RICHARD MONTGOMERY Professor of Mathematics B.A., Sonoma State University Ph.D., University of California, Berkeley My Page My Courses

Phone: 831-459-4841 Office: McHenry #4120 Office Hours: By Appt. Email: rmont@ucsc.edu

RESEARCH INTERESTS

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 solved 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.                 

SELECTED PUBLICATIONS

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).

R. Montgomery with Persi Diaconis and Susan Holmes: Dynamical Bais in the Coin Toss, SIAM Review, v. 49, no. 2, 211, May 1 (2007).

R. Montgomery with Alex Castro, Roberto Manduchi and X. Shi: Rotational Invariant Operators Based on Steerable Filter Banks, IEEE Signal Processing Letters, v. 13, no. 11, Nov. (2006).

R. Montgomery with A. Chenciner and J. Fejoz:  Rotating Eights I: The three Gamma_i families, Nonlinearity 18, 1407-1424, (2005).

R. Montgomery: Hyperbolic Pants Fit a Three-body Problem, Ergodic Theory and Dynamical Systems, v. 25, 921-947, June. (2005).

R. Montgomery: "A Tour of Subruemannian Geometries, Their Geodesics and Applications", Mathematical Surveys and Monographs, v. 91, American Math Society, Providence , Rhode Island, 2002.

R. Montgomery: Infinitely Many Syzygies (pdf) appeared in Archives for Rational Mechanics, v.164 (2002), 311-340, 2002.

A. Chenciner, J. Gerver, R. Montgomery and C. Simo: Simple choreographies of N bodies: a preliminary study. (pdf) appeared in Geometry, Mechanics, and Dynamics, volume in honor of the 60th birthday of J.E. Marsden, P. Newton, P. Holmes, A. Weinstein, ed., Springer-Verlag, (2002).

R. Montgomery: A New Solution to the Three-Body Problem. Notices of the American Mathematical Society, 471-481, May. (2001).

R. Montgomery with M. Zhitomirskii: Geometric Approach to Goursat Flags., Ann. Inst. H. Poincare Anal. Non Lin/'ears, vol. 12, no. 4 459-493, (2001).

R. Montgomery with A. Chenciner: A remarkable periodic solution of the three-body problem in the case of equal masses., Annals of Mathematics, v. 152, 881-901, Nov. (2000).

R. Montgomery: The braid group and action-minimizing periodic orbits. Nonlinearity, Vol 11, 363-376 (1998).

R. Montgomery: The connection for a family of completely integrable systems whose holonomy is the classical adiabatic angle (Berry's phase). Comm. Math. Phys. v.120, 269-294 (1998).

M. Kazarain, R. Montgomery and B. Shapiro: Characteristic classes for the degenerations of two-plane fields in four dimensions. Pac. J. Math., v. 179, 2, 355-370 (1997).

R. Montgomery: The geometric phase of the three-body problem. Nonlinearity, v. 9, 1-20 (1996).

H. Berg, K. Ehlers, R. Montgomery and A. Samuel: Do cyanobacteria swim using travelling surfaces waves? Proceedings of the National Academy, Biophysics section, v. 93, no.16, 8340-8343 (1996).

R. Montgomery: Hearing the zero locus of a magnetic field, Communications in Mathematical Physics, v.168, No. 3 (1995).

R. Montgomery: Hearing the Zero Locus of a Magnetic Field. (pdf) Communications Math. Physics, v. 168, no. 3. 651-675, (1995).

R. Montgomery: Abnormal minimizers. SIAM J. Control and Optimization, v. 32, no. 6, 1605-1620 (1994).

R. Montgomery: Gauge theory and control theory, nonholonomic motion planning. (J. Canny and Z. Li, editors), Kluwer Acad. Press, 343-378 (1993).

R. Montgomery: Isoholonomic problems and some applications. Comm. Math. Phys. v.128, 565-592 (1990).

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