Richard Montgomery, Mathematics Professor,
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Ph.D. Students
(ammended Sept. 2012:)For nearly 15 years my primary mathematical obsession has been the
planar zero-angular momentum three body problem.
The basic question within that problem is still open after 344 years of work.
Arbitrarily close to a bounded (eg. periodic) solution, does there exist an unbounded solution?
My methods are primarily those of differential geometry, so I
might be called an applied differential geometer. Calculus of variations,
dynamical systems, a bit of Lie group theory, and a smidge of topology
often arise in my papers. Algebraic geometry
has been sneaking in, due to the influences of blow-up on my work
with Zhitomirskii and the birth of a K3 inside the planar 4 body problem.
A big influence on my career has been
`classical' gauge theory: the geometry of a principal bundle with connection.
Following the physicists
Shapere, Wilczek and Guichardet, I explored the connections
between gauge theory and questions in everyday (not high energy) physics and control such as how does
how a cat, dropped from upside down, with zero angular momentum?
Idealizing the cat to consist of only three mass points led me deep into the jungle of the
three-body problem, where I have stumbling about in wonderment ever since.
Overview of research periods
1982-1988. Symplectic and Poisson reduction. What is the reduced space of the cotangent bundle of a principal bundle?
1986-1998. Falling cats. The isoholonomic problem. Realization
that the isohol. problem is one of optimal control. Subriemannian geometry, culminating in
the `abnormal geodesic' and a book titled `A tour of SubRiemannian Geometry'.
1999-2012 and on. Beginning with the rediscovery of Cris Moore's figure eight solution to the three body problem,
Chenciner and I helped open up a mini-industry of `choreography' solutions to the N-body problem. My most general result here is the theorem that with the exception of Lagrange's orbit
every zero angular momentum negative energy solution to the three body problem has instants of collinearity, or `syzygies'.
2002- 2011. Various problems and the interstices of singularity theory, geometry of plane-fields (distributions), and algebraic geometry,
culminating in a book with Misha Zhitomirskii: `Points and Curves in the Monster Tower'.
Alex Castro (2010), UCSC.
Vidya Swaminathan (2008), UCSC
William C. McCain (2007), UCSC
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
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updated, Sept. 12, 2012