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Research Interests

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Singularities of Stationary Integral Varifolds

In the 1960’s F. Almgren investigated the problem of constructing an unstable minimal submanifold, i.e. an unstable critical point of the n - dimensional area functional in an (n + k) -dimensional Riemannian manifold. This is a hard problem and initially he was only able to construct a type of weak, measure-theoretic solution, called a stationary integral n-varifold. These geometric objects can have a singular set, a set of points at which it is not smooth and embedded. There are a wealth of interesting questions about the regularity and singularities of stationary integral varifolds which fall into the realms of geometric measure theory:

  • Does the singular set of a stationary integral n - varifold have zero n - dimensional measure? (i.e. is it regular almost everywhere?)

  • What possible tangent cones can a stationary integral n - varifold have?

  • Does each point of a stationary integral n - varifold have a unique tangent cone?

  • One can even ask about whether or not the singular set itself has any structure of its own: In cases where the singular set is lower-dimensional (less than n), is it rectifiable? Or perhaps it resembles a smooth variety? 

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Nodal Sets

The study of the zero sets or 'nodal sets' of solutions to elliptic problems is closely related to the study of minimal surfaces, both through the existence theory and the regularity theory and many interesting questions emerge from ths interplay. But there are also many interesting problems about nodal sets that are less related to minimal surfaces: An old result of H. Whitney states that any closed subset of Euclidean space is the zero set of some smooth function. If however we insist that the smooth function vanishes to only finite order, then the zero set has some structure: It is countably (n-1) - rectifiable. And, near to a point at which the function vanishes to order K, the zero set is contained in K graphs of W^{1,p} functions (for some p > 1). What is the optimal such statement for the nodal set of a smooth function with finite vanishing order? Can we say something like "...contained in CK graphs of C^1 functions"?
 

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