Santa Clara University

Mathematics and Computer Science department


SCU Mathematics/CS Colloquium Series

Winter 2011

All talks will be at 3:50 PM.  There will be refreshments before each talk in O'Connor 31 at 3:40 PM.


Tuesday January 11, O'Connor 207

Speaker: Klavdija Kutnar, University of Primorska and The Ohio State University

Title: Hamiltonian Cycles and Paths in Vertex-transitive Graphs

Abstract: A path (cycle) containing every vertex in a graph is called a Hamilton path (Hamilton cycle, respectively). Hamilton cycles have been studied extensively in graph theory for their own sake, because of connections with the four color problem, and the traveling salesman problem. A graph is called vertex-transitive if for any pair of vertices u and v there exists an automorphism mapping u to v. In 1969, Lovasz asked whether every finite connected vertex-transitive graph has a Hamilton path, thus tying together two seemingly unrelated concepts: traversability and symmetry of graphs. With the exception of the complete graph on two vertices, only four connected vertex-transitive graphs that do not have a Hamilton cycle are known to exist. These four graphs are the Petersen graph, the Coxeter graph and the two graphs obtained from them by replacing each vertex by a triangle. The fact that none of these four graphs is a Cayley graph has led to a folklore conjecture that every Cayley graph has a Hamilton cycle. (A Cayley graph is a graph whose automorphism group admits a regular subgroup.) Both of these two problems are still open. However, a considerable number of partial results are known. In this talk an overview of these results will be introduced. A special emphasis will be given to recent results concerning the existence of Hamilton cycles in cubic Cayley graphs arising from groups having (2,s,3)-presentation.

Tuesday January 25, O'Connor 207

Speaker: Vladimir Drobot, San Jose State University

Title: Golomb Rulers

Abstract: What is the smallest number of inch marks on a ruler that allow us measure all integral distances? This question motivates our survey of Golomb rulers, perfect rulers, and minimal rulers -- different types of rulers for allowing the most measurements with the smallest number of marks.

Tuesday February 8, O'Connor 207

Speaker: Tamsen McGinley, Santa Clara University

Title: A Bijective Proof of the Hook Formula

Abstract: Young Tableaux are arrays of numbers (the content) in rows of boxes (the shape) satisfying certain combinatorial constraints. Such tableaux have been essential in the solution of many enumerative problems in algebra, representation theory, and elsewhere. In 1954, Frame, Robinson and Thrall gave a formula for the number of standard Young tableaux of a given shape in terms of L-shaped subregions, or hooks, of the tableau. The formula seemed tantalizingly simple, and suggested an elementary bijective proof should exist to prove it. However, over the years, many proofs of the "hook formula" have been found, of varying complexity, but none had been considered satisfactory. The theorem is combinatorial, and the "right" proof should be bijective. In 1997, Novelli, Pak and Stoyanovskii found a bijective proof, which will be given in this talk.

Tuesday February 15, O'Connor 207

Speaker: Carl Yoshizawa, Genomic Health, Inc.

Title: Personalized Medicine and the Role of Statistics

Abstract: Are you wondering how to translate your strengths and training in mathematics into a career? This colloquium will describe the role of molecular diagnostics in the field of personalized medicine, the explosion in information resulting from tremendous advances in technology, and the challenges and opportunities this provides for people with strengths in the mathematical sciences. The hypothesis testing framework, false positives and multiple testing, false discovery rates, identification power and regression toward the mean will be among several topics that will be discussed in a conceptual fashion. Knowledge of statistics is not required.

Tuesday February 22 , O'Connor 207

Speaker: Matthias Beck, San Francisco State University

Title: Discrete Volume Computations for Polytopes: An Invitation to Ehrhart Theory

Abstract: Our goal is to compute the volume of certain easy (and fun!) geometric objects, called polytopes, which are fundamental in many areas of mathematics. Although polytopes have an easy description, e.g., using a linear system of equalities and inequalities, volume computation is hard even for these basic objects. Our approach is to compute the discrete volume of a polytope P, namely, the number of grid points that lie inside P, given a fixed grid in Euclidean space such as the set of all integer points. A theory initiated by Ehrhart implies that the discrete volume of a polytope has some remarkable properties. We will exemplify Ehrhart theory with the help of several families of polytopes whose discrete volumes are connected with some of our friends in various mathematical areas, such as binomial coefficients, Eulerian, Stirling, and Bernoulli numbers.

This talk will be accessible to anybody who has finished the basic calculus and linear algebra courses. In particular, we will not assume that the audience knows the terms mentioned in this abstract, such as the concept of a polytope.

Thursday February 24, O'Connor 206

Speaker: Ivars Peterson, MAA Director of Publications and Communications

Title: Geometreks

Abstract: Few people expect to encounter mathematics on a visit to an art gallery or even a walk down a city street (or across campus). When we explore the world around us with mathematics in mind, however, we see the many ways in which mathematics can manifest itself, in streetscapes, sculptures, paintings, architectural structures, and more. This illustrated presentation offers illuminating glimpses of mathematics, from Euclidean geometry and normal distributions to Riemann sums and Moebius strips, as seen in a variety of structures and artworks in Washington, D.C., San Francisco, Toronto, New Orleans, and many other locales.

 

If you have a disability and require a reasonable accommodation,
please call/email Rick Scott 408-554-4460/rscott at scu dot edu (or
use 1-800-735-2929 TTY—California Relay).

Abstracts of previous talks are available here.
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