Physical & Biological Sciences Division
Distinguished Professor
Faculty
Office of Sponsored Projects
Physics Department
McHenry Library
McHenry Building Room #4120
Mathematics Department
B.A., Sonoma State University
Ph.D., University of California, Berkeley
Richard Montgomery is best known for work in two areas: the classical N‑body problem and sub‑Riemannian geometry. In the first area, he and Alain Chenciner (IMCEE Paris) rediscovered the figure eight orbit of Cris Moore and rigorously prove of this solution to the three-body problem, a solution in which which three equal massed stars or planets chase each other around a figure-eight shaped curve. Their paper is mentioned in the best-selling science fiction book "The Three Body Problem'' by Cixin Lin. The figure eight work led to the discovery of a huge variety of "choreography orbits'' N-body solutions in which a number of equal masses chase each other around various aesthetically pleasing curves. Their result and the techniques introduced by the paper drove much work in mathematical celestial mechanics from 2000 to 2015. In the second area he is best known for his counterexample and his book "A Tour of SubRiemannian Geometry''. The counterexample, a certain three-dimensional geometry with a geodesic which does not satisfy the previously accepted geodesic equation, changed the face of subRiemannian geometry. One of his most cited papers is on coin flipping and was written with Persi Diaconis and Susan Holmes (Stanford). This team showed that a flipped coin is more likely to land heads up if starts heads up, and tails up if it starts tails up. This paper and the two research areas just described have their roots in his early work of his on how a cat changes shape in mid-air in order to land on its feet.
Since 2000 Richard Montgomery's work has been primarily in two areas: (I) the N-body problem and (II) the geometry of distributions.
I) In 1997 I became obsessed with the 3-body problem from celestial mechanics, a special case of the classical N-body problem, and one of the oldest problems in mathematics and physics. A version of the N-body problem was formulated by Newton, and he sloved it exactly for N=2 in his Principia, recovering Kepler's 3 laws. In a precise sense, Poincare proved at the turn of the last century that the 3-body problem cannot be solved exactly "is not integrable". Like Galois's impossibility proof, Poincare's 'impossible' lead to an enormous body of work, perhaps most notably as the root of the "chaos theory" popular in the late 70s (a.k.a. nonlinear dynamics). my best work to date is probably my 2000 paper with Chenciner. You can read more about it HERE:
http://www.scholarpedia.org/article/N-body_choreographies
My methods include calculus of variations, some Lie group theory, and the geometry of the "shape space" of similarity classes of triangles.
II) By a"distribution" I mean a linear sub-bundle of the tangent bundle of manifold. My interest began through exploring connections (a pun) between how cats land on their feet, how micro-organisms swim and gauge theory, connections initiated by A. Shapere and F. Wilczek (a recent Nobel laureate, for something else). In the cat problem the distribution is defined by the condition "total angular momentum equals zero". If one imposes a metric on the distribution planes and looks for the shortest paths ("best ways to land on one's feet") one arrives naturally at "subRiemannian Geometry".
A very special type of distribution arises out of the problem of curves in the plane, the kth order derivatives of such a curve, compactified, form a kind of universal space for a certain type of distributions called "Goursat distributions". I have written a series of papers and one book on these, inspired by my collaborator, singularity theorist M. Zhitomirskii. In about 2006, we discovered the spaces for these distributions had been studied by algebraic geometers under the name of "Semple Tower" and "Nash blow-up". The semple tower and Nash blow-up for surfaces in space, as opposed to curves in a plane is poorly understood and is one of the focusses of my research since 2008.
Most of the other papers listed here can be characterized as "applied differential geometry" or "mechanics". I enjoy working with people from other disciplines.
I have taught `Classical Geometries' umpteen times and still enjoy it.
This class is aimed primarily at those who might teach geometry and high school
but our high performing graduate school bound junior and seniors enjoy it also.
It is taught in an interactive style and aimed at reversing the tide
of stripping geometry out of the high school curricululum.
I typically teach this course, one other upper division undergraduate course
and two graduate courses per year.
American Mathematical Society Fellow
R. Montgomery with M . Zhitomirskii: Points and Curves in the Monster Tower, Memoirs of the AMS, v. 205 (2010).
- R. Montgomery with Gil Bor: $G_2$ and the Rolling Distribution, L'Enseignement Mathematique, v. 55, 157-196, (2009).
- R. Montgomery with V. Swaminathan and M. Zhitomirskii: Resolving Singularities Using Cartan's Prolongation, Journal of Fixed Point Theory and Applications (Arnol'd volume); v. 3, no. 2, Sept. (2008).
- R. Montgomery with Duncan Ralph and Onuttom Narayan: Exact Identities for Nonlinear Wave Propagation, Phys Rev E., v. 77, 056219, May (2008).
- R.Montgomery with Alex Castro: The Chains of Left-invariant CR-Structures on SU(2), Pac. J. Math., v. 238, no. 1, 41-71, (2008).
