Cal Poly SLO Mathematics Department Colloquia, 2001-2002

Colloquium Schedule

2001-2002 Colloquia

Recurrence and Topology in Dynamical Systems
John Alongi, Mathematics Department, Cal Poly

Abstract
Henri Poincare observed that to understand the long run behavior of the solutions of a system of ordinary differential equations, one must understand the solutions of the system which eventually return near themselves infinitely often -- a phenomenon which we informally refer to as recurrence. But, how should we define recurrence formally? My talk will motivate and develop several topological definitions of recurrence for orbits of flows on nonempty compact metric spaces culminating with Charles Conley's idea of chain recurrence.

On Operators Commuting with Toeplitz Operators Modulo the Finite Rank Operators
Caixing Gu, Mathematics Department, Cal Poly

Abstract
A linear operator, on an infinite dimensional Hilbert space, can be represented by an infinite matrix. The matrix representations of a Toeplitz operator have constant entries on their diagonals, that is, T(f) has i,j, entry f(i-j) where f(i-j) is the i-j Fourier coefficient of f.

A finite rank operator has finite dimensional range. An inner function g is an analytic function inside the unit disk with absolute value 1 on the unit circle, e.g. a power of z. In this talk we will prove that if a linear operator X is such that X - T(g)* X T(g) is finite rank for every inner function g, then X is the sum of a Toeplitz operator and a finite rank operator. This solves a variation of an open question. Effort will be made to make the talk accessible to undergraduate students with some linear algebra and complex analysis background.

The Secret of Brunelleschi's Cupola
Mario Martelli, Department of Mathematics, Claremont McKenna College

Abstract
Brunelleschi was awarded the construction of the cupola of the Cathedral of Firenze, since he proposed to do it without centering. There was apprehension about the stability of his design, but Brunelleschi dispelled it by building a model, which he then destroyed. The base of the cupola is a regular octagon and nobody has ever found how Brunelleschi was able to build it without centering.

Over the years many architects and engineers proposed solutions which were not valid. The models built according to these solutions collapsed. Recently an architect and a former student of mine proposed a new theory based on a fundamental geometrical idea. I will illustrate their point of view particularly regarding the way it explains how to lay (the so-called corda blanda) the bricks forming the cupola.

Nonlinear Operations on Schwartz Distributions by Means of Nonstandard Analysis
Hans Varneave, University of Ghent, Belgium

Abstract
In this talk, we show how nonstandard analysis allows us to describe Schwartz-distributions as pointwise nonstandard functions, hence allowing us to define nonlinear operations on them. We discuss the relations with Colombeau's theory, but also give alternatives.

DNR-Based Instruction in Mathematics
Guershon Harel, Mathematics Department, U.C. San Diego

Abstract
How can we motivate students to learn mathematics? How can we help students retain what they learn? Why do students hold misconceptions we have never taught them explicitly? How can we help students remove misconceptions they currently hold, and is it possible to avert the occurrence of new ones? And in particular, how can we help students acquire good habits of mind in doing mathematics? These are important and difficult questions in mathematics education, which are on the minds of many teachers and instructors, teacher leaders, curriculum developers, and researchers who study the process of mathematics learning and teaching. Unfortunately, there are no definite answers to these questions, even if approached empirically, because the answers usually depend on one's philosophical orientation. In this talk I will not describe the different philosophies that might entail different instructional approaches. Rather, I will depict a system of instructional principles aimed at helping teachers in all levels build coherent vision of the processes of learning and teaching.

The system does not dictate routines for how to teach effectively - such routines do not exist - rather, it provides effective ways of thinking about learning and teaching. The system, called DNR, is the product of years of research into specific questions such as: What is students' knowledge of fundamental ideas in elementary mathematics (e.g., fractions and elementary algebra) as well as in advanced mathematics (e.g., linear algebra, axiomatic geometry, and real analysis)? What sorts of experiences seem effective in shaping students' knowledge of these ideas? Are there promising instructional approaches that can help students gain and retain deep understanding of these ideas, and in turn gain and retain effective ways of mathematical thinking such as problem solving heuristics and appreciation for mathematical rigor? The three leading principles in the system are labeled: Duality, Necessity, and Repeated-Reasoning. These principles form a system in the sense that they complement each other in addressing students' intellectual needs and in addressing curricular needs.

3-Manifolds, Geometry, and Algebra Part II
Matthew White, Mathematics Department, Cal Poly

Abstract
This talk is a continuation of last year's discussion of 3-manifold topology. Our focus in this case will be on hyperbolic 3-manifolds; that is, those manifolds locally isometric to hyperbolic 3-space. Work by Thurston, Gromov, and many others over the last three decades has provided a great deal of insight into the structure of these manifolds. Accordingly, theorems can be proved used techniques that are both accessible and geometrically pleasant. We will give short sketches for some of these results. Finally, we will discuss their possible generalization to some open problems.

Comparing Writing with Interviews and Exams as Assessments of Students' Understanding of the Concept of the Derivative
Gwen Fisher, Mathematics Department, Cal Poly

Abstract
This study compares the use of open-ended writing tasks with interviews and traditional exams to assess students' understanding of the concept of the derivative. The data consists of 13 students' responses to two one-hour interviews, 11 writing tasks, and 3 in-class exams. An example of a writing task is, "What does the derivative have to do with limits and vice-versa?" The theoretical framework for the data analysis depicts the derivative as a ratio, as a limit, and as a function, each of which can be conceptualized in any of the four representations: numeric, graphical, symbolic, and contextual. Data analysis is both qualitative and quantitative. The results show that daily writing tasks are reliable assessments for some, but not all, forms of students' understanding that are not assessed by traditional exams.

On the Metamorphosis of Vandermonde's Identity
Don Rawlings, Mathematics Department, Cal Poly

Abstract
We perform the controversial procedure of monomial insertion on Vandermonde's identity. Our method (madness?) reveals an amusing cache of binomial identities. As an application, we compute the expected time it takes for an n-component system to completely crash.

A blend of combinatorics, algebra, and probability, our talk will be accessible to a broad audience.

Knots, 3-coloring, and the Nakanishi Conjecture
Jim Hoste, Mathematics Department, Pitzer College

Abstract
A fundamental question of knot theory is how to distinguish between two different knots. One of the simplest techniques that can be used involves coloring the diagrams of the two knots in a certain way. This procedure is easy to describe, yet has deep connections to group representations, knot polynomials such as the Jones polynomial, and to the "3-twist" conjecture of Y. Nakanishi.

The Euclidean Algorithm Revisted
John W. Petro, Mathematics Department, Western Michigan University

Abstract
In this talk we shall quickly review the Euclidean Algorithm, which is used to compute the greatest common divisor "gcd(A,B)" of a pair of integers A and B. We shall show that using elementary ideas from linear algebra we can easily represent gcd(A,B) as a linear combination of A and B. We shall then explore some interesting related problems. For instance, how many algorithm steps would one expect to use to compute gcd(A,B)? What is the worst case scenario? What would be the most efficient algorithm to compute gcd(A,B,C) for three integers A, B, and C? What is the probability that gcd(A,B) would be 1 for randomly chosen A and B? What about gcd(A,B,C) for randomly chosen A, B, and C?

This talk will be especially appropriate for undergraduates. All mathematics majors are encouraged to attend.

The Complexity of Computing Knot Genus
Joel Hass, Department of Mathematics, U.C. Davis

Abstract
In joint work with Ian Agol and Bill Thurston, we have shown that the problem of computing the genus of a knot in a 3-manifold is NP-complete. I'll explain what these terms mean and how the result fits into topology and complexity theory.

Values of Zeta Functions at Negative Integers
Jamie Pommersheim, Pomona College

Abstract
The Riemann zeta function is one of the most important functions in number theory. The calculus student can easily verify that the series will converge if s > 1, but the series diverges if s is less than or equal to 1. Nevertheless, the miracle of analytic continuation allows us to extend zeta(s) to a function on the entire complex plane. In particular, for positive integers n, the values zeta(-n) are well-defined! These values turn out to be rational numbers which are closely related to the Bernoulli numbers. After studying the basics of the Riemann zeta function, we will introduce a natural generalization, namely the zeta function zeta sub K (s) of a number field K. We will talk about recent techniques for finding the values zeta sub K (-n) of these more general zeta functions in the case where K is a quadratic number field.

When Topology Meets Chemistry
Erica Flapan, Pomona College

Abstract
Stereochemistry is the study of the 3-dimensional structure of molecules, and topology is the study of those properties of geometric objects that are invariant under continuous deformation. It is not obvious that these two fields have anything in common. In fact, not long ago there was little communication between researchers in these two areas. Prior to forty years ago, analyzing the topological properties of existing molecular structures was not very difficult, because as topological objects, the graphs of all of the molecular structures known at the time could be deformed into a plane. Thus understanding the stereochemistry of a molecule only required the evaluation of its geometry and not its topology. Recently, knots and links and other non-planar molecules have been synthesized whose structures and properties come from their topology as well as their geometry. These molecules are often large enough that they no longer have the rigidity that is characteristic of small molecules, so understanding their deformations is an important part of understanding their structure. In this talk we will discuss the symmetries of such flexible molecules.

Solving Two-Way Diffusion Equations
Jorge Aarao, Claremont-McKenna

Abstract
There was once an equation - like heat -
Whose solutions were not quite as neat.
They exist. They're unique.
But which arcane technique
Will make our solutions complete?

We will talk about the heat equation and one of its cousins, the generalized Fokker-Planck equation, which models plasma equilibrium, and is a two-way diffusion equation - that is to say, time could run backwards. Along the way we will briefly discuss what do we mean when we say we will "solve" an equation.

Random Matrices and Connections to Determinants of Convolution Operators
Estelle Basor, Cal Poly

Abstract
The classical Szego-Kac-Widom formula yields an asymptotic expansion for determinants of convolution operators. In my talk I will show how this formula gives information about problems in random matrix theory. Conversely, random matrix theory gives insight about computing determinant asymptotics of other operators of the form I + K where K is a "sufficiently small" operator.

Cal Poly's Summer 2001 Research Experience for Undergraduates
Jonathan Shapiro, Mathematics Department, Cal Poly

Abstract
I will tell stories about and show pictures from the REU (Research Experience for Undergraduates) program we ran for the first time last summer. I will describe the research conducted by my two students and myself. This research involved questions concerning the continuity of the norm of composition operators. Specifically, we ask and partially answer the question: is the map from the space of analytic self maps of the unit disk to the norm of the induced composition operator (on the Hardy Space H^2) continuous? We give examples involving related questions of continuity as well.

When, and How Can You Approximate f(x) by Convolutions?
Sandy Grabiner, Pomona College

2000-2001 Colloquia
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1998-1999 Colloquia