### The Claremont Colleges Algebra/Combinatorics Seminar Fall 2007

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

email: Gizem.Karaali@pomona.edu

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Calendar and Upcoming Seminars

Our Next Speaker | Upcoming Seminars | Abstracts
Abstracts
• Hilbert's 10-th problem, heights, and search bounds for rational points on varieties I & II
Lenny Fukshansky (Claremont McKenna College)
Hilbert's 10-th problem (in its modern formulation) asks if there exists an algorithm that, given a Diophantine equation (or a system of such equations), can decide whether it has a (non-trivial) solution in integers. In a famous work by Matijasevich (based on the previous work by Davis, Putnam, and Robinson) in 1970 this question was answered negatively. This result has inspired a large amount of work on various generalizations of Hilbert's 10-th problem, where one searches for solutions in rings larger than Z, for instance over a number field. In general, a basic expectation is that the answer still remains negative, however for equations of small degree it is possible to suggest an algorithm. The approach that I want to discuss comes from arithmetic geometry and is based on the notion of height of points in a projective space. Starting from just a search for rational points on some very simple varieties, one can rather quickly get to some more general effective results related for instance to the algebraic theory of quadratic forms. In the first talk I will introduce the problem, the machinery, and discuss the case of linear equations. In the second talk, I will concentrate on the case of quadratic forms. If I time allows, I will also briefly discuss what can be done for higher degree.
• Algebraic structures from knots
Sam Nelson (Pomona College)
We will see some examples of algebraic structures which arise in knot theory, primarily motivated by the search for invariants of knots and links. Examples include quandles, racks, biquandles and virtual biquandles.
• Exact and Asymptotic Dot Product Representations of Graphs
Kim Tucker (Harvey Mudd College)
The rich subject of geometric representations of graphs has, at its heart, a straight-forward basis: given a graph G, assign a geometric structure to each vertex of the graph such that the adjacency of any two vertices can be determined via a symmetric function of the two associated structures. The wide variety of geometric representations derives from the choice of the class of geometric structures and the choice of function used to determine adjacency. (A classic example is that of an interval representation of a graph where intervals of the real line are assigned to vertices). In this talk, we examine two types of graph representations in which vectors in d-dimensional Euclidean space are assigned to vertices so that the dot products of the vectors corresponding to a pair of vertices indicate whether or not the vertices are adjacent. These dot product representations are motivated, in part, by their application to networks, especially to social networks. We define two invariants, the exact and asymptotic dot product dimensions, of a graph and discuss their values for various classes of graphs. We establish a variety of properties of these graph invariants, including their relationships to other well-known invariants such as the independence number and the matching number (in the case of bipartite graphs). To conclude, we present full characterizations of the sets of graphs that have exact dot product dimensions equal to 0, 1, 2, and 3 and relate these characterizations to polynomial time algorithms for recognition for any fixed dimension. (This is joint work with Edward R. Scheinerman at Johns Hopkins University.)
• Cluster algebras and Invariant Theory
Sebastian Zwicknage (University of California, Riverside)
Cluster algebras were introduced by S. Fomin and A. Zelevinsky in 2001 and since then have made appearances in many areas of mathematics, from statistical physics to combinatorics to the study of quantum groups. In this talk I will introduce the concept of cluster algebras via some examples. I will then show how cluster algebras can be used to give a new formulation of the Fundamental Theorems of Classical Invariant Theory (drawing from the examples used to introduce cluster algebras).
• Vertex Algebras and Chiral Cohomology
Emrah Paksoy (Pomona College)
In this talk I will introduce the notion of a vertex algebra following the approach of Lian-Zuckerman. We will construct some examples and use them to obtain a new class of cohomology theories namely, chiral cohomology. It is an unfortunate fact that chiral cohomology will coincide with the deRham cohomology but finally we will define the equaivariant version in which some surprises are awaiting.
• Covering and pairwise generating finite groups
Attila Maroti (University of Southern California)
Let G be a non-cyclic finite group that can be generated by two elements. Let \sigma(G) be the minimal number m so that G is the union of m proper subgroups, and let \mu(G) be the maximal number k such that there exists a subset X of G of size k such that any distinct pair of elements of X generates G. It is easy to see that \mu(G) \leq \sigma(G). In this talk we will investigate these numbers for various groups G. The motivation comes from a conjecture of Blackburn saying that \mu(G)/\sigma(G) tends to 1 as the sizes of the finite simple groups G tend to infinity.
• SAGE and Symmetric Functions
SAGE is a free, open-source computer algebra that aims to be a replacement for Mathematica, Maple, Matlab, and Magma and was started by William Stein, a number theorist at the University of Washington. Recently, a lot of support has been added to SAGE for working with combinatorial objects such as partitions, permutations, tableaux, etc. Additionally, SAGE now has fairly good support for symmetric functions, which are "polynomials" in infinitely many variables which are invariant under any permutation of those variables. In this talk, I will give a brief introduction to SAGE and give an overview of symmetric functions and how to work with them in SAGE. I will also talk briefly about some generalizations of symmetric functions including Hall-Littlewood polynomials and Macdonald polynomials.
• Representations of twisted loop algebras and their block decompositions
Prasad Senesi (University of California, Riverside)
We investigate the category of finite-dimensional representations of a twisted affine Kac-Moody algebra. This category is not semisimple, so it is natural to search for its block decomposition. We will begin by describing the block decomposition of an Abelian category and giving some examples of categories of representations which are not semisimple. We will then provide a parametrization of the blocks of the category of finte-dimensional modules for an algebra of type A_3^{(2)}, and describe how this decomposition is related to the decomposition corresponding to the untwisted algebra of type A_3^{(1)}.
• Combinatorial Trigonometry (and a method to DIE for)
Art Benjamin (Harvey Mudd College)
Many trigonometric identities, including the Pythagorean theorem, have combinatorial proofs. Furthermore, some combinatorial problems have trigonometric solutions. All of these problems can be reduced to alternating sums, and are attacked by a technique we call D.I.E. (Description, Involution, Exception). This technique offers new insights to identities involving binomial coefficients, Fibonacci numbers, derangements, zig-zag permutations, and Chebyshev polynomials.
• Dynamic Modular Curves
Michelle Manes (University of Southern California)
Consider a rational map $\phi$ on the projective line, from which we form a (discrete) dynamical system via iteration. A fundamental question in arithmetic dynamics is the uniform boundedness conjecture of Morton and Silverman, which states that there is a constant independent of $\phi$ (depending only on its degree) giving an upper bound for the number of $K$-rational preperiodic points of $\phi$. This is a deep conjecture, and no specific case of it is known. I have proposed a specific version of the conjecture: that in the case of a degree-2 rational map and $K = \mathbb{Q}$, the upper bound is 12. In this talk, which assumes no previous knowledge of arithmetic dynamics, I will describe why this question is so difficult and sketch work that has been done to date, including giving justification for my refined uniform boundedness conjecture. The techniques used so far, which have clear limitations, involve constructing algebraic curves parameterizing maps $\phi$ together with points of period $n$ for varying $n$ (so-called dynamic modular curves).

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