Algebra-Geometry-Combinatorics Seminar
This seminar meets every Friday in the SFSU Mathematics Department (in Thornton Hall 211--directions to campus). For further information, please contact the seminar organizers (Federico Ardila, Matthias Beck, Joseph Gubeladze, and Serkan Hosten).
Dates for Spring 2008: Feb 29 Mar 6 Mar 7 Mar 14 Mar 19 Mar 21 Apr 4 Apr 11 Apr 18 Apr 25 May 2 May 9 May 14
February 29, 2008
2:10 p.m.
Brandon Rhoades
University of Minnesota
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Cyclic Sieving, Promotion, and Representation Theory
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Let X be a finite set, C = 〈c&rang be a finite cyclic
group acting on X, and X(q) &isin Z[q] be a polynomial over the
integers. Following Reiner, Stanton, and White, we say that the
triple (X, C, X(q)) exhibits the cyclic sieving phenomenon if
for any integer d &ge 0, the number of fixed points of
cd is equal to
X(ζd), where &zeta is a
primitive |C|th root of unity. We
prove a pair of conjectures of Reiner et al. concerning cyclic
sieving phenomena where X is the set of standard tableaux of a fixed
rectangular shape or the set of semistandard tableaux with fixed
rectangular shape and uniformly bounded entries and C acts by jeu de
taquin promotion. Our proofs involve modeling the action of
promotion via irreducible
GLn(C)-representations
constructed using the dual canonical basis and the Kazhdan-Lusztig
cellular representations.
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3:10 p.m.
Adriano Garsia
University of California, San Diego
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Some outstanding open problems in the theory of Macdonald polynomials
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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 $\mu$.
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March 6, 2008
Pamela Fong Symposium
Richard Stanley
Massachusetts Institute of Technology
| TITLE: A survey of plane tilings.
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ABSTRACT: We will survey some of the highlights of the theory of plane
tilings, focusing on tiling a bounded region of the plane with
finitely many tiles. A standard example, though not very
mathematical, is a jigsaw puzzle. We consider such questions as the
following: (1) Is there a tiling? (2) How many tilings are there? (3)
About how many tilings are there? (4) Is a tiling easy to find? (5) Is
it easy to prove or to convince someone that a tiling doesn't exist?
(6) What does a typical tiling look like? We point out some
interesting connections between tilings and such topics as computer
science, continued fractions, probability theory, and mathematical
logic.
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4:10 PM in HSS 317
March 7, 2008
Pamela Fong Symposium
Richard Stanley
Massachusetts Institute of Technology
| TITLE: Increasing and decreasing subsequences.
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ABSTRACT: A subsequence of a permutation of 1,2,..., n is *increasing* if its elements appear in increasing order. For instance, 4578 is an
increasing subsequence of 43571826. A *decreasing subsequence* is
similarly defined. We will survey the subject of increasing and
decreasing subsequences, focusing on what can be said about the
longest increasing and longest decreasing subsequence of a
permutation. Topics will include (a) relationship to Young tableaux
and the famous RSK algorithm, (b) the asymptotic behavior of the
length of the longest increasing subsequence (due to Baik, Deift, and
Johansson), (c) connections with random matrix theory, and (d) an
extension of the theory from permutations to complete matchings.
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4:10 PM in SCI 210
March 14, 2008
3:10 p.m.
Sangwook Kim
George Mason University
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Flag enumeration of matroid base polytopes
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For a matroid M on [n], the matroid base polytope Q(M) is the convex polytope in R^n whose vertices are the incidence vectors of the bases of M. In this talk, we study flag structures of matroid base polytopes. We describe faces of matroid base polytopes in terms of matroid data, and give conditions for hyperplane splits of matroid polytopes. Also, we apply this to the cd-index of a matroid base polytope of a rank 2 matroid.
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March 19, 2008
3:10 p.m.
Frank Sottile
Texas A&M University
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Gale duality for complete intersections
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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.
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March 21, 2008
2:10 p.m.
Bruce Sagan
Michigan State University
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Monomial Bases for NBC Complexes
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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.
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3:10 p.m.
Benjamin Braun
University of Kentucky
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The Complex of Non-Crossing Diagonals of a Polygon
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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.
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April 4, 2008
2:10 p.m.
Luis Serrano
University of Michigan
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Noncommutative P-Schur functions, the shifted plactic monoid, and applications.
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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.
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3:10 p.m.
Helene Barcelo
Arizona State University
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The discrete fundamental group of the associahedron.
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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.
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April 11, 2008
2:10 p.m.
Francesco Brenti
Universita' di Roma "Tor Vergata"
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Parabolic Kazhdan-Lusztig and R-polynomials for quasi-minuscule quotients
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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.
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3:10 p.m.
Kelli Talaska
University of Michigan
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Boundary measurements in planar networks
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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.
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Apr. 18, 2008
3:10 p.m.
Mohamed Omar
University of California, Davis
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Combinatorial Approaches to the Jacobian Conjecture
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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.
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April 25, 2008
3:10 p.m.
Jeff Doker
University of California, Berkeley
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Matroid Polytopes as Minkowski Sums
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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.
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May 2, 2008
2:10 p.m.
Nantel Bergeron
York University
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Towers of algebras, Combinatorial Hopf algebras and Dual graded graphs.
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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).
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3:10 p.m.
Lauren Williams
Harvard University
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Combinatorics and statistical physics: a story of hopping particles
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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.
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May 9, 2008
3:10 p.m.
Tristram Bogart
Queen's University, Canada
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A Tropical Approach to Rational Curves on General Hypersurfaces
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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.
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May 14, 2008
2:10 p.m.
Maree Afaga
San Francisco State University
Previous semesters
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