Illinois Section of the MAA

Abstracts of Talks at the 1997 ISMAA Annual Meeting

Forty Years of Sphere Eversions

George Francis, University of Illinois at Urbana-Champaign
1:30p.m., Friday March 21, 1997, Room 207

Unlike celebrated conjectures which resist proof long enough to become famous and then discreetly fade into the background, here is an unexpected theorem about a theoretically visible event, turning a sphere inside out. Yet ever since it was proved in the late 50s, people have devised ingenious ways of really seeing what Smale's theorem says is possible. The eversions of Shapiro-Thom-Phillips in the 60s, Morin-Pugh-Max(70s), Morin-Apery-Denner(80s), Thurston-Levy and Kusner-Sullivan (90s) are milestones along the evolution of topological methods and computer-graphical techniques applied to this remarkably durable problem in mathematical visualization.

Our illustrated review of historical and contemporary sphere eversions culminates with premieres of the latest minimax eversions. Based on the Banchoff-Max quadruple-point theorem and starting from Bryant-Kusner immersed minimal surfaces, John Sullivan's experiments with Brakke's Surface Evolver on graphical supercomputers rediscover the classical eversions within the aesthetic economy of geometrical optimization.

Innovations in Undergraduate Teaching: Encouraging or Allowing use of the TI-92

Brian Keller, Iowa State University; Donald Porzio, Northern Illinois University; Sandra Spalt Fulte, Mathematics Coordinator, Quincy Public Schools
3:00p.m., Friday March 21, 1997, Room 221

This panel, moderated by Rosemary Schmalz of Eastern Illinois University, will discuss use of the TI-92 calculator in the mathematics classroom, designing appropriate examinations and questions of equity.

The History of Mathematics and its Use in Teaching

Herbert E. Kasube, Bradley University; Andrew S.Leahy, Knox College; Diann R. Porter, University of Illinois at Chicago
3:00p.m., Friday March 21, 1997, Room 223

During the Summer of 1996 the panelists attended the Institute in the History of Mathematics and its Use in Teaching in Washington, DC. They will present some of the ideas from the IHMT and discuss how the history of mathematics can enrich the students' experiences.

Trends in Mathematics: Why I Love Perron-Frobenius

Hans Schneider, The University of Wisconsin, Madison
3:00p.m., Friday March 1, 1997, Room 201

A brief excursion into the history, generalizations and applications of the theory of nonnegative matrices.

Actuarial Careers and Preparing Students for Them

Carl C. Cowen, Purdue University, West Lafayette, IN
3:00p.m., Friday, March 21, 1997, Room 225

An overview of how we can prepare student for acturial careers, both in a regular mathematcs department and in a specialized actuarial program. We will take a peek at the new exam system due 5/2000.

MythMath: From Bumblebees to Poincaré

Ivars Peterson, Science News
ISMAA Annual Banquet, Friday March 21, 1997, Clock Tower Inn

Watching out for myth and misinformation is important not only in browsing the web but also in teaching and explaining mathematics.

Totally Electronic Journals, Paperless Classes, and Equivisibility of Ellipsoids

Thomas Banchoff, Brown University
8:30a.m., Saturday March 22, 1997, Room 207

Interactive computer graphics on the internet makes it possible for mathematicians to carry out and communicate research and teaching in completely new ways, changing how we think about journals, classroom interaction, and collaborative research and teaching. This interactive presentation will feature examples from classical and higher dimensional geometry.

Innovations in Undergraduate Teaching: Calculus Reform Concerns

Wally Dodge, New Trier High School; Dan Gardner, Elgin Community College; Don Porzio, Northern Illinois University
9:45a.m., Saturday, March 22, 1997, Room 201

The panel, moderated by Mary Beth Dever of Benedictine University, will describe pros and cons of calculus reform by those experienced in the use of calc-reform texts or materials.

Issues in Undergraduate Education: Mentoring Panel

Dr. William Heaps, NASA, Goddard Space Flight Center; Julie French, student, Millikin University; Jan Baird, Nims Associates, Inc.
9:45a.m., Saturday, March 22, 1997, Room 221

Students and mentors from business, industry, and government will discuss the feasibility, convenience, problems, and experiences of someone not in academics mentoring a college student with particular attention to using e-mail as the communication vehicle.

Trends in Mathematics: From Black-Scholes to Risk Holes: A Look at the Mathematics of Option Pricing

John Price, Maharishi International University
9:45a.m., Saturday, March 22, 1997, Room 223

The mathematics of options pricing is full of surprises. To price an option on a stock you can completely ignore the expected value of the stock price and only consider its standard deviation. This and other features of the theory can be explained with basic discrete mathematics. Recent developments include the role of fractals in understanding convergence of discrete structures to Black-Scholes continuous models.

A Linear Algebra Course Front Loaded with Concepts with a Supporting Computer Lab

Lawrence Neff Stout and Robin Sue Sanders, Illinois Wesleyan University
9:45a.m., Saturday, March 22, 1997, Room 225

Our Linear Algebra course starts with an introduction to all of the concepts (vector spaces, linear transformations, matrices for linear transformations, eigenvalues, determinants, inverses, cannonical forms, and inner products) in the context of Reals^2 where the only difficulties are conceptual.

We then turn to vector spaces and linear transformations in general and ask about spans, kernels, and images. These questions motivate the need to develop good methods for solving systems of equations. For finite dimensional vector spaces, bases give us a way to make linear transformations concrete with matrices. Row reduction of the matrices then allows us to answer questions about the dimension of the kernel and image of a linear transformation.

Our course is accompanied by a computer lab using Mathematica. We will bring along animations which show how eigenvalues give information about iteration in Reals^2 and a notebook on how row reduction can be used to answer questions about linear transformations from F^n to F^m where F is the rationals, reals, complexes, or a finite field Z_p.

A Project on Cricles In Space

Carl C. Cowen, Purdue University, West Lafayette, IN
9:45a.m., Saturday, March 22, 1997, Room 225

The project of deciding if a collection of points in space do or do not lie on a circle has been successful in my recent linear algebra classes. The project forces students to integrate several parts of the course and material from other courses.

Innovations in Undergraduate Teaching: Internet Use in the Mathematics Classroom

John Emert, John Emert, Ball State University
10:50a.m., Saturday, March 22, 1997, Room 207

The goal of the Mathematical Archives is to provide organized Internet access to a wide variety of mathematical resources, with primary emphasis on materials which are used in the teaching of mathematics. Currently, the Archives is particularly strong in its collection of educational software. This workshop, in a web-friendly computer lab, will introduce by example the education resources and organized collection of links which comprise the Mathematics Archives.

Project NExT: What has it done for me lately?

Jim Marshall, Illinois College; Stacey Rodman, Augustana College; Sharon Robbert, Trinity Christian College; Lanette Poteete-Young, Judson College
10:50a.m., Saturday, March 22, 1997, Room 221

Since its inception in 1994, Project NExT (New Experiences in Teaching) has helped new math teachers break into the profession. The panel discussion will present several perspectives on how it has helped the participants and why it may be a good idea for you to encourage your new faculty to participate.

Trends in Mathematics: Why Not?: The Ties Between Knot Theory and Just About Everything

Charles Delman, Eastern Illinois University
10:50a.m., Saturday, March 22, 1997, Room 201

Assuming no prior knowledge on the part of the audience, we will motivate the study of knots with a historical survey, from the initial work of Lord Kelvin and his contemporaries to exciting present day developments and applications.

Student Papers I: Problem Solving in "The 11th Hour"

Joan Lind, Augustana College
10:50a.m., Saturday March 22, 1997, Room 225

Puzzles can often be viewed as mathematical problems. I will be using mathematical techniques to find solutions to puzzles such as those found in the computing game "The 11th Hour" (created by Graeme Devine and Rob Landerous).

Student Papers I: Coding Theory"

Christine Leroux, Northern Illinois University
11:10a.m., Saturday March 22, 1997, Room 225

This talk will introduce coding theory and its applicatons, including error detecting, error correcting, and equivalent codes.

Student Papers I: Adventures in Digital Watermarking

Scott Carver, Northern Illinois University
11:30a.m., Saturday March 22, 1997, Room 225

With the explosive growth of the Internet and the ease with which data can be copied and downloaded, protecting ownership rights to computer images, video and sound clips has become a cause for serious concern. This talk describes how mathematics can be use to make (and break!) invisible "watermarks" in computer images to prevent others from stealing them.

Student Papers II: An Exporation of the Gauge Integral

Kirk B. Larsen, North Central College
11:00a.m., Saturday March 22, 1997, Room 227

Traditionally, the Riemann Integral is the first integral we learn in our introductory calculus courses. We eventually discover that it has some limitations, such as that not every derivative is Riemann integrable. We will introduce the Generalized Riemann Integral, which is also known as the Guage Integral. After presenting some basic concepts and working through an example, we will prove the Fundamental Theorem of Caculus, with respect to the Gauge Integral:

If F:[a,b] -> R is differentiable on [a,b], then F' is integrable on [a,b] with the definite integral of F' on [a,b] equal to F(b) - F(a).

Notice that any derivative is integrable!

Paul Erdös: The Master of Collaboration

Jerrold Grossman, Oakland University
12:00 noon, Saturday, March 22, 1997, Room 207

Paul Erdös (1913-1996) wrote over 1400 books and articles, about two thirds of them in collaboration with others. Without getting into the specifics of his results, this talk will look at Paul's life and work, as well as some interesting properties of the collaboration graph with Erdös at it center.

September 18, 1997