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Mailing address: TMCC, 7000 Dandini Blvd., Reno NV. 89512

Thesis Advisor: Richard Montgomery, UC Santa Cruz

Class Information: CLASSES

Interests outside math: Rock Climbing (DNB Middle Cathedral, El Capitan, Half Dome, Zodiac on El Capitan, Nose Route on El Capitan) , painting, and bicycle touring (photo Alaska).

Research interests: My primary field of research is applications of differential geometry. I am currently interested in two lines of research: microswimming and nonholonomic mechanics. Some of my publications are available here.

Microswimming. My PhD thesis and first papers were concerned with the self-propulsion of microorganisms. My collaborators on this research include Jair Koiller, Richard Montgomery, Joaquim Rodruiges, Howard Berg, Aravi Samuel, and Marco Raupp. I approach the problem using the differential geometric model introduced by the physicists Al Shapere and Frank Wilczek. The fact that a swimming microorganism cannot exert net forces or torques on the surrounding field defines a connection on the principle fiber bundle whose total space P consists of all located shapes the organism can assume and whose base space P/SE(3) consists of all unlocated shapes. The horizontal directions are vector fields along the outer membrane of the organism that, taken as boundary conditions for the Stokes equations, lead to no net forces or torques on the fluid.The vertical direction are vector fields on the outer membrane corresponding to rigid translations and rotations. To swim an organism moves in horizontal directions to generate vertical motion!

The machinery behind microswimming is not always clear. The intuition we develop living in the macro-world sometimes gets in the way of uncovering it. (An aircraft carrier coasts for miles after stopping its engines at cruising speeds while an e-coli coasts a distance small compared to the diameter of a hydrogen atom when it "throws in the clutch." - See Random Walks in Biology by Howard Berg for the computation.) E-coli swim by rotating a rigid helical filament called a flagellum. Protozoa such as paramecia and opalina swim using coordinated beats of cilia that form traveling waves passing down (or up!) the body of the organism. The swimming strategies of some other organisms are not understood. A famous example is a strain of cyanobacteria called synechococcus that lives in the Atlantic ocean. This organism is one or two micrometers in diameter and swims at about 10 body lengths per second. It has no apparent moving parts. The only mechanism for its motility that has not been ruled out is one I posed as a graduate student in 1995. I conjectured that it must swim by passing small amplitude tangential compression waves along its outer membrane. Here are some animations, made on Maple, illustrating this mechanism: ANIMATIONS.

Update on Synechococcus: An electron microscopy study of the outer membrane has revealed that motile strains of Synechococcus have a crystalline outer shell (CS) penetrated by a profusion of spicules (SP) extending from the inner membrane. We conjecture that CS-SP interactions could generate high frequency vibrations leading to acoustic streaming. See my list of publications for a recent paper describing the details.

Nonholonomic Mechanics. Over the last three or four years, I have become interested in nonholonomic mechanics. This is a branch of classical mechanics that deals with mechanical systems with velocity constraints that cannot be realized as constraints on position. We generally assume that every point in the configuration space can be reached by some path satisfying the velocity constraint. Common examples of nonholonomic systems include coins and balls that roll without slipping. A ball can be rolled without slipping or twisting on a table from one position and orientation to any desired position and orientation. A beautiful example of a nonholonomic constraint is illustrated by an ice skater. She is free to skate where ever she wants on the ice but she can only move in the direction her skate is pointing. Nonholonomic constraint often give rise to counter-intuitive phenomena in dynamics. If an object is spun on the ground we expect it to just rotate until it comes to a rest because of friction. Frictional forces, however, give rise to nonholonomic constraints that can lead to more interesting behavior as is illustrated by the rattleback.

I use the affine connection point of view to write the equations of motion for mechanical systems with nonholonomic constraints. This point of view was introduced by Cartan in his address to the International Congress of Mathematicians in 1928. A mechanical system with an nonholonomic constraint is described by an n-dimensional manifold M endowed with a Riemannian metric < , > representing the kinetic energy, and a rank m distribution H representing the nonholonomic constraint. Any trajectory r must be tangent to H, i.e. r'(t) must lie in H(r(t)). The evolution of the system is governed by the nonholonomic geodesic equations which are obtained by computing the acceleration of the trajectory using the Levi-Civita connection associated to < , >, orthogonally projecting the result onto H, then setting this expression equal to 0.

My recent work, done in collaboration with Jair Koiller, involves two lines of research: first, we use the Cartan equivalence method used to study the geometry underlying nonholonomic systems, and second we study the question of whether, under an appropriate change of time scale, a nonholonomic system can be put into Hamiltonian form. The second question is true for some systems (a rubber ball, with center of mass at the geometric center but with unequal moments of inertia, that rolls without slipping or twisting on a plane) and false for others (a sled with a blade at the front whose center of mass is behind the blade). A necessary condition for a system to be "Hamiltonianized" is that it posess an invariant measure.