W. G. Pritchard Lab Seminar - 109 Boucke Building

	     **October 24, 2001**
	
Geometry, Topology, and Hydrodynamics.

Augustin Banyaga

Department of Mathematics
Penn State University

Abstract:

The goal of this talk is to give an elementary introduction to some
Riemannian Geometry and Differential Topology involved in Hydrodynamics. 

The configuration space of an incompressible fluid filling a domain M
is the group  Diff_V(M) of diffeomorphisms of M preserving a volume
element V. The fundamental equations are the Euler equations. Their
solutions (fluid flows) are geodesics on Diff_V(M)  for the right
invariant metric defined by the energy function. Viewed from this
perspective, Hydrodynamics appears as the study of the Riemannian
Geometry of the "infinite dimensional Lie" group Diff_V(M). This goes
back to Arnold in 1966.

More recently, J. Etnyre and R. Ghrist discovered a connection
between hydrodynamics and the topology of contact manifolds. This
connection allows one to prove interesting results on steady fluid
flows. For instance, they  prove the existence of smooth steady
fixed-point free solutions to the Euler equations on all 3-manifolds
and all subdomains of R^3 with torus boundaries.