Description
Mathematics Colloquium.
Speaker: Aaron Charles Naber.
Affiliation: Northwestern University.
Date and Time: Monday 06 January, 11:00 am - 12:00 noon.
Venue: Ramanujan Hall, Department of Mathematics.
Title: Analysis of Geometric Nonlinear Partial Differential Equations.
Abstract: It is well known that solutions of a linear elliptic equation,
for instance a harmonic function $\Delta u = 0$, are automatically smooth.
Indeed, one has many ways to quantify this and understand precise
estimates on solutions. In the context of nonlinear equations, this need
not be the case and one may be forced to deal with a singular set
$Sing(u)$. Though dimension estimates on singular sets have been
understood since the 60's, it is only in the last few years that the
structure of the singular sets is starting to come into focus. This talk
will give an introduction to all of this, including a rough idea of the
methods and results. We will study the case of nonlinear harmonic maps as
it requires the least background, however these methods and extensions
have been used lately in many areas (e.g. minimal surfaces, Yang-Mills,
Einstein manifolds, nodal analysis and Q-valued harmonic maps, to name a
few).v