October 13, 2014
The idea of holonomy
The notion of 'holonomy' in mechanical systems has been around for more than a century and gives insight into daily operations as mundane as steering and parallel parking and in understanding the behavior of balls (or more general objects) rolling on a surface with friction. A sample question is this: What is the best way to roll a ball over a flat surface, without twisting or slipping, so that it arrives at at given point with a given orientation?
In geometry and physics, holonomy has turned up in many surprising ways and continues to be explored as a fundamental invariant of geometric structures.
In this talk, I will illustrate the fundamental ideas in the theory of holonomy using familiar physical objects and explain how it is also related to group theory and symmetries of basic geometric objects.
Following Monday's lecture, there will be a reception in honor of Professor Bryant at the Koenig Alumni Center, 1202 University Avenue (the SE corner of Broadway and University).
October 14, 2014
Convex billiards and non-holonomic systems
Given a closed, convex curve C in the plane, a billiard path on C is a polygon P inscribed in C such that, at each vertex v of P, the two edges of P incident with v make equal angles with the tangent line to C at v. (Intuitively, this is the path a billiard ball would follow on a frictionless pool table bounded by C.) For 'most' convex curves C, there are only a finite number of triangular billiard paths on C, a finite number of quadrilateral billiard paths, and so on.
Obviously, when C is a circle, there are infinitely many closed billiard n-gons inscribed in C, but, surprisingly, the same is true when C is an ellipse. (This is a famous theorem due to Chasles.)
The interesting question is whether there are other curves, besides ellipses, for which this is true. In this talk, I'll discuss these phenomena and show how they are related to the geometry of nonholonomic plane fields (which will be defined and described).
| A North Carolina native, Robert Bryant received his PhD in mathematics in 1979 at the University of North Carolina at Chapel Hill, working under the direction of Robert B. Gardner. After serving on the faculty at Rice University for seven years, he joined the faculty of Duke University in 1987, where he held the Juanita M. Kreps Chair in Mathematics until moving to the University of California at Berkeley in July 2007. He served there on the faculty and as the Director of the Mathematical Sciences Research Institute from 2007 until 2013, after which, he returned to Duke University as the Phillip Griffiths Professor of Mathematics. He has held visiting positions at universities and research institutes around the world.
His research interests center on exterior differential systems and the geometry of differential equations as well as their applications in Riemannian geometry, special holonomy, and related areas.
He serves on the editorial boards of the Duke Mathematical Journal, Communications in Analysis and Geometry, and the Journal of the AMS. He is currently the President-elect of the American Mathematical Society and will assume the office of the AMS Presidency in February 2015, for a term of 2 years. He is a fellow of the American Academy of Arts and Sciences and a member of the National Academy of Sciences.
In his spare time, he enjoys reading, playing the piano, rock climbing, and other physical activities.
This Lecture Series is funded by an endowment given by Professor Ira M. DeLong, who came to the University of Colorado in 1888 at the age of 33. Professor DeLong essentially became the mathematics department by teaching not only the college subjects but also the preparatory mathematics courses. Professor DeLong was a prominent citizen of the community of Boulder as well as president of the Mercantile Bank and Trust Company, organizer of the Colorado Education Association, and president of the charter convention that gave Boulder the city manager form of government in 1917. After his death in 1942 it was decided that the bequest he made to the mathematics department would accumulate interest until income became available to fund DeLong prizes for undergraduates and DeLong Lectureships to bring outstanding mathematicians to campus each year. The first DeLong Lectures were delivered in the 1962-63 academic year.