined "critical group" of the voltage graph describing the covering. For double covers, the voltage graph is a signed graph with critical group defined in terms a Laplacian matrix that appears in work of Zaslavsky. One can generalize this to a notion of "double coverings" of signed graphs, and use this to reinterpret a result of H. Bai on the p-primary structure of the critical groups of n-cubes for odd primes p. Here is his report, and the ensuing arXiv preprint with Reiner.
Sicong Zhang studied combinatorial aspects of cluster algebras motivated from the physics literature. In particular, string theorists such as A. Hanany and R.-K. Seong study certain families of quivers and construct duals for them given as tilings of a torus, known as a brane tiling. Zhang investigated several such examples, including a six-vertex quiver associated to the dP_3 lattice. Certain subgraphs of this tiling were previously studied by C. Cottrell-B.Young and M. Ciucu after being introduced by J. Propp under the name Aztec Dragons. Zhang proved that a certain infinite sequence of cluster variables associated to this quiver has the property that their Laurent polynomial expansions can be expressed, under a suitable weighting scheme, in terms of perfect matchings of these subgraphs. Here is his report.
Rohit Agrawal and Vladimir Sotirov examined a real cone inside the group algebra of the symmetric group Sn, introduced by Stembridge, dual to the cone of monomial-positive immanants of n-by-n matrices. Stembridge showed that this cone has finitely many extreme rays for n at most 5, and asked if there are finitely many in general. Agrawal and Sotirov present some general relations among the generators of the cone, and use this to exhibit its (finitely-many) extreme rays for n=6. Here is their report.
Rohit Agrawal, Vladimir Sotirov and Fan Wei
made more explicit a bijection of Cools et al.
between rectangular standard Young tableaux
and G-parking functions as representatives for chip-firing
groups on certain graphs G. They then used this to prove
that, under the bijection, the evacuation involution on
tableaux corresponds to vertical reflection of the graph.
Here is their report,
and their
arXiv preprint.
(Fan Wei later received the 2012 AWM Alice T. Schafer Prize
in part for this work.)
Francisc Bozgan attempted to prove a conjectural Jacob-Trudi-style determinant formula for the dual stable Grothendieck polynomials of Lam and Pylyavskyy, corresponding to a partition. He has so far has proven it in the case where the partition has at most two columns in its Ferrers diagram, using the notion of elegant fillings. Here is his report.
Jehanne Dousse investigated a question suggested by this recent theorem of John Stembridge, motivated by the digraphs governing Kazhdan-Lusztig cell representations of Coxeter groups: for a fixed integer polynomial p(x), there are only finitely many strongly-connected digraphs whose adjacency matrix A satsifies p(A)=0. For quadratic p(x), Dousse classifies these digraphs completely. For cubic and higher degree p(x), she gives a necessary condition. She also analyzes the solutions of maximal size for some particular families of polynomials, using known results on strongly regular graphs and the directed line graph construction. Here is her report.
Daniel Hess and Benjy Hirsch showed that the simplicial complexes of strongly and weakly separated subsets of {1,2,..,n}, after removing cone points, have the homotopy types of an (n-3)-sphere and a point, respectively. Furthermore, they show that one has equivariant homotopy equivalences with respect to a natural Z/2Z x Z/2Z-action. Here is their report, and their arXiv preprint, which has appeared in Topology and its Applications (160 (2013), pp. 328-336).
In-Jee Jeong proved explicit formulas for certain cluster variables in cluster algebras derived from planar bipartite graphs, when one performs particular sequences of mutations. The formulas turn out to be generating functions for perfect matchings of certain subgraphs of the original graph. Here is his report.
Shiyu Li investigated patterns generated by sequences of quiver mutations using the theory of cluster algebras. Starting with a certain cyclic quiver, she demonstrated relations between the sequences obtained via mutations and the Fibonacci numbers. Here is her report.
David B Rush and Danny Shi showed that for any minuscule poset P, one has a cyclic sieving phenomenon for the triple (X,X(q),C) in which X is the set of order ideals of P or of P x [2] (where [2] is a 2-element chain), X(q) is the q-count for the orders by cardinality, and C is the cycle group generated by the action on order ideals or antichains of P considered by Duchet, Brouwer-Schrijver, Fukuda, Cameron-FonDerFlaass, and Panyushev at various levels of generality. Their proof for P is case-free, and uses the theory of minuscule heaps and fully commutative elements, while the proof for P x [2] uses the classification of minuscule posets. Here is their report, and their arXiv preprint; the paper itself is to appear in J. Algebraic Combinatorics.
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.
Some related preprints
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