ASPECTS OF FINSLER GEOMETRY
31 October - 1 November 2013
Conference Room 7.15, Level 7, Ingkarni Wardli
(see on the map
A North Carolina native, Robert Bryant received his PhD in mathematics in 1979 at the University of North Carolina at Chapel Hill, working under Robert B. Gardner. After serving on the faculty at Rice University for seven years, he moved to 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. In July of 2013, he returned to Duke, rejoining the Mathematics Department.
He has held numerous visiting positions at universities and research institutes around the world. During the 2001-02 academic year, he visited MSRI as a Clay Mathematics Visiting Professor, and he was in residence at MSRI during the Fall 2003 term as a co-organizer of the program in Differential Geometry.
His research interests center on exterior differential systems and the geometry of differential equations as well as their applications to Riemannian geometry, special holonomy, and mathematical physics.
In 2002, he was appointed by then-President Bush to serve on the Board of Directors of the Vietnam Education Foundation, and he currently serves on the International Committee for the National Mathematics Center of Nigeria. He has served as the Director of the MSRI, the Park City/IAS Mathematics Institute and as a Vice President of the American Mathematical Society. He is a fellow of the American Academy of Arts and Sciences, the American Mathematical Society and a member of the National Academy of Sciences. See here for more details.
Titles and Abstracts:
Lecture 1: The origins of Finsler geometry in the calculus of variations
Abstract: In Riemann's famous 1854 address "On the hypotheses that lie at the foundations of geometry", he proposed that our notions of geometry were based on optimization principles, i.e., that what we perceive as 'straight lines' are actually just the curves that minimize some functional on curves in space and that 'geometry' is just the study of the appropriate functional.
It took some time before this notion in this generality was taken seriously, with the fundamental work being done in the early 20th century by Finsler, Cartan, and Chern. By that time, so-called 'Riemannian geometry' (a very special case of what Riemann originally had in mind) had undergone extensive development and much effort was put into forcing the general 'Finsler' case to fit the pattern of the special case of 'Riemannian' geometry, with somewhat unsatisfactory results.
In this talk, I will outline an approach to Finsler geometry that starts from the point of view that the central importance of the subject is to uncover the properties of the solutions to the calculus of variations problem that underlies Finsler geometry and will explain how this leads to a more natural understanding of Finsler geometry than one often encounters in its expositions.
Lecture 2: Finsler manifolds of constant flag curvature
Abstract: Continuing in the natural development of Finsler geometry, we consider the Finslerian analog of the problem that Riemann solved for Riemannian geometry in his 1854 lecture: Finding and describing the examples of constant curvature. In the more general case, the curvature, which is the main invariant that governs the behavior of geodesics, is a more subtle concept than in the Riemannian case, and the classification of the examples of constant curvature is, correspondingly more delicate. On the other hand, it turns out that the greater richness of the family of Finsler manifolds of constant flag curvature means that it has close connections with other kinds of geometry, such as complex geometry and manifolds endowed with other interesting structures. I will describe what is known about this classification problem and some of these connections, especially with complex and Kahler geometry, in this lecture.
Lecture 3: Recent developments in special holonomy manifolds (DG Seminar)
Abstract: One of the big classification results in differential geometry from the past century has been the classification of the possible holonomies of affine manifolds, with the major first step having been taken by Marcel Berger in his 1954 thesis. However, Berger's classification was only partial, and, in the past 20 years, an extensive research effort has been expended to complete this classification and extend it in a number of ways. In this talk, after recounting the major parts of the history of the subject, I will discuss some of the recent results and surprising new examples discovered as a by-product of research into Finsler geometry. If time permits, I will also discuss some of the open problems in the subject.
Lecture 4: The geometry of rolling surfaces and non-holonomic mechanics (Colloquium)
Abstract: In mechanics, the system of a sphere rolling over a plane without slipping or twisting is a fundamental example of what is called a non-holonomic mechanical system, the study of which belongs to the subject of control theory. The more general case of one surface rolling over another without slipping or twisting is, similarly, of great interest for both practical and theoretical reasons. In this talk, which is intended for a general mathematical audience (i.e., no familiarity with control theory or differential geometry will be assumed), I will describe some of the basic features of this problem, a bit of its history, and some of the surprising developments that its study reveals, such as the unexpected appearance of the exceptional group G_2.
There will be no registration fees: all are welcome. However, if you are interested in attending, kindly send an e-mail to Pedram Hekmati
by 28 October 2013, with your name, position, affialiation and e-mail.
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