Colloquium Schedule
1999-2000 Colloquia
- Generalized Alexander Modules of Knots and Links
Shelly Harvey, Rice University
Abstract
Given any knot, there is the classical Alexander module of a knot which is characterized by its Alexander polynomial. This module is simply the first homology of the infinite cyclic cover of the knot complement seen as a module over the ring of Laurent polynomials with coefficients in Z, Z[t,t^(-1)]. We can generalize this procedure by taking G-covers (where G is some group) to get a module over ZG. We will define the generalized Alexander modules for knots, state some new results for knots and talk about some of the problems that we encounter for links. In particular, we will discuss how these generalized Alexander modules behave under concordance of links. This talk will focus on the speaker's current research.
About the Speaker
Shelly Harvey received her undergraduate degree in mathematics at Cal Poly San Luis Obispo, and is now working on her doctorate at Rice University. - Joint Mathematics, Physics and Chemistry
Colloquium
Exotic Atoms and Molecules
Dimitar D. Bakalov
Quantum Theory Project - Institute for Theoretical Chemistry, University of Florida at Gainesville
Abstract
When slowed down in matter, most elementary particles form Coulomb bound states by replacing the nucleus or one of the electrons of the target atoms or molecules. The comparison of high precision spectroscopy measurements of the long-lived states of such exotic atoms and molecules with accurate theoretical predictions of QED can provide valuable information on the fundamental properties of the elementary particles. These include, among other things, data on the electromagnetic structure of protons, the magnetic moment of antiprotons, and values of the weak interaction constants of protons, all of which are order(s) of magnitude more precise than those presently available.
About the Speaker
Dimitar D. Bakalov is a Professor at the Institute for Nuclear Research of the Bulgarian Academy of Sciences, and is presently a Fulbright Scholar at Quantum Theory Project-Institute for Theoretical Chemistry, at the University of Florida. He has been a long term visitor at many international institutions including: CERN (Switzerland), UINR (Dubna, Russia), ITAMP, Harvard, INFN (Italy). He is an author of more than 50 research papers. - Prime Power Conjecture
John L. Hayden, Bowling Green State University
Cal Poly Visiting Professor
Abstract
An affine plane of order n is a finite geometry consisting of n^2 points and n^2 + n lines. The prime power conjecture asserts n must always be a prime power for such geometries.
Most known examples of affine planes have automorphisms called elations that allow the points to be coordinated as (x,y) where x,y are elements from a group G, |G| = n. We study the incidence matrix A of an affine plane admitting a Cartesian group G and prove that n = 2^r when the roots of A are real. Additional results are obtained when the roots of A lie in certain cyclotomic number fields. - The Best Way to Knock 'em Down
Art Benjamin, Mathematics Department, Harvey Mudd College
Abstract
"Knock 'em Down" is a game of dice that is so easy to learn that it is being played in classrooms around the world as a way to develop students' intuition about probability. However, as our analysis will show, lurking underneath this deceptively simple game are many surprising and highly unintuitive results.
Disclaimer! Professor Benjamin takes no responsibility for any scams or "get rich quick" schemes that students may learn by applying the ideas of this talk! - A Channel Assignment Model: The Span Without a
Face
Jeff Mintz, Aaron Newcomer, and J.C. Price, Mathematics Department, Cal Poly
Abstract
We were asked to find an efficient method for assigning radio channels to a grid. We wrote a computer program to assist in making these assignments and to help determine the span. For the two cases we were asked to study, we found that the span was nine in the case that neighboring channels had to differ by two (k = 2) and the minimum of 3k + 3 and 2k + 7 for the more general case. In each case, we created an appropriate tiling of the plane. We concluded the paper with studies of some generalizations of the given problem.
This is a presentation by one of Cal Poly's Math Modeling Teams of their Outstanding Award-winning solution paper in the 2000 Mathematical Contest in Modeling.
- Spectra of Compact Composition Operators on Function Spaces
Over Bounded Symmetric Domains in Several Complex Variables
Dana Clahane, Mathematics Department, U. C. Irvine
Abstract
In 1975, James Caughran and Howard Schwartz proved the elegant result that for a holomorphic self-map of the open unit disk in the complex plane, the spectrum of the map's induced composition operator on the Hardy space is the set containing 0,1, and all possible products of the derivative of the self-map evaluated at the fixed point of the map, if the composition operator is compact. Along the way, these Caughran and Schwartz proved the uniqueness and existence of this fixed point, which also happens to be the unique constant function that the iterates of the map converge to on compact subsets of the disk.
In 1983, Barbara MacCluer extended these results to the open unit ball of n-dimensional complex space. We show that one can extend MacCluer's result on the spectrum to the more general case of hardy and weighted Bergman spaces of irreducible bounded symmetric domains. We also show that for a large class of domains and function spaces over these domains, compactness of composition operators on these spaces implies existence of a unique fixed point for the self-map. - Boundedness in Metric Space
Gerald Beer, Department of Mathematics, Cal State Los Angeles - Zeros of Harmonic Polynomials
Dr. Don Sarason, Mathematics Department, U.C. Berkeley
Abstract
The fundamental theorem of algebra states that a polynomial with complex coefficients in a complex variable, of positive degree N, has N roots, counting multiplicities. Is there a comparable theorem for harmonic polynomials, that is, polynomials with complex coefficients in two real variables that satisfy Laplace's equation (equivalently, functions in the complex plane writable as sums of analytic polynomials and conjugate analytic polynomials)? Perhaps surprisingly, the question was taken up only comparatively recently, and much remains unknown. This talk will describe the current state of knowledge. The talk will be elementary, accessible to anyone who has studied basic complex analysis. - Hyperbolic Saddle Trajectories for Non-autonomous Differential
Equations
Professor Ning Ju, California Institute of Technology
Abstract
The hyperbolic equilibrium of autonomous differential equations is a well-known basic concept for the study of the stability of the dynamical systems generated by the original differential equations. The extended concept for the non-autonomous differential equations is interesting but not well understood. Recent study shows that this concept is useful for the study of the dynamical behavior of the solutions to non-autonomous differential equations, for example for the study of the transport and mixing problems of aperiodic fluid flows. In this talk, we will show, under some conditions, how to define this new concept mathematically and how to find them explicitly. - What Did Archimedes Do Besides Cry Eureka?
Dr. Sherman Stein, Mathematics Department, U.C. Davis
Abstract
This talk will begin with a brief discussion of the life of Archimedes, followed by a survey of his major accomplishments, in particular, his work on areas and volumes, centers of gravity, and the equilibrium of floating objects, that is, the design of ships so that they don't topple over easily.
Dr. Stein's three most recent books are Archimedes: What Did He Do Besides Cry Eureka, Strength in Numbers,and Algebra and Tiling(with S. Szabo). Algebra and Tilingwas awarded the Beckenbach Prize from the Mathematical Association of America. - Special Undergraduate Colloquium
The First-Digit Phenomenon
Dr. Ted Hill, Mathematics Department, Georgia Tech
Abstract
Discovered over a century ago, the First Digit Phenomenon predicts that in "random data sets" of real numbers, the leading (non-zero) significant digit is not uniformly distributed, as would be expected, but rather follows a specific logarithmic distribution now known as Benford's Law. This survey talk will mention some of the colorful history of the problem, empirical evidence, classical proofs of various sorts, and modern applications such as detection of tax fraud, which has received wide attention in the popular press (New York Times, Wall Street Journal, etc.)
A new probability limit law explaining the appearance of Benford's law in many data sets will be described. This talk will be aimed at the non-specialist. - Eigenvalues of Random Matrices over Finite Fields
Dr. Kent Morrison, Mathematics Department, Cal Poly
Abstract
Construct a random matrix of size n by n by selecting the n^2 entries independently from a finite field with each element of the field equally probable. This talk will consider questions such as the probability that there are k eigenvalues in the base field, counted with multiplicity, and the expected number of eigenvalues in the base field. It is possible to answer these questions in the limit as the size of the matrix goes to infinity. - Seeing is Believing: Interactive Software for Visualizing
Calculus and its Applications to Economics and Business
Dr. Jean Marie McDill, Mathematics Department, Cal Poly
Abstract
Our students live in a visual world. Most instructors have a strong visualization of the concepts of calculus. Dr. McDill is involved in an NSF project to design and develop interactive computer software that communicates these visualizations to the students. There are many tools that will work in any calculus course, although the applications are selected from economics and business.
Business calculus is notorious for being unappreciated by economics and business majors who see it as rigid and irrelevant. Even well-prepared students have trouble making the transition from the discrete to the continuous domain that is inherent in calculus. The purpose of this project is to address these problems by creating truly interactive software that emphasizes visualization, applications, data and curve fitting. The user can change parameters with sliders and see immediate results in linked models and graphs. The graphics are both eye-catching and carefully focused on the concepts and applications to be learned.
This talk is suitable for anyone with an interest in mathematics, economics or finance. - The Eccentric Circles of Claudius Ptolemy
Carol Day, Professor of Mathematics, Thomas Aquinas College
Abstract
By the application of geometrical theorems and trigonometrical calculations to extensive observations made by himself and others, Ptolemy produced models of the planetary motions which are superior in some respects to the models of Nicholas Copernicus. By distinguishing three centers of circular motion, he was able to reproduce some of the features of an elliptical orbit. Copernicus, being more of a purist in respect to the circles, was forced to adopt much more cumbersome models to rival the old hypotheses in accuracy. The true merit of Copernicus' explanation was its unification of all the planetary orbits into a coherent system. Ptolemy's account of the individual orbits was better. - Holomorphic Composition Operators
Professor Bernard Russo, Department of Mathematics, University of California, Irvine
Abstract
A survey of known results and open problems concerning boundedness and compactness of composition operators over the Bergman and Hardy spaces of complex domains, with special attention to the unit disk and the unit ball in several variables. - Mathematics and the Internet
Don Hartig, Mathematics Department, Cal Poly
Abstract
Professor Hartig will describe a course that he developed for math majors at the Universidad de las Islas Baleares in Palma de Mallorca last spring. The students learned about the history of the internet and the world wide web and enough HTML (HyperText Markup Language) to create web pages (and sites) focusing on topics in mathematics. Various student projects will be demonstrated.
Time permitting, he will also describe how Math 501 (Methods of Applied Mathematics I) is developing this quarter. This course is team taught by Hartig and Professor Russ Cummings of the Aeronautical Engineering Department in a multimedia classroom with heavy reliance on the internet and tightly integrated with Maple worksheets. - The Fisher-Hartwig Conjecture: A Historical
Account
Estelle Basor, Mathematics Department, Cal Poly
Abstract
The Fisher-Hartwig Conjecture concerns an asymptotic formula for finite Toeplitz determinants with singular generating functions. The original motivation for the formula came from the Ising model in statistical mechanics. Dr. Basor will describe the history of the conjecture, its current status, connections with operator theory, and also some generalizations.
1998-1999 Colloquia