Office: 263 Baskin Engineering
Phone:831- 459-4841
E-mail: rmont@math.ucsc.edu
| Richard Montgomery | |
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math prof,
Office: 263 Baskin Engineering Phone:831- 459-4841 E-mail: rmont@math.ucsc.edu
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Research Interests
written around 2004:
For the last five years or so I have been obsessed by the three-body problem.
The bulk of my research could be
described as applied differential geometry. The main
tools I use are differential geometry, and the calculus
of variations, mixed with a bit of group theory and dynamical systems.
An underlying theme of my work has been
gauge theory, i.e. principal bundles with connections.
Following initial work of the physicists
Shapere and Wilczek and earlier work
of Guichardet, I have explored how gauge structures arise naturally, not in high
energy physics, but in
`mundane' problems of mechanics and control theory.
One such problem is that of the orientation
of three bodies moving in space or the plane.
Another is that of a falling cat (or, if you prefer, a robot) dropped upside down. The cat flips itself right side up, even though its angular momentum is zero. It does this by changing its shape. In terms of gauge theory, the shape space of the cat forms the base space of a principal SO(3)-bundle, and the statement "angular momentum equals zero" defines a connection on this bundle.
The `isoholonomic problem' arising in gauge theory led me naturally into the field Subriemannian geometry, the geometry called Carnot-Caratheodory by Gromov and his school. See my book with the AMS above.
My initial research was in symplectic and Poisson reduction. In my thesis I analyzed the reduction of the cotangent bundle of a principal bundle by its group. In the mid and late 80s I did a bit of work on understanding and making rigorous various approximations used for nonlinear hyperbolic partial differential equations, specifically, particle mechanics (also called adiabatic ) approximations to the nonlinear Schrodinger and Yang-Mills-Higgs equations, and the modulation equations used in studying high frequency limits of certain integrable PDE like the sine-Gordon equation.
Ph.D. Students
Andrew Klingler, 1999.
thesis: Stochastic Calculus and Eigenvalue bounds for Geometric Laplacians.
Alex Golubev, 1999. (co-advised with Viktor Ginzburg.) thesis: A Gray's theorem for Engel Structures.
Cesar Castilho, 1998.
(co-advised with Viktor Ginzburg.)
thesis: The Motion of a Charged Particle on a
Riemannian Surface under a Non-zero Magnetic Field
email: castilho@dmat.ufpe.br.
professor at: Departamento de Matematica
Universidade Federal de Pernambuco
Recife, PE 50540-740
Brazil
Kurt Ehlers, 1995.
thesis: The Geometry of Swimming and Pumping at Low Reynolds Number
email: kehlers@scs.unr.edu
Reno Community College.
Girija Mittagunta, 1994.
(co-advised with Tudor Ratiu.)
thesis:
Reduced Spaces for Coupled Rigid Bodies and Their Relation to Relative Equilibria
Patrick Tantalo, 1993, (UCSC)
(co-advised with Tudor Ratiu.)
thesis:
Geometric Phases for the Free Rigid Body with Variable Inertia Tensor.
Now a lecturer in Computer Science/ Engineering, UCSC.
Gil Bor, 1991 (Berkeley), ( unofficial student,
co-advised with Jerry Marsden.) thesis: Non-self dual solutions to the Yang-Mills equations
over the four-sphere.
Now at CIMAT, Guanajuato , Mexico.
email: gil@fractal.cimat.mx