The REU coordinator for Summer 2015 is myself, Vic Reiner.
Here are answers
to some frequently asked REU questions.
Here are the reports from the REUs, in reverse chronological order. An asterisk indicates the REU coordinator.
Colin Aitken worked on hyperdeterminants of higher tensors over a field F, which generalize determinants of 2-tensors or matrices, and were studied by Cayley, Gelfand-Kapranov-Zelevinsky, and others. Aitken studied the set of nondegenerate 3-tensors in F^{2} ⊗ F^{k} ⊗F^{k+1}, that is, those having nonvanishing hyperdeterminant. Letting GL_{k} denote the invertible k x k matrices over F, he showed that the set of such nondegenerate 3-tensors carries a transitive action of GL_{_k} x GL_{k+1} acting in the second and third tensor positions, with the kernel of the action isomorphic to F^{x}. Thus it is a homogeneous space (GL_{_k} x GL_{k+1})/F^{x}, which identifits its topology when F is a topological field, and shows that it has cardinality q^{k^2} (q-1)^{2k} [k]!_{q} [k+1]!_{q} when F is a finite field having q elements. This last fact generalizes an unpublished result of Musiker and Yu. Here is his report.
Christian Gaetz, Michelle Mastrianni, Hailee Peck, Colleen Robichaux, David Schwein, and Ka Yu Tam (mentored by Rebecca Patrias) examined the K-theoretic analogue of RSK insertion and the K-Knuth equivalence, focussing on unique rectification targets: the increasing tableaux whose row words have no other tableaux row words within their K-Knuth equivalence class. They give several new examples of such tableaux, along with an algorithm to determine if two words are K-Knuth equivalent. Their poster on this topic won an Outstanding Presentation Award in the MAA Undergraduate Poster Session at the 2015 Joint Mathematics Meetings. Here is their arXiv preprint.
Christian Gaetz, Kyle Meyer, Ka Yu Tam, Max Wimberley, Zijian Yao and Heyi Zhu examined Dennis White's conjecture (from arXiv:0903.2831) on what he calls the (n,k)-Schur cone: the positive cone within the symmetric functions of degree n spanned by all products of Schur functions indexed by partitions with at most k parts. White showed for k=2 that the extreme rays of the cone can only come from products which are nested in a certain sense, and conjectured that these nested products actually are all extreme, proving this extremeness for a subclass of them. The students provide a simplified proof for this subclass, and proofs for some other subclasses, lending further support to the conjecture. They also collect some further data on the extreme rays of the (n,k) Schur functions for higher values of k. Here is their report and their arXiv preprint.
Jacob Haley, David Hemminger, Aaron Landesman, and Hailee Peck generalized a 2013 result of Barot and Marsh, who had shown that mutating an orientation of the Dynkin diagram of a finite Weyl group W allows one to correspondingly "mutate" the usual Coxeter presentation for W into a new and different presentation. Their generalization shows how to "mutate" the usual presentation of the (generalized) braid group for W into a new and different presentation. They also conjecture an analogous lifting to braid groups of a result of Felikson and Tumarkin for affine Weyl groups W. Here is their report and their arXiv preprint.
David Hemminger, Aaron Landesman, and Zijian Yao defined a new construction on a graded poset P called its edge poset E(P). They conjectured, and proved in many interesting cases, that if one starts with a a Boolean algebra B_{n} of rank n and forms the quotient by any subgroup G of the symmetric group, the quotient poset B_{n}/G which is well-known to be Peck, will have has its edge poset E(B_{n}/G) also Peck. This was motivated by a special case of a recent unimodality result of Pak and Panova, where B_{n}/G is the poset of Ferrers diagrams inside a rectangle. Here is their report, and their arXiv preprint.
Aaron Landesman proved a combinatorial conjecture by Stasinski and Voll that arose in their work on zeta functions for classical groups. The result asserts that a signed generating function sum for a certain length-like statistic over descent classes in the hyperoctahedral group has a simple q-multinomial-like product formula. Here is his arXiv preprint.
Kyle Meyer showed that the number of ways of tiling a planar figure using only horizontal 1-by-l and vertical m-by-1 polyominoes is #P-complete if max(l,m) is at least 3 (and min(l,m) is at least 2). This lies in between two known results: a result of Beauquier, Nivat, Remila and Robson showing that deciding the existence of such a tiling is NP-complete, and work of Kasteleyn showing that counting tilings by 1-by-2 and 2-by-1 dominos is the same as counting perfect matchings and can be done in polynomial time. Here is his report .
Mariya Sardali, Max Wimberley and Heyi Zhu studied quivers with vertices labeled 1,2,..,n which are periodic with period 2, in the sense that performing quiver mutation at vertex 1 and then at vertex 2 is isomorphic to the original quiver after relabeling (1,2,...,n) as (n-1,n,...,1,2,...,n-2). Among other things, they classifed all such period 2 quivers with 6 nodes, and found some infinite families of period 2 quivers of larger sizes. Here is their report .
Zijian Yao proved a conjecture of Max Glick regarding Schwartz's pentagram map on polygons. Schwartz had shown that when one performs n-1 iterates of the pentagram map, starting from an axis-aligned 2n-sided polygon, all of the vertices collapse to a single point; Glick had conjectured that this point is the barycenter of the original polygon. Yao proved this, along with variations for analogues of the pentagram, both in higher dimensions and in lower dimensions. Here is his report, and his arXiv preprint.
Josh Alman, Cesar Cuenca and Jiaoyang Huang studied recurrences that exhibit Laurent phenomenon. Some of the most famous among these are the Somos and the Gale-Robinson recurrences. Their approach is based on finding period 1 seeds of Laurent phenomenon algebras of Lam-Pylyavskyy. The main results include a complete classification in rank three, finding many new interesting families of examples, and generalizing a result of Fordy and Marsh to binomial seeds that do not fit into the setting of cluster algebras. Here is their report, and the arXiv preprint.
Josh Alman, Carl Lian and Brandon Tran proved a host of new results on circular planar electrical networks, following de Verdiere-Gitler-Vertigan and Curtis-Ingerman-Morrow. They construct a poset of electrical networks with n boundary vertices, and prove that it is graded by number of edges of critical representatives. They conjecture that the poset is Eulerian and prove an initial result in this direction. Also, adapting methods of Callan and Stein-Everett, they answer a variety of enumerative questions; notably, they give the number of elements of the poset (i.e., the number of equivalence classes of underlying graphs of electrical networks, or equivalently the number of cells in the space of response matrices). Alman, Lian and Tran also study positivity phenomena of the response matrices arising from circular planar electrical networks, extending work of Kenyon-Wilson and Postnikov. In doing so, they introduce electrical positroids, and discuss a naturally arising example of a Laurent phenomenon algebra, as studied by Lam-Pylyavskyy. Here is their report, and two resulting arXiv preprints, Part I being more on the electrical posets, and Part II more on the positivity phenomena. These two preprints were later combined in a single paper, that appeared in J. Combin. Theory Ser. A. (2015).
Cesar Cuenca studied the cluster algebras associated to double wiring diagrams that were introduced by Fomin and Zelevinsky, aiming toward the conjecture that any of their cluster variables is Schur positive when it is evaluated on a generalized Jacobi-Trudi matrix. Already an interesting special case of this conjecture is that of cluster variables which are one mutation step away from a double wiring diagram, and he proves it in the case where that one step is a mutation at a chamber of degree at most 6, using theory of Temperley-Lieb immanants. Here is his report.
Jiaoyang Huang, Andrew Senger, Peter Wear, and Tianqi Wu examined partition identities that appeared in a recent paper of Buryak and Feigin. Using algebraic geometry, the latter authors had shown the following two statistics on the set of partitions of n are equidistributed: the number of cells of the Ferrers diagram whose hook difference (the difference of its leg and arm lengths) is one, versus the largest part of the partition that appears at least twice. Haung, Senger, Wear and Wu prove this combinatorially in some special cases. They propose refinements of the Buryak and Feigin result and prove them in some cases combinatorially. In addition, they obtain a new formula for the q-Catalan numbers which naturally leads them to define a new q,t-Catalan number with a simple combinatorial interpretation. Here is their report and their arXiv preprint.
Megan Leoni, Seth Neel and Paxton Turner studied combinatorial aspects of cluster algebras motivated from the physics literature, building off of previous work from 2012 REU student Sicong Zhang. In particular, they explored new algebraic properties of the well-studied del Pezzo 3 (dP3) quiver and geometric properties of its corresponding brane tiling. This includes a factorization formula for the cluster variables arising from a large class of mutation sequences (called τ-mutation sequences) and construction of a new family of subgraphs of the dP3 brane tiling which they call Aztec Castles. Leoni, Neel, and Turner proved that for each τ-mutation sequence (which also corresponds to a path in the Affine A2 Coxeter Lattice), there is an Aztec Castle whose weighted number of perfect matchings agrees with the Laurent expansion for the associated cluster variable. Here is their report, and the arXiv preprint, which will appear in J. Physics A.
Abby Stevens and Shannon Gallagher studied a family of Laurent phenomenon algebras that are close to Ptolemy cluster algebras. It turns out that if if you "break" an arrow in a type A mutation class quiver, the resulting LP algebra is of finite type, and in some sense is still controlled by triangulations of a polygon. They prove the structural properties for a specific family of such algebras. Here is their report
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 F_{q} when one chooses a primitive element σ, how many m-dimensional F_{q}-subspaces W have the property that the n different translates W,σW,σ^{2}W,...,σ^{n-1}W "split" the extension as a direct sum? Chen and Tseng prove Ghorpade and Ram's conjecture that this number is [n]_{q^m} q^{m(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, which will appear in the journal Finite fields and their applications.
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, which appeared in Discrete Mathematics (2014).
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 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.
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 the
published version
(Elec. J. Combin. 20(3):P33, 2013) written with Gregg Musiker.
(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 appeared in J. Algebraic Combinatorics (37 (2013), pp. 545--569), and is also discussed in R.M. Green's book "Combinatorics of minuscule representations" (Section 11.3).
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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