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# Can a violently
nonisotropic space be useful, interesting and beautiful?

### Roger Brockett

Division of Engineering and Applied Sciences, Harvard University

###
Thursday, November 2, 2000

102 Bradley Hall, 4 pm

Tea 3:30 pm, Math Lounge

**Abstract: **Throughout this century the model of space
provided by a riemannian manifold has dominated a large part of the
literature on geometrical analysis. One distinguishing characteristic
of such spaces is that they are, qualitatively speaking, isotropic.
That is to say, if there are local coordinates *x_1,x_2,...x_n* then
the distance is locally approximated by a square root of the sum of
squares, with suitable weights. Models for spaces that are not
isotropic in this sense have received much much less attention.
However, questions arising in control theory, the study of diffusion
processes and, in less specific way, small scale physics, have
suggested that there are compelling reasons to move beyond
qualitatively isotropic models toward the investigation of spaces in
which different directions have qualitatively different metric
properties. Perhaps the best developed line of work in this direction
goes under the name of subriemannian geometry. Existing work already
shows this subject to be both tractable and mathematically rich. In
this talk I will introduce some of the main ideas and then will go on
to discuss a set of questions that take the subject in a new
direction, related to problems of current interest in control theory
and classical mechanics.

This talk will be accessible to undergraduates.