
Date Time 
Location  Speaker 
Title – click for abstract 

01/23 3:00pm 
MILN 216 

Organizational Meeting 

01/30 3:00pm 
MILN 317 
Marcelo Aguiar TAMU 
The Hopf monoid of generalized permutehedra
Joyal's notion of species constitutes a good framework for the study of certain algebraic structures associated to combinatorial objects. We briefly review the notion of "Hopf monoid" in the category of species and then illustrate it with several examples. We introduce the Hopf monoid of generalized permutahedra (the latter are certain polytopes recently studied by Postnikov, Reiner and Williams). Our main result is an explicit antipode formula for this Hopf monoid. We explain how reciprocity theorems of Stanley on graphs and of Billera, Jia and Reiner on matroids can be deduced from this result.
The talk borrows from joint works with Swapneel Mahajan and with Federico Ardila.
Note: this talk may serve as an introduction to the working algebra seminar that will meet on Fridays at 2. 

02/06 3:00pm 
MILN 317 
J.M. Landsberg TAMU 
On the geometry of holographic algorithms
Many counting problems, such as determining the number of perfect matchings of a planar graph, have algorithms that resolve them "quickly", i.e., in a polynomial number of steps with respect the the input data (e.g. number of edges in the graph). Others can be checked quickly, but there is no known algorithm for solving them quickly, such as counting the number of perfect matchings in an arbitrary graph. Whether or not the lack of a good algorithm for the second class of problems is just due to our ignorance or because no such algorithm exists, is a central, if not the central question in complexity theory. Recently L. Valiant has developed algorithms for quickly solving some such problems, which he calls "holographic algorithms". Jason Morton and I have interpreted some of Valiant's work geometrically, and the range of applicability of our version of Valiant's method hinges on a question in combinatorics that I will present. 

02/13 3:00pm 
MILN 317 
Aaron Lauve TAMU 
Hopf structures on binary trees (variations on a theme)
We discuss several algebraic structures that can be placed on the multiplihedra {M_{n}}, a family of polytopes defined by Stasheff in the study of higher categories and homotopy theory. The structures we find are best understood by considering the multiplihedra's relationship to two other families of polytopes more familiar to combinatorists: the permutahedra {P_{n}} and the associahedra {A_{n}}. These objects were given the structure of Hopf algebra by MalvenutoReutenauer and LodayRonco, respectively. In this talk, we give M the structure of Pmodule and AHopf module algebra in a manner respecting the cellular maps from P to M to A. We also give a basis of coinvariants for a second Hopf module structure over A. (This is joint work with F. Sottile and S. Forcey.) 

02/19 1:00pm 
MILN 317 
Sonia Natale U. Nacional de Córdoba – Argentina 
Simple Hopf algebras and finite groups
We will discuss the simplicity of some semisimple Hopf algebras arising from finite groups through cocycle deformations. 

02/20 1:00pm 
MILN 317 
Nicolás Andruskiewitsch U. Nacional de Córdoba – Argentina 
On the classification of finitedimensional pointed Hopf algebras
We give an overview of the present state of this problem, illustrated by the main examples coming from the theory of quantum groups. 

02/27 3:00pm 
MILN 317 
Martin Malandro Sam Houston State University 
Fast Fourier Transforms for Inverse Semigroups
The classical fast discrete Fourier transform has found a wealth of applications ranging from seismic analysis to the fast multiplication of large numbers. There is a general theory of Fourier transforms for finite groups, in which the Fourier transform on the cyclic group of order n is the classical Fourier transform. In this talk we will generalize the notion of the Fourier transform to finite inverse semigroups (which are grouplike objects that capture local symmetries) and provide a general framework for constructing fast Fourier transforms on these objects. We will also discuss fast Fourier transforms for specific inverse semigroups of interest and potential applications. We will proceed from a representationtheoretic point of view and emphasize the algebra involved in the construction of these algorithms.
This talk should be accessible to grad students with an interest in algebra (I will begin by defining the classical discrete Fourier transform, representations, inverse semigroups, etc.) 

03/06 3:00pm 
MILN 317 
Kyle Petersen University of Michigan 
Promotion and cyclic sieving
Let X be a finite set acted upon by a cyclic group C. The cyclic sieving phenomenon (CSP), due to Reiner, Stanton, and White, is the name given to the special situation when there exists a polynomial f that precisely encodes the Corbit structure of X. (Usually f is a certain qenumertor for the elements of X.) As a toy example, let X be all 2subsets of a 4 element set, and let C act by shifting mod 4. There are two orbits: {2,3} → {3,4} → {1,4} → {1,2}, and {1,3} → {2,4}. Now let f = [{4}choose{2}]_{q} = 1 + q + 2q^{2} + q^{3} + q^{4}. This is the standard qanalogue of the binomial coefficient: it counts lattice paths in a 2by2 box by area. If we plug in powers of 4th roots of unity, that is, powers of i=√1, we recover the fixedpoint set sizes. We get f(i) = 0, f(1) = 2, f(i^{3}) = 0, f(1) = 6, which tells us there are 2 subsets fixed by two shifts and 6 fixed by four shifts, whereas one or three shifts fix nothing. I will present several examples of the CSP related to Schützenberger's action of jeu de taquin promotion. This can be visualized nicely on rectangular Young tableaux with two or three rows, as shown in joint work with Pylyavskyy and Rhoades. Very recently, in joint work with Serrano, instances of CSPs for certain shifted tableaux and for reduced expressions of the longest element of the hyperoctahedral group have emerged. 

03/13 3:00pm 
MILN 317 
John Irving Saint Mary's University, Nova Scotia 
Transitive Factorizations in the Symmetric Group
The aim of this talk is to introduce the audience to the combinatorics of "transitive factorizations" of permutations. A transitive factorization is a decomposition of a given permutation into an ordered product of others, where the factors act transitively on the underlying set of symbols. The special case in which the factors are transpositions dates back to Hurwitz, who recognized that this can serve as a combinatorial model for certain branched coverings of Riemann surfaces. Much has been learned since then, but despite many glimpses at beautiful combinatorial structure, still surprisingly little is known about the nature of these factorizations. We will survey various enumerative problems in this area, beginning with Hurwitz's original problem and moving towards recent related work on "star factorizations". 

03/27 3:00pm 
MILN 317 
Kia Dalili University of Missouri 
Asymptotic behavior of value semigroups
Let ν be a valuation dominating a noetherian local domain, and let S = {ν(r)  r ∈ R} be its value semigroup and Γ = S – S its value group. The possible value groups Γ have been extensively studied and classified classically, the value semigroup S however is much less understood. In this talk we will briefly look at the known results classifying value groups and some well known constraints on the value semigroups then we will look at growth rate of the value semigroup and its asymptotic behavior to obtain new constraints on possible value semigroups. This is joint work with S.D. Cutkosky and O. Kashcheyeva. 

03/30 2:00pm 
MILN 216 
Bill Schmitt George Washington University 
The free splice of matroids
Given matroids M(A) and N(B) such that the restrictions M(A ∩ B) and N(A ∩ B) are equal, an amalgam of M and N is a matroid L on A ∪ B such that LA = M and LB = N. Amalgams might not exist, and even when they do, there might be no freest amalgam. In this talk we consider the "twisted" version of this situation, in which the restriction N(A ∩ B) is equal to the contraction M/(A – B). In this case we define a splice of M and N to be a matroid L on A ∪ B such that LA = M and L/(A – B) = N. In contrast to the situation for amalgams, it turns out splices always exist and, furthermore, there is a freest splice, which we call the free splice of M and N. (In the case of disjoint A and B, the free splice is the free product, introduced by Henry Crapo and William Schmitt in 2005.) We show how to construct the free splice, as a certain Higg's lift, and give cryptomorphic descriptions of it, in terms of bases, independent set, rank function, flats and cyclic flats. We show that minors of free splices are free splices of higgs lifts of corresponding minors, and we characterize, in terms of cyclic flats, matroids that are irreducible with respect to free splice. We describe the associativity properties of free splice and examine its interaction with some other matroid operations. This is joint work with Joe Bonin. 

04/03 3:00pm 
MILN 317 
Svetlana Poznanovik TAMU 
Major index for 01fillings of moon polyominoes
It is a classical result by MacMahon that the inversion number and the major index have the same distribution over the set of rearrangements of a fixed multiset. Recently, a major index was defined for matchings and set partitions and it was shown that it has the same distribution with the number of crossings. I will present a definition of a major index for fillings of moon polyominoes with zeros and ones. This statistic, when specialized to certain shapes, reduces to the major index for words and set partitions. Moreover, I will show that it is equally distributed with the number of northeast chains, which are a natural generalization of inversions and crossings. 

04/10 3:00pm 
MILN 317 
Philip Matchett Wood Rutgers University 
On the probability that a discrete complex random matrix is singular
Consider a square matrix with n rows and n columns, where each entry is filled in independently at random +1 or 1, each with probability 1/2. What is the probability that this matrix is singular, as a function of n as n becomes large?
Matrices with +/1 entries serve as a test case for studying discrete random matrices, as opposed to continuous random matrices. Discrete random matrices often appear in practice, for example, when studying a large matrix computation with a computer, since the representation in the computer of the matrix (and any noise) is inherently discrete. The first exponential upper bound on the probability that a +/1 random matrix is singular was proved by Jeff Kahn, Janos Komlos, and Endre Szemeredi in 1995, and Terence Tao and Van Vu made a another breakthrough in 2006 by using a structural result from additive combinatorics. This talk will discuss a recent improvement on the approach of Tao and Vu that leads to the best know upper bounds on the probability that such a matrix is singular. We will also discuss generalizations to other types of discrete random matrices and mention connections with other problems involving random matrices. Joint work with Jean Bourgain and Van Vu. 

04/17 3:00pm 
MILN 317 
Daniel Redelmeier TAMU 
Hyperpfaffians and their applications to combinatorics
In this talk we will discuss the hyperpfaffian, which is an extension of the pfaffian to either multidimensional arrays or tensor algebras. We will spend the first part of the talk examining the different definitionsused for the hyperpfaffian, as there are several nonequivalent forms. Then we will examine uses of the hyperpfaffian in combinatorics. The first is the HyperpfaffianCactus theorem, which is related to the classical matrixtree theorem and the later pfaffiantree theorem. Second we will look at hyperpfaffian orientations, which can be used to count perfect matchings on a hyperpfaffian. Finally we will look at hyperpfaffian rings/ideals, and specifically look at the fact that unlike the pfaffian ideal, the hyperpfaffian ideal is not an algebra with straightening law under reasonable assumptions. 

04/21 1:00pm 
MILN 317 
Igor Pak University of Minnesota 
MacMahon's master theorem and its generalizations
MacMahon's Master Theorem is a classical combinatorial result celebrated by its applications to binomial identities. In this talk I will present an algebraic and a direct bijective proof of the theorem. I will then discuss various noncommutative generalizations and how they fit together. 

04/24 3:00pm 
MILN 317 
Fernando Rodriguez Villegas University of Texas 
Rational and Algebraic Hypergeometric Functions
In the late 80's Beukers and Heckmann classified all algebraic hypergeometric functions in one variable. The key ingredient in their proof is the signature of a quadratic form fixed by the monodromy group. I will explain how this quadratic form is related to an old construction due to Bezout and how we may reformulate their result in terms of the padic valuation of Taylor coefficients. Using this, in joint work with E. Cattani and A. Dickenstein, we were able to classify rational twovariable hypergeometric functions. 

05/01 3:00pm 
MILN 317 
Joerg Feldvoss University of South Alabama 
Complexity, Varieties, and Representation Type
In this talk I will define the complexity of a finitedimensional module over an associative algebra and explain that for certain algebras over an algebraically closed ground field the complexity can be realized as the dimension of an affine variety. I will present several applications of these concepts which were introduced originally for modular representations of finite groups by Alperin and Carlson in the late seventies and early eighties. Then I will define the representation type of an associative algebra and state the trichotomy theorem of Drozd. If time permits, I will conclude by explaining how to use the affine varieties to prove a wildness criterion for certain selfinjective algebras. This is joint work with Sarah Witherspoon. 