The Claremont Colleges
Algebra/Number Theory/Combinatorics Seminar
Spring 2008

Tuesdays 12:15 - 1:10 PM
Millikan 208
Pomona College, Department of Mathematics
610 N. College Ave. (Corner of 6th and College Ave.)
Claremont, CA 91711

For more information contact: Gizem Karaali
email: Gizem.Karaali@pomona.edu

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    12:15 p.m., Millikan 208
Our Next Speaker | Upcoming Seminars | Abstracts
Calendar and Upcoming Seminars
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Abstracts
  • A random construction for permutation codes and the covering radius
    Cheng Yeaw Ku (California Institute of Technology)
    We analyse a probabilistic argument that gives a semi-random construction of a permutation code on n symbols with distance n-s, and a bound on the covering radius for sets of permutations in terms of a certain freuency parameter. This is joint work with Peter Keevash.
  • Division Algebras and Wireless Communication
    Al Sethuraman (California State University, Northridge)
    Within the last five years, division algebras have emerged as the perfect tool for coding for wireless communication with multiple antennas. In this talk we will describe some key features of division algebras and then describe how they are applied to wireless communication. Along the way, we will touch on questions on algebraic lattices that arise in this application.
  • Algebraic Voting Theory
    Mike Orrison (Harvey Mudd College)
    If the results of your election procedure can be realized as a matrix-vector product, then the representation theory of the symmetric group can probably say something interesting about the way you are voting.  In this talk, I'll explain why this is the case by (introducing and) using only a handful of simple combinatorial objects (e.g., tabloids) and some basic ideas from representation theory (e.g., Schurıs Lemma) to recast and extend some well-known results in the field of voting theory.  This is joint work with Zajj Daugherty, Alex Eustis, and Greg Minton.
  • Liar Games on a General Channel
    Kathryn Nyman (Loyola University Chicago)
    We imagine a game in which Carole thinks of a number between 1 and n, and Paul tries to determine this number by asking Carole questions. The twist comes in when we allow Carole to lie up to k times according to a "channel of allowable lies". We look at a strategy of packings and coverings through which Paul can win the game for a given n. This is joint work with Robert Ellis.
  • Combinatorics of orthogonal polynomials
    Pallavi Jayawant (Bates College)
    In recent decades, combinatorial methods have been effectively used in the study of orthogonal polynomials such as Hermite, Charlier, Laguerre, etc. I will talk about the use of graphs as combinatorial models for some of these polynomials. I will then focus on the Charlier polynomials and demonstrate the use of the graphical model to develop generating function identities for these polynomials. I will end with a discussion of other families of orthogonal polynomials and possibilities for future work.
  • An unexpected application of algebraic number theory to operator algebras
    Marta Asaeda (University of California, Riverside)
    An operator algebra is an algebra of bounded linear operators on a Hilbert space. Subfactors are pairs of operator algebras with certain properties, and traditionally their study belonged to the area of functional analysis. However since the discovery of the Jones polynomial, the field evolved dramatically, with new connections to other areas of mathematics such as low dimensional topology, representation theory, mathematical physics, and topological quantum field theory. Subfactors with certain properties had been conjectured to exist. Nine years ago, Haagerup and I proved that two of the conjectured subfactors do exist. In this talk, I will describe my recent result with Yasuda that proves that none of the others exist except one. The proof uses the classical theory of ramification of prime ideals in field extension.
  • A six generalized squares theorem
    Paula Tretkoff (Texas A&M)
    In 1770, Joseph Louis Lagrange proved that every positive integer can be expressed as the sum of four squares of integers. Adrien-Marie Legendre improved on the theorem in 1798 by stating that a positive integer can be expressed as the sum of three squares if and only if it is not of the form 4^k(8m + 7). His proof was incomplete, leaving a gap which was later filled by Carl Friedrich Gauss. We consider a generalization of Lagrange's result which has applications to Polynomial Identity algebras (PI algebras). This work is joint with Amitai Regev. No background is needed to follow the talk, in particular none in Number Theory or in PI algebras.
  • A Variation on the Tennis Ball Problem
    Naiomi Cameron (Lewis and Clark College)
    We consider a variation on the Tennis Ball Problem studied by Mallows-Shapiro and Merlini, et al. The $s$-Tennis Ball problem is the following: At first turn, you are given $s$ balls labelled $1,2,\dots,s$, where $s$ is a fixed positive integer. You toss one of them out of the window onto the lawn. At the second turn, balls numbered $s+1,s+2,\ldots,2s$ are given to you and now you toss any of the $2s-1$ remaining balls onto the lawn. This continues for $n$ turns. We consider two questions. First, how many different combinations of balls on the lawn are possible after $n$ turns? Second, what is the sum of the labels of the balls on the lawn, over all distinct possibilities, after $n$ turns? In the case where $s=2,$ the solution to the orginal problem is the well known Catalan numbers. The variations discussed in this talk yield the Motzkin numbers and other related sequences.
  • New Commutativity Conditions for Rings (2000 ­ 2005)
    Jim Pinter-Lucke (Claremont McKenna College)
    I will briefly review the older conditions that were developed prior to 2000 and then concentrate on the recent results. These results were inspired by results in Groups, which I will also review. In particular we will look at a proof that uses a Ramsey Theory result to show that a group in which XY = YX for all n-set X, Y is Abelian.
  • Generating Functions of Rational Polyhedra and Dedekind-Carlitz Polynomials
    Matthias Beck (San Francisco State University)
    We study higher-dimensional analogs of the Dedekind-Carlitz polynomials, c(u,v;a,b) := Sum_{k=1, ..., a-1} u^{k-1} v^{floor(kb/a)} , where u and v are indeterminates and a and b are positive integers. These polynomials satisfy the reciprocity law (u-1) c(u,v;a,b) + (v-1) c(v,u;b,a) = u^{a-1} v^{b-1} - 1 , from which one easily deduces many classical reciprocity theorems for the Dedekind sum and its generalizations, most notably by Hardy and Berndt-Dieter. Dedekind-Carlitz polynomials appear naturally in generating functions of rational cones. We use this fact to give geometric proofs of the Carlitz reciprocity law. Our approach gives rise to new reciprocity theorems and a multivariate generalization of the Mordell-Pommersheim theorem on the appearance of Dedekind sums in Ehrhart polynomials of 3-dimensional lattice polytopes. (I will not assume familiarity with Dedekind sums or discrete geometry and I will carefully define all the terminology used above.) This is joint work with Christian Haase (Freie Universit"at Berlin) and Asia Matthews (Queens University).
  • Representations of Hopf algebras associated to $S_n$
    Andrea Jedwab (University of Southern California)
    The Frobenius-Schur indicator has proven to be a very useful invariant in Hopf algebras, both in classification problems and representation theory. The indicator measures whether a finite dimensional representation of a semisimple Hopf algebra $H$ is self dual or not, and distinguishes between those self dual representations that admit a symmetric non-degenerate $H$-invariant form and those whose form is skew-symmetric. The symmetric group is known to only admit representations with indicator $1,$ meaning they are all orthogonal. We study two dual families of Hopf algebras arising from factorizations of the symmetric group, and determine the indicators of the corresponding representations.
  • Chains of Sublattices in a Product of Finite Distributive Lattices: A Conjecture from the 1984 Banff Conference on Graphs and Order
    Jonathan Farley (California Institute of Technology)
    Let L be a finite distributive lattice. Let Sub_0(L) be the lattice of sublattices of L and let l_[Sub_0(L)] be the length of the shortest maximal chain in Sub_0(L). It is proved that, if K and L are non-trivial finite distributive lattices, then l_*[Sub_0(K\times L)]= l_*[Sub_0(K)]+l_*[Sub_0(L)]. A conjecture from the 1984 Banff Conference on Graphs and Order is thus proved.
  • Non-Minimal Factorization in Numerical Monoids
    Jay Daigle (Pomona College)
    A monoid is an algebraic structure with associativity and identity but no inverses; a numerical monoid is a submonoid of the natural numbers under addition. The factorization properties of numerical monoids are connected to many important areas in additive number theory, including the Frobenius problem. In this talk we will discuss several recent results about factorization in numerical monoids. We will then consider a generalization of factorization that allows factoring with respect to non-irreducible elements, and identify pathological structures that can and cannot occur.
  • Computational Geometry of Positive Definite Quadratic Forms
    Achill Schuermann (University of Magdeburg)
    In this accessible talk we start from the sphere packing and the sphere covering problems in Euclidean $d$-space, and are led to the geometry of positive definite quadratic forms, an old and rich subject lying at the intersection of Number Theory and Discrete Geometry. We explain how classical theories can be used computationally to solve the lattice restricted sphere packing and covering problems. We extend these theories in order to support the search for optimal periodic point sets, with the goal to resolve one of the major open questions in the field: Do there exist periodic sets which give better sphere packings or better sphere coverings than any lattice in the same dimension? Along the way, we discover several new phenomena and connections to other fields of mathematics.
  • Hyperdeterminants, secant varieties, and tensor approximations
    Vin de Silva (Pomona College)
    Whereas the Eckart-Young theorem for matrices gives a conclusive answer to the problem of finding a best low-rank approximation to a given matrix, there is no analogous theorem for tensors/hypermatrices of higher order. Indeed the problem turns out to be ill-posed in essentially all non-trivial cases, because of the failure of tensor rank to be upper-semicontinuous. In this talk we will dissect the somewhat unexpected structure of the rank function for real $2 \times 2 \times 2$ hypermatrices. The Cayley hyperdeterminant is our chosen scalpel. This is joint work with Lek-Heng Lim.
  • Regressive functions on pairs
    Andres Caicedo (California Institute of Technology)
    Let [N]^k denote the collection of (unordered) k-size sets of natural numbers. The well-known Ramsey theorem states that if A is finite, then any function f:[N]^k --> A admits an infinite homogeneous set H, that is, f restricted to [H]^k is constant. A function f:[N]^k --> N is regressive iff f(u) < min(u) whenever min(u) > 0. These functions do not necessarily admit homogeneous sets, but Kanamori and McAloon showed that they admit infinite min-homogeneous sets H (so f(u)=f(v) whenever min(u)=min(v)>0). They were interested in these functions for their connection with provability in Peano Arithmetic. For k=2, their results imply that the corresponding ``regressive Ramsey numbers'' have rate of growth at least Ackermannian. I present a combinatorial proof that these numbers grow precisely like Ackermann's function. My argument considerably improves all previously known bounds. I also discuss some open problems.
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