Dartmouth 2003 Kemeny Lecture Series MATH

Robert Bryant

Duke University

will give the following series of lectures

Geometry Old and New: From Euclid to String Theory

Monday, May 12, 2003

7:00 - 8:00 pm

104 Wilder Hall

(Reception To Follow)

Abstract: The dream of geometers since the time of Euclid has been to explain the phenomena of the physical world in terms of very simple concepts and objects (such as circles, lines, spheres, Platonic solids, and so on) that display a high degree of symmetry. As our understanding of the physical world has become more sophisticated, it has become evident that symmetry plays a more profound role than is apparent at first glance. This deep connection turns out to hinge on our ability to formulate principles governing physical processes as principles of ``least action'', as was explained by Emmy Noether less than 100 years ago. In this talk, I will describe some of the ideas and the history of the development of this mathematical understanding of the relationship between symmetry (for example, the principle of invariance) and conserved quantities (such as energy and momentum) and how it still shapes our attempts to develop a ``theory of everything''. I will attempt to explain why string theory seems to require our universe to be a space of much higher dimension than is apparent to our senses. No mathematics beyond vector calculus will be assumed.

Note: This talk will be accessible to undergraduates.

NB: A PDF version of this announcement (suitable for posting) is also available.

New Zoll metrics on the 2-sphere

Tuesday, May 13

4:00 - 5:00 pm

102 Bradley Hall

(Tea 3:30 pm Math Lounge)

Abstract: It is classical that, on the standard round 2-sphere in 3-space, the geodesics are the great circles and, hence, are all closed. This is a very special situation since, for the `generic' metric on the 2-sphere, almost none of the geodesics close.

A metric on the 2-sphere for which all of the geodesics close is nowadays called a Zoll metric, after Otto Zoll, who about a century ago, used some ideas of Darboux to construct an explicit family of metrics on the 2-sphere with the property that all of the geodesics of any member of this family are closed. Each of Zoll's examples has an axis of symmetry, but Funk conjectured (and, much later, Victor Guillemin proved) that the Zoll metrics that are `near' the round metric in a suitable sense can be parametrized by the odd functions on the 2-sphere, thus providing a potentially enormous family of Zoll metrics on the 2-sphere. However, other than examples with an axis of symmetry, none of these have been found explicitly before now.

In this talk, I will review the now-classical story outlined above and will use some ideas that go back to Jacobi (who showed that the geodesics on the ellipsoid form a completely integrable system) to construct some new explicit examples of Zoll metrics, in fact, ones with no rotational symmetry.

Note: This talk will be accessible to graduate students.

NB: A PDF version of this announcement (suitable for posting) is also available.

Finsler metrics of constant flag curvature

Thursday, May 15

4:00 - 5:00 pm

102 Bradley Hall

(Tea 3:30 pm Math Lounge)

Abstract: Finsler geometry is a generalization of Riemannian geometry in which one generalizes from having a smoothly varying (positive definite) inner product on the tangent space at each point to having a smoothly varying strictly convex Banach norm on the tangent space at each point. Many problems in the calculus of variations for curves are naturally formulated as geodesic problems on Finsler manifolds and, indeed, this is where the main interest in this geometry arises.

Thus, associated to each Finsler geometry, there is its family of geodesics. The first natural invariant one defines in Finsler geometry is the invariant that governs the variation of these geodesics, i.e., the Jacobi fields. A Finsler manifold is said to have constant flag curvature $c$ if its Jacobi operator along any geodesic is conjugate to that along a geodesic in a Riemannian space form of constant sectional curvature $c$. In contrast to the Riemannian case, a Finsler manifold of constant flag curvature does not have to be homogeneous or even Riemannian.

In this talk, I will review the basic about Finsler manifolds and their local invariants and discuss what is known about the existence and generality, both local and global, of Finsler metrics of constant flag curvature. Among the more recent interesting results have been the determination of these structures in dimension 2 and, via some recent results of LeBrun and Mason, the proof that the Riemannian round 2-sphere is the only reversible Finsler metric of constant flag curvature +1 on the 2-sphere.

Note: This talk will be accessible to specialists only.

NB: A PDF version of this announcement (suitable for posting) is also available.