TBD —  2008
    Professor Jacob Lurie

  Oct 4—Oct 6, 2005
    David Vogan

 
  Mar 28—Apr 1, 2005
    Stuart Antman

  Sep 28—Oct 1, 2004
    Pierre-Louis Lions

 
  Apr 17–23, 2004
    Maxim Kontsevich

  Feb 24–26, 2004
    Hillel Furstenberg

Ordway Visitors

  2008–09
  2007–08
  2006–07
  2005–06
  2004–05
  2003–04
  2002–03
  2001–02
  2000–01

Dynamical Systems Seminar: An Innocuous-Looking ODE with Curious Dynamics

A typical steady-state problem from physics, e.g., one governed by an elliptic PDE, has the abstract form f(u,a) = 0 where u is the unknown and a is a set of parameters. The fundamental mathematical problem for such an equation is to determine how the number and behavior of solutions depend on the parameter a. One way to interpret such results is to regard the parameters as slowly (quasistatically) varying with time. This might be the only feasible method of interpretation when the dynamical version of the equation is intractable. One purpose of this talk is to justify the validity of a quasistatic approximation for a simple physical problem governed by a deceivingly simple ODE. The justification depends on the analysis of the problem not only for slowly varying parameters, but also for rapidly varying parameters. The analysis shows that the solutions exhibit a very rich behavior.

Monday, March 28, 3:35 pm VinH 16

Many of the difficulties presented by the nonlinear partial differential equations from solid mechanics are inherently geometrical, reflecting that the equations must (i) describe one-to-one deformations of regions of Euclidean space, and (ii) meet certain invariance requirements, which complicate the geometrical description. This lecture treats geometrically exact problems governed by quasilinear parabolic-hyperbolic systems in which there is but one independent spatial variable. The main emphasis is on how standard methods of nonlinear analysis, like the Faedo-Galerkin method, must be significantly modified to accommodate the intrinsic difficulties of solid mechanics.

Wednesday, March 30, 3:35 pm VinH 16

A material body is incompressible if every deformation of it locally preserves its volume, in particular, if the Jacobian determinant of every continuously differentiable deformation of it is identically 1. Since the nonlinear PDEs of evolution for such 3-dimensional bodies have largely resisted analysis, it is useful to have effective theories for slender bodies governed by equations with but one independent spatial variable. This lecture shows that the actual construction of one such very attractive theory requires the solutions of a sequence of first-order PDEs (by the method of characteristics). Although the resulting equations are more complicated than those for bodies not subject to the constraint of incompressibility, they have novel regularity properties not enjoyed by the latter. The governing equations for an elastic body can be characterized by Hamilton's Principle. The ODEs governing travelling waves for these equations can also be characterized by Hamilton's Principle, but the kinetic and potential energies for these ODEs do not correspond to those of the PDEs. These ODEs admit periodic travelling waves with wave speeds that are are supersonic with respect to some modes of motion and subsonic with respect to others.

Thursday, March 31, 3:35 pm, VinH 16