P657 | 31247 | Townsend, J.

COURSE ANNOUNCEMENT: ELEMENTS OF DIFFERENTIAL GEOMETRY AND TENSOR ANALYSIS WITH APPLICATIONS TO PERCEPTION PSYCHOLOGY AND BRAIN SCIENCES [P657] & COGNITIVE SCIENCE [Q700] Instructor: James T. Townsend Second Course: Fall 2010 READING: Required Texts: 1. An Introduction to Differential Manifolds and Riemannian Geometry, by W. M. Boothby, 2003; Academic Press. 2. Riemannian Manifolds, by M. P. Do Carmo, Birkhauser, 1992, Birkhauser, Boston. Recommended: Solid Shape by J. J. Koenderink, 1990, MIT press. DESCRIPTION OF COURSE: This two-term course builds on multivariate calculus to naturally enter into the arena of differential geometry and ultimately the beautiful theory of topological and differential manifolds. The first phase was offered in FALL, 2009 while the second phase will be offered in FALL, 2010. I. Term 1 was grounded in the familiar Euclidean milieu but foundations were laid that took us in the first several weeks from covariant and contravariant form theory (from Cartan’) up through mappings and differentiation on surfaces and a first introduction to manifolds. During the latter half of the first term we aimed on learning more about forms, moving onto such topics as curvature studied from the point of view of tensor theory (introduced earlier) and the concept of so-called shape operators (pointing out the similarities and differences of the Cartan ‘form’ vs. the Levi- Civita tensor calculus approaches to this topic). Then we are in a position to re-study surfaces as entities as manifolds, independent of their ‘graphs’ in Euclidean space, which leads inexorably to Riemannian geometry. II. Term 2 will view the material from Term 1 in more general and elegant manifold theory and will be offered more in a seminar form than the first Term. Some potential topics: A. Diffeomorphisms now within a more general theory of differentiable mappings, including immersions, submersions, and embeddings. B. A look at geodesics through parallel transport in Riemannian manifolds will be accorded. C. If time, we will take a peek at how local properties around a point in a space can sometimes be extended to important aspects of the larger manifold. D. Tangent-to-fiber bundle theory. E. Contributions of Lie group/algebra approaches. F. Integration on manifolds. G. More on curvature.