Psychology and Brain Sciences | Topical Seminar: Elements of Differential Geometry & Tensor Analysis with Applications to Perception
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.