Summer REU's (Research Experiences for Undergrads)

Vic Reiner
Upcoming REU info

We are planning on mentoring REUs in Summer 2012. The application deadline is February 16, 2012.
All applications must be submitted through the MathPrograms listing, which lists the necessary application materials.
(Please ignore the old MathJobs ad which was recently switched over,
as it has conflicting application deadline date information, and will not let you submit anything!)

Past REU info

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.

• Summer 2011 (co-mentored with Gregg Musiker and Pasha Pylyavskyy)
  participants: Rohit Agrawal, Francisc Bozgan, Jehanne Dousse, Daniel Hess, Benjy Hirsch, In-Jee Jeong, Shiyu Li, David B Rush, Danny Shi, Vladimir Sotirov, Fan Wei
  Group photo

  • Rohit Agrawal and Vladimir Sotirov examined a real cone inside the group algebra of the symmetric group S_n, 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.

  • 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.

  • 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.

• Summer 2010 (co-mentored with Dennis Stanton): CSP, up-maps, plane partitions, and binary trees
  participants: Charles Chen, Alan Guo, Xin Jin, Gaku Liu, Rik Sengupta, and Warut Suksompong
  PhD student REU TAs: Alex Csar, Jia Huang, Alex Miller, Nathan Williams
  Group photo
  • Alan Guo proved a 2006 conjecture of Sen-Peng Eu on a cyclic sieving phenomenon (CSP) occurring for noncrossing connected graphs. Here is his arXiv preprint, which appeared in Elec. J. Combinatorics.
  • Gaku Liu tried various attacks on the (still) notorious Borwein "+--" conjecture, and documented these in this report.
  • Rik Sengupta and Warut Suksompong introduced an interesting and well-behaved weakening of the Tamari order on binary trees, called the "comb order". The point of the comb order relates to several important but hard-to-compute functions of a pair binary trees
    • their distance in the rotation graph,
    • their meet and join in the Tamari order, and most importantly,
    • their number of common "parse words" as in the recent work of Cooper, Rowland and Zeilberger attempting to implement an approach to the 4-color Theorem pointed out by Kauffman.
    All of these functions become easy to compute when the two binary trees have an upper bound in the comb order. Here is their report, and arXiv preprint.
  • Charles Chen, Alan Guo, Xin Jin and Gaku Liu dug deeper into the "up maps" U_r: R_r --> R_r+1 expressing multiplication by x+y+z in the graded ring R=k[x,y,z]/(x^A,y^B,z^C). These maps have been examined recently by various commutative algebraists, such as Li and Zanello. Chen, Guo, Xin and Liu refined, re-proved, and generalized some of Li and Zanello's striking results, such as the fact that when A=a+b,B=a+c,C=b+c, the middle up map U_m for m=a+b+c-2 has determinant equal (up to sign) to the number of plane partitions in an a x b x c box. Their work generalizes this to some of the other symmetry classes of plane partitions (classes, 3,6,8). They also prove refined statements about how the Smith normal form of the up maps U_r with r below the middle value m are determined by the Smith normal form for the middle map U_m, via an interesting lemma on Smith normal forms of minors of Toeplitz matrices. Here is their report, and arXiv preprint, to appear in J. Commutative Algebra.

• Summer 2009
  • (no REU-- we took a break while Reiner was on sabbatical)

• Summer 2008 (co-mentored with Dennis Stanton): q-series, invariant theory, and critical groups
  participants: Kevin Carde, Joe Loubert, Andy Manion, Aaron Potechin, and Adrian Sanborn
  • Kevin Carde, Joe Loubert, Aaron Potechin, and Adrian Sanborn proved a conjecture of Guo-Niu Han on hook length expansions, which interpolates between two such classical hook expansions. Their report is here, and here is the ensuing arXiv preprint.
  • Andy Manion and Aaron Potechin resolved all of the conjectures left open by our REU work from the summers of 2003, 2004 on the structure of critical groups of line graphs. This led to them writing up the results in Sections 5,6 of this paper (joint with Berget, Maxwell, Reiner).
  • Aaron Potechin answered affirmatively a question in invariant theory, showing that "seaweed" subgroups of GL_n(F_q) always have invariant rings which are polynomial algebras. This work was the starting point for a vast generalization by Jia Huang that appears in this paper.

• Summer 2007 (co-mentored with Dennis Stanton): Sieving phenomena, alternating sign matrices, and holes in semigroups
  participants: Alex Cloninger, Fraser Chiu, Evgeny Demkhin, Gabriel Kreindler, Yuncheng Lin, Noah Stephens-Davidowitz
  • 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.

• Summer 2006 (co-mentored with Dennis Stanton): One-crossing partitions and differential posets
  participants: Mike Bergerson, Alex Miller, Andrew Pliml, Paul Shearer, Nick Switala
  • 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.

• Summer 2005 (co-mentored with Dennis Stanton): Partitions, plane partitions, and sieving phenomena
  participants: Omar Abuzzahab, Matt Korson, Martin Man-chun Li, Seth Meyer, Alex Miller
  • 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.

• Summer 2004
  • (No formal REU, although Andy Berget was supported to work on his previous projects, along with Molly Maxwell)

• Summer 2003: Critical groups and parking functions
  participants: Adil Ali, Ramon Calderer, Andy Berget, Jesse Plautz
  • 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 suggested a similar relation between their critical groups. His report is here.

• Summer 2002: Trees, determinants and Pfaffians
  participants: Scott Hirschman, Brian Jacobson, Minseung Kim, Andy Niedermaier
  • 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.).

• Summer 2001: Tree groups of shifted graphs
  participants: Hyung Kim, Hans Christianson
  • 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.)

• Summer 2000- 2 projects:
  • Tree groups of shifted graphs-
    participants: Paul Bendich, Tristram Bogart

    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.

  • One-parameter families of algebraic curves (co-mentored with Joel Roberts)-
    participant: Rory Mulvaney

    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

• Brian Jacobson went on to solve completely the problem of computing the critical group of a threshold graph, as part of an IT honors undergrad thesis. The other part of his thesis proves a conjecture of Kuperberg about the critical group of a planar graph and its isomorphism with a certain Kasteleyn-Percus matrix for the graph. A .pdf version of his thesis is here.

• David Treumann wrote an IT undergrad honors thesis about functoriality of the critical group, that is, in what situations to expect a homomorphism between the critical group of two graphs. A .pdf version of his thesis is here.

• Alex Miller wrote an IT undergrad honors thesis continuing his work on the Smith normal forms of the up-down and down-up maps in differential posets. A .pdf version of his thesis is here. This work led to a paper with V. Reiner that appeared in the journal ORDER.

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