Luke Rogers
Department of Mathematics
593 Malott Hall
Cornell University.
Ithaca, NY, 14853-4201
U.S.A.
luke(the usual symbol)math.cornell.edu
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Luke Rogers luke(the usual symbol)math.cornell.edu |
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I am an H.C. Wang Assistant Professor in the Mathematics Department at Cornell University. Previously I was a Lecturer in the Department of Mathematics at the State University of New York at Stony Brook and before that a graduate student at Yale University. My mathematical interests include analysis on fractals, Sobolev spaces and quasiconformal mappings These have connections to harmonic analysis, potential theory, complex analysis and geometric measure theory. In addition to my academic work I am an avid rock-climber, occasional cyclist and hiker, and all-around outdoor enthusiast. Look here for some of my interests.
My thesis work was about universal extension operators for Sobolev spaces, which are operators that extend functions from any Sobolev space on a domain (locally they are defined on any function that is locally integrable) to the corresponding Sobolev space on the ambient Euclidean space, with estimates. The main result of my thesis extended methods of E. Stein and P.W. Jones to show that a universal extension operator exists for Sobolev spaces on locally-uniform domains. If you are familiar with the "cone-condition" for boundary points of a domain, which is often assumed when studying boundary value problems, then you can think of locally uniform domains as a generalization in which there is a "twisting cone" at every boundary point. The basic example of a twisting cone is the region between two logarithmic spirals.
Subsequently I have generalized the results of my thesis to consider Sobolev spaces on domains satisfying a weaker condition that is more measure-theoretic than geometric. The condition is usually called Ahlfors (or Ahlfors-David) regularity. It says that if we take a ball of radius r (for r between 0 and 1) around any point in the domain, then the intersection of the ball and the domain has measure at least Cr^n. This condition has been shown to be necessary for the existence of a bounded linear extension operator by Hajlasz, Koskela and Tuominen.
While at Cornell I have been working on a number of interesting problems in analysis on fractals. All of this work is joint with Bob Strichartz, and different projects have involved a number of other people, including Kasso Okoudjou (University of Maryland), Alexander Teplyaev (University of Connecticut) and Erin Pearse (Cornell) as well as my undergraduate advisees Michael Barany (Cornell) and Jessie DeGrado (Cornell).
One of the main problems we have addressed is related to the structure of smooth functions on certain fractal sets. This structure is very different to that of smooth functions on Euclidean spaces, not least because on fractals the product of smooth functions is almost never smooth! What we have been doing is constructing analogues of some tools of classical analysis (including smooth bump functions, partitions subordinate to open covers, distributions, etc.) in the fractal setting. These should be useful for studying differential equations on fractal structures. I see this as one possible step in understanding dynamical processes and change in physical systems, because much of the world around us is (at least approximately) fractal in nature. So far we have quite complete results for the existence of smooth bump functions, and have a solution to the smooth partitioning problem in the post-critically finite (p.c.f.) case. This lets us define distributions on p.c.f. fractals and establish their basic properties. The tools involved in the proofs are both analytic and probabilistic.
There are still a great many things to do in this area. Bob Strichartz, Erin Pearse and I are currently working on some questions related to tempered distributions on unbounded versions of gasket fractals, and there are a great many natural questions about the structure of the operators corresponding to differential equations. I am also co-supervising (with Bob Strichartz) an undergraduate student (Jessie DeGrado) who is investigating some aspects of the harmonic gradients introduced by Teplyaev. My other undergraduate advisee, Michael Barany, is looking at a problem on the existence of Dirichlet forms on an infinite gasket structure. Dirichlet forms are basic to the construction of Laplacians on fractals and form a founding pillar of analysis in this setting. The work Michael is doing is particularly related to that of Sabot and of Metz on non-linear Perron-Frobenius theory.
Modulation spaces are spaces of functions with a some phase space localization measured by the modulation norm. As such they are well adapted to studying the evolution of the phase space structure of solutions of partial differential equations. It is well known that L^2 quantities can be used to describe energies, and many well-known PDE (eg the wave equation) have a conservation of energy property expressible in these terms. On the other hand, L^p properties for p different than 2 are not usually conserved. Instead one may wish to look at phase-space localization as expressed using the modulation space norm. The results for unimodular multipliers (including those for the wave and Schroedinger equations) are in a paper I collaborated on with Arpad Benyi, Kasso Okoudjou and Karlheinz Grochenig. This is the initial step for a number of projects related to modulation spaces and PDE.
This semester I am teaching Math 424, Fourier Analysis and Wavelets. If you are a student in this course, I would welcome your feedback on my teaching, which can be submitted anonymously using Blackboard .