Stephen Preston

Stephen Preston

Associate Professor
Ph.D., SUNY Stony Brook, 2002

Math 302
Mailing address:
Stephen Preston
Department of Mathematics
University of Colorado at Boulder
Campus Box 395
Boulder, CO 80309-0395
Research interests:
Differential equations, Riemannian geometry, continuum mechanics
Personal home page:

My research is in partial differential equations and Riemannian geometry, especially the geometric approach to the ideal Euler equations of fluid mechanics pioneered by Vladimir Arnold in 1966. The idea is to look at fluid flows as geodesics on an infinite-dimensional manifold (the group of volumorphisms) and use the curvature to understand stability of fluid flows. My dissertation research involved clarifying the relationship between curvature, Eulerian stability, and Lagrangian stability. Recently I have been studying the nature of conjugate points on the volumorphism group, which behave strikingly differently between two- and three-dimensional fluids.

Select Publications:

For ideal fluids, Eulerian and Lagrangian instabilities are equivalent.  GAFA  14 (2004), no. 5, 1044-1062.

Nonpositive curvature on the area-preserving diffeomorphism group,   J. Geom. Phys. 53, (2005) no. 2, 226-248.

Singularities of the exponential map on the volume-preserving diffeomorphism group, (with D. G. Ebin and G. Misiolek).  GAFA  16  (2006), no. 4, 850-868.

On the volumorphism group, the first conjugate point is always the hardest. Comm. Math. Phys. 267 (2006), no. 2, 493-513.

Intervals of conjugate points on the volumorphism group,   Preprint.