Mathematics Graduate Students Organization

Seminar:

Date: April 8

This talk is about general (non-linear) 2nd order scalar PDE in the plane, i.e. equations of the form

F(x, y, z, z_x, z_y, z_{xx}, z_{xy}, z_{yy}) = 0,

from a GEOMETER's point of view.

WARNING #1: My apologies to the analysts, but this talk contains NO analysis.
WARNING #2: I actually won't even talk about how to solve such PDE...

Rather, I'll talk about the problem of equivalence: When are two PDE locally equivalent via a change of variables? This is now a problem of geometry. Changing variables to simplify an ODE or PDE underlies most solution techniques that we know. Understanding this equivalence problem is a first step before one can address general solution methods, especially for highly nonlinear problems for which little is known.

What does it means for a PDE to be elliptic, parabolic, hyperbolic when the equation is fully non-linear? The answer is simple to state, and not too much of a stretch if you're familiar with the linear theory, but the geometric reasoning for this trichotomy delves into some lovely differential geometry and Lie group theory. In particular, I'll describe: (1) what the geometric study of PDE has to do with the conformal / symplectic geometry of surfaces in the Lagrangian-Grassmannian, and (2) the geometric origins of the form of the generalized Monge-Ampere equation.