David Bressoud's Speaking Topics
Proofs and Confirmations: The Story of the Alternating Sign Matrix Conjecture
What is the role of proof in mathematics? Most of the time, the search for proof is less about establishing truth than it is about exploring unknown territory. In finding a route from what is known to the result one believes is out there, the mathematician often encounters unexpected insights into seemingly unrelated problems. I will illustrate this point with an example of recent research into a generalization of the permutation matrix known as the "alternating sign matrix." This is a story that began with Charles Dodgson (aka Lewis Carroll), matured at the Institute for Defense Analysis, drew in researchers from combinatorics, analysis, and algebra, and ultimately was solved with insights from statistical mechanics. This talk is intended for a general audience and should be accessible to anyone interested in a window into the true nature of research in mathematics.
Stories from the Development of Real Analysis
Analysis is what happened to calculus in the 19th century as mathematicians discovered that their intuition of how to apply calculus was failing them, especially as their repertoire of infinite series expanded. The conceptual difficulties that they encountered are precisely where we should expect our own students to have trouble. Understanding how these controversies were resolved illuminates many of the definitions, axioms, and theorems that baffle our students. This talk will focus on one or more of three broad issues that arose during this century and that caused both controversy and confusion as they were straightened out: What do we mean by convergence of a series of functions and when, for the purposes of calculus, can we treat an finite sum of functions as if it were a finite sum? How did our modern understanding of the Fundamental Theorem of Calculus arise, and what does it really say? And how did we get the Heine-Borel Theorem?
Calculus as a High School Course
Over the past quarter century, 2- and 4-year college enrollment in first semester calculus has remained constant while high school enrollment in calculus has grown tenfold, from 50,000 to 500,000, and continues to grow at 6% per year. We have reached the cross-over point where each year more students study first semester calculus in US high schools than in all 2- and 4-year colleges and universities in the United States. There is considerable overlap between these populations. Most high school students do not earn college credit for the calculus they study. This talk will present some of the data that we have about this phenomenon and its effects and will raise issues of how colleges and universities should respond.
The Truth of Proofs
Mathematicians often delude themselves into thinking that we create proofs in order to establish truth. In fact, that which is "proven" is often not true, and mathematical results are often known with certainty to be true long before a proof is found. I will use some illustrations from the history of mathematics to make this point and to show that proof is more about making connections than establishing truth.
