Some outstanding open problems in the theory of Macdonald polynomials

Let $P_{n,k}$ be the collection of k-bounded partitions of n. The Macdonald polynomials in the family $\{H_{\mu}(x;q,t)\}$ (for $\mu \in P_{n,k}$) are jig-saw puzzles put together from bi-colored pieces decomposing the Hall-Littlewood polynomials in the family $\{H_{\mu}(x;0,t)\}$. In this talk we will present some computer data showing how the elements of the family $\{H_{\mu}(x;q,t)\}$ can be obtained from one another by changing bi-colors. This process is beautifully explained by some (still unproved) representation theoretical conjectures regarding the bi-graded $S_n$ module $M_{\mu}[x,y]$ which is the linear span of all the derivatives of $\Delta_{\mu}(x,y) = \det(x_i^{r_j} y_i^{s_j})_{i,j = 1}^n$, where $(r_1,s_1), \ldots, (r_n,s_n)$ are the southwest corners of the cells of the Ferrers diagram of $\mu$.

March 6, 2008

Pamela Fong Symposium

Richard Stanley

Massachusetts Institute of Technology

March 7, 2008

Pamela Fong Symposium

Richard Stanley

Massachusetts Institute of Technology

March 19, 2008

3:10 p.m.

Frank Sottile

Texas A&M University

Gale duality for complete intersections

Gale duality for polynomial systems is an elementary reformulation of a system of polynomial equations as a system of equations involving rational master functions in the complement of a hyperplane arrangement. Some properties of the original system are easier to understand in the Gale dual system. In this talk, I will describe this Gale duality, look at some examples of this construction, and give some elementary consequences. This is joint work with Frederic Bihan.

March 21, 2008

2:10 p.m.

Bruce Sagan

Michigan State University

Monomial Bases for NBC Complexes

Let G be a graph whose edge set $E=\{e_1,...,e_q\}$ has been totally ordered. A broken circuit of $G$ is a cycle whose smallest edge has been removed. Consider the NBC complex, $\Delta$, consisting of all subsets of $E$ which do not contain a broken circuit. Let $R$ be the Stanley-Reisner ring of $\Delta$, which is obtained from the polynomial ring $F[x_1,\ldots,x_q]$ (where $F$ is a field) by modding out the monomials corresponding to nonfaces of $\Delta$. Jason Brown gave an explicit description of a homogeneous system of parameters for $R$ in terms of fundamental edge-cuts in $G$. So $R$ modulo this h.s.o.p. is a finite dimensional vector space. We conjecture an explicit monomial basis for this vector space in terms of the circuits of $G$ and prove that the conjecture is true for several families of graphs.

3:10 p.m.

Benjamin Braun

University of Kentucky

The Complex of Non-Crossing Diagonals of a Polygon

Given a convex polygon P with n vertices, it is well known that there is an associated simplicial complex T(P) with vertices given by diagonals in P and facets given by triangulations of P. A theorem of C. Lee states that T(P) can be realized as the boundary complex of a polytope called the associahedron. We will investigate the homotopy type of T(P) for non-convex polygons using tools from discrete Morse theory. This work is joint with Richard Ehrenborg.

April 4, 2008

2:10 p.m.

Luis Serrano

University of Michigan

Noncommutative P-Schur functions, the shifted plactic monoid, and applications.

A symmetric function is a polynomial in $Q[x_1,x_2,...]$ which remains invariant when the variables get permuted. Symmetric functions form an algebra $\Gamma$, which arises in other fields of mathematics, such as representation theory, algebraic geometry, etc. An important basis for $\Gamma$ as a vector space is formed by Schur functions, which multiply according to the Littlewood-Richardson rule. Fomin and Greene have developed a theory of Schur functions in noncommutative variables which gives rise to generalizations of the Littlewood-Richardson rule. In this talk, we will develop an analogous theory of functions in noncommutative variables, for an important subalgebra of $\Gamma$; that generated by the P-Schur functions.

3:10 p.m.

Helene Barcelo

Arizona State University

The discrete fundamental group of the associahedron.

Over the last 50 years, combinatorial simplicial complexes have been extensively studied. One approach has been to study the homotopy groups of posets of faces of such simplicial complexes. More recently, Kramer and Laubenbacher introduced a notion of discrete homotopy, called A-theory, that has given surprising results. For example, Babson et al. showed that the k-th discrete fundamental group of the order complex of the Boolean lattice is nothing other than the (classical) fundamental group of the complement (over the real numbers) of the k-equal arrangement. In order to prove this result one is led to study the discrete fundamental group of the permutahedron. In a similar spirit, we study the discrete fundamental group of the type-A associahedron. In particular we show that this is a free group on (n+2) choose 4 generators. We believe that this notion of discrete homotopy will shed a new light on the study of polytopes, in particular on type-B (and other) associahedra.

April 11, 2008

2:10 p.m.

Francesco Brenti

Universita' di Roma "Tor Vergata"

Parabolic Kazhdan-Lusztig and R-polynomials for quasi-minuscule quotients

Kazhdan-Lusztig and R-polynomials were first defined by Kazhdan and Lusztig in [Invent. Math., 53 (1979), 165-184]. Since then they have found numerous applications, especially to representation theory and to the geometry of Schubert varieties. In 1987 Deodhar introduced parabolic analogues of these polynomials. These are related to their ordinary counterparts in several ways, and also play a direct role in other areas, including geometry of partial flag manifolds and the theory of Macdonald polynomials. In this talk I will study these polynomials for the quasi-minuscule quotients of the symmetric group. More precisely, I will first show how these quotients are closely related to ``rooted partitions'' and then I will give explicit, closed combinatorial formulas for the polynomials. For the Kazhdan-Lusztig ones these are based on a class of rooted partitions which seems to be new and imply that they are always (either zero or) a power of q, and that they are not combinatorial invariants. The results presented imply those for the Hermitian symmetric quotients obtained in [F. Brenti, Pacific J. Math., 207 (2002), 257-286]. I will conclude with a general conjecture. This is partially based on joint work with Federico Incitti and Mario Marietti.

3:10 p.m.

Kelli Talaska

University of Michigan

Boundary measurements in planar networks

For a planar directed graph G, Postnikov's boundary measurement map sends positive weight functions on the edges of G onto the appropriate totally nonnegative Grassmann cell. We establish an explicit formula for this map by expressing each Pluecker coordinate as a ratio of two combinatorially defined polynomials in the edge weights, with positive integer coefficients.

Apr. 18, 2008

3:10 p.m.

Mohamed Omar

University of California, Davis

Combinatorial Approaches to the Jacobian Conjecture

The Jacobian Conjecture is a long-standing open problem in algebraic geometry. Though the problem is inherently algebraic, it crops up in fields throughout mathematics including perturbation theory, quantum field theory and combinatorics. We survey the major contributions made toward combinatorial approaches to resolving the conjecture, mainly those by Singer and Wright. We present new proofs of their theorems and motivate their constructions. We also resolve the Symmetric Cubic Linear case, and present new conjectures whose resolution would prove the Jacobian Conjecture to be true.

April 25, 2008

3:10 p.m.

Jeff Doker

University of California, Berkeley

Matroid Polytopes as Minkowski Sums

Lots of familiar combinatorial objects can be described in terms of things called Matroids, and every Matroid can be represented as a polytope. It turns out that these Matroid polytopes, as well as some other related polytopes, can be decomposed into nice Minkowski sums of simplices.

May 2, 2008

2:10 p.m.

Nantel Bergeron

York University

Towers of algebras, Combinatorial Hopf algebras and Dual graded graphs.

Bergeron and Li have introduced a set of axioms which guarantee that the Grothendieck groups of a tower of algebras A_0+A_1+... can be endowed with the structure of graded dual Hopf algebras. Hivert and Nzeutzhap, and independently Lam and Shimozono constructed dual graded graphs from primitive elements in Hopf algebras. In this paper we apply the composition of these constructions to towers of algebras. We show that if a tower gives rise to graded dual Hopf algebras then we must have dim(A_n)=r^n n! where r = dim(A_1).

3:10 p.m.

Lauren Williams

Harvard University

Combinatorics and statistical physics: a story of hopping particles

The asymmetric exclusion process (ASEP) is a simple but rich model from statistical physics concerning particles hopping on a 1-dimensional lattice: it serves as a primitive model for traffic flow and appears in a sequence alignment problem in computational biology. This talk will provide a gentle introduction to the ASEP followed by connections of the ASEP to combinatorics, including the totally non-negative part of the Grassmannian and combinatorial Hopf algebras.

May 9, 2008

3:10 p.m.

Tristram Bogart

Queen's University, Canada

A Tropical Approach to Rational Curves on General Hypersurfaces

In the 1980's, Herbert Clemens made a series of conjectures about the dimensions of spaces of rational curves on general complex hypersurfaces in projective space. The most general of these conjectures is that there are only finitely many rational curves of degree d on a general quintic threefold in P^4. He proved that a general hypersurface of degree 2n-1 in P^n contains no rational curves, a statement generalized and strengthened by Geng Xu and by Claire Voisin. In ongoing joint work with Ethan Cotterill, we develop a new approach to these questions via tropical geometry. A tropical curve is a graph embedded in R^n in such a way that at each vertex, the primitive integer edge directions add up to zero. The curve is rational if the graph is a tree. The tropical hypersurface of a polynomial f is the polyhedral complex obtained as the codimension one skeleton of the normal fan of a certain subdivision of the Newton polytope of f. Since tropicalization preserves inclusion, the tropical analogue of Clemens' theorem would imply the original theorem. Magnus Vigeland recently produced a family of tropical surfaces in R^3 of degree d that contain no tropical lines when d is at least 4; our goal is to show that these same surfaces contain no tropical rational curves when d is at least 5. Such a proof, combinatorial in flavor, would have the benefits of being constructive and probably characteristic-free. Our current result is that Vigeland's surfaces contain no tropical rational curves that are generic in a certain sense.

May 14, 2008

2:10 p.m.

Maree Afaga

San Francisco State University