Renowned Mathematician Addresses YC Students

Dr. Sylvain Cappell of Courant Institute Visits Campus

On Tuesday, January 27, the mathematics, physics and computer science students of Yeshiva College were privileged to hear a lecture by Dr. Sylvain Cappell, one of the world's most eminent mathematicians. The lecture, which was sponsored by the YC Math Club and the Office of the Dean, was entitled "Integration vs. Addition". Preceding the lecture, a dinner was served in Belfer Commons during which students were afforded the opportunity to ask Dr. Cappell questions and discuss topics of personal interest.

Dr. Cappell is a topologist. Loosely speaking, topology is the study of normal geometric figures as they undergo continuous transformations. A continuous transformation can stretch, shrink, bend, or twist an object, but it cannot tear it or attach to parts of it that were not previously attached. Topology also includes knot theory, which studies different properties of knots. Dr. Cappell studies topology in all dimensions. Most topologists specialize in either higher or lower dimensions since the third and fourth dimensions differ in many respects from higher dimensions.

Dr. Cappell currently works at the Courant Institute of Mathematical Science, a prestigious mathematical research program. He divides his time between research, administrative work, and teaching. The most time consuming of his duties is his research. In casual conversation during the dinner preceding the lecture, Dr. Cappell mentioned at least five papers he was either in the process of working on or had just recently published. In his administrative work he has had a great deal of influence on the development of the Institute. As he put it, "At this point, most of my colleagues were probably hired by me."

In his lecture, Dr. Cappell discussed several very basic but, to date, unsolved mathematical problems. One problem he alluded to is the "The Circle Problem." This problem was originally addressed by the famous nineteenth century mathematician, Gauss. The problem is to find how well the number of lattice points inside a circle approximates its area. "The Circle Problem" deals with a circle centered at the origin of a Cartesian coordinate plane. It is reasonable to assume that the number of lattice points inside a circle reasonably approximates its area, but it has taken centuries to develop a reasonable idea of how great the error factor in this estimate is and a precise answer is as yet only a conjecture.

Dr. Cappell also mentioned an unsolved problem in computer programming which is being addressed by mathematicians. In addition, he dealt with the problem of summing sequences of functions of integers.

Finally, Dr. Cappell mentioned the problem of generalizing Pick’s theorem. Pick’s theorem deals with the problem of finding the area of a polygon (many sided figure) the vertices (angles or corners) of which are lattice points. In the 1870’s Pick developed an insightful formula to accomplish this by counting lattice points inside and on the boundaries of such a polygon. However, this solution is only valid for two dimensional figures. Dr. Cappell recently published a paper involving some very powerful theorems about the calculus of complex variables which presents the complete solution of this problem in n dimensions.

Dr. Cappell concluded, "Anyway, I’ve taken you, I hope, on a little tour which shows you that counting, which is something you grew up with as one of the simplest of mathematical processes, already has problems which we need new ideas on, and for new ideas, mathematics needs more young people." The lecture was followed by a question and answer period.

Dr. Rosenfeld, Dr. Karpishpan, Dr. Otway, Dr. Cwilich, and Dr. Kaplan, members of the YC math and physics faculty, all attended the lecture. All were very pleased. Dr. Cwilich was especially impressed by the clarity of the presentation. Ben Franz, one of the students who attended the lecture, remarked, "Dr. Cappell is clearly one of the more fascinating and illuminating personalities we have heard speak at YU and it would be a pleasure to hear him speak again."