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Tondeur Lecture in Mathematics: Jeff Cheeger (NYU)

Event Type
Department of Mathematics
245 Altgeld Hall
Mar 6, 2013   4:00 pm  
Steve Bradlow

The 2013 Tondeur Lectures in Mathematics will be presented by Jeff Cheeger (Courant Institute of Mathematical Sciences, New York University) entitled "Quantitative Behavior of Singular Sets For Certain Geometric PDE's" at 4 p.m. on March 5, 6 and 7.  A reception will be held following the March 5 lecture in 314 Altgeld Hall.  Abstract: Since solutions of nonlinear elliptic and parabolic partial differential equations

need not be smooth, a basic issue is to understand the size of the singular set $\S$. We will describe a method, developed jointly with Aaron Naber,

for improving  known lower bounds on the Hausdorff codimension of $\S$, to more effective

estimates.  Specifically, in each unit ball, we  bound  the volume of a tube of radius $r$  around

a set which is larger than $\S$. If a point $x$ lies outside this $r$-tube and the

ball $B_{r/2}(x)$ is rescaled to unit size, then on the rescaled ball, the appropriately

rescaled solution satisfies definite $C^k$ bounds. (These bounds are independent of $r$.)

Cases to which the method applies include, Einstein manifolds, harmonic maps,

minimal submanifolds and mean curvature flow. Also, in the context of linear elliptic

equations, there are improved estimates on the set of points at which the gradient vanishes.

In the first lecture, for the cases of noncollapsed Einstein manifolds, minimizing harmonic maps, and minimizing hypersurfaces, we will give some background and a statement of the main results.  The Einstein case, which is representative, will be discussed in more detail in the next two lectures.

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