My Research Interests

My research interests are mostly related to analysis on spaces which lack smoothness properties. These spaces include Euclidean domains (and pieces of manifolds) which have highly irregular boundaries as well as spaces for which the intrinsic structure is very non-Euclidean.

There are several approaches to analysis to metric-measure spaces that do not have a Euclidean structure.  In order to get started one usually makes some assumptions that provide a large class of “well-behaved” functions.  Most of the work I have done recently is in what is usually called “Analysis on Fractals”, though parts of it should perhaps more accurately be called “Analysis on Metric-Measure-Dirichlet spaces”or “Analysis on self-similar spaces”.  In this approach one either constructs or assumes the existence of a Dirichlet form, which one should think of as being an abstract version of the L^2 norm of the Euclidean gradient, and therefore (by general functional-analytic considerations) a Laplacian operator.  The “well-behaved functions” one considers are those in the domain of the Dirichlet form (finite energy functions – a sort of Sobolev space), or in the domain of the Laplacian (“differentiable functions”) or some power of the Laplacian (“smooth of some finite order”).  One can then try to build a theory that parallels the usual calculus in Euclidean spaces for this class of functions and operators, as well as studying the associated differential and partial differential equations.  Eventually one would like to be able to analyze the behavior of solutions on spaces that approximate structures which occur in nature.  To give just one example, one could ask what solutions of the wave equation look like on a percolation network (such as a distribution of oil or gas in a rock formation); I emphasize that we are a long way from being able to give a good answer to this question!

This type of analysis involves a blend of harmonic analysis, potential theory, functional analysis and probability theory.  To get started one needs the Dirichlet form.  In the cases where the resulting potential theory will give point sets positive capacity (and therefore finite energy functions will be continuous) the form can be constructed as a limit of forms on graphs, which often gives a concrete way to compute with interesting functions.  This approach is especially useful on self-similar sets because self-similarity gives a relationship between the local and global analytic structures which is a little like that in Euclidean spaces (which are about as self-similar as it is possible to be!).  There are quite a lot of analytic results where some sort of self-similarity plays an important role, and several of my papers are on results of this type.  Once one leaves the self-similar setting many things become more difficult.  There are some things that can be done by purely functional analytic methods if one knows strong estimates for the heat kernel associated to the Dirichlet form (for example my paper with Strichartz and Teplyaev on Smooth Bumps contains a result of this type) but constructing Dirichlet forms and giving explicit descriptions of finite energy functions is much more difficult.

There are a lot of interesting problems in this area, so I will not try to mention all of them.  Among the things I am thinking about are geometric structures on metric-measure-Dirichlet spaces (in particular Riemannian structures associated to the form, as in my paper with Ionescu and Teplyaev on Derivations and Dirichlet forms) and the associated question of developing non-commutative geometries and eventually quantum field theories on self-similar spaces.  I am looking at several open problems about smooth functions and their properties on self-similar sets with resistance-type Dirichlet forms (and on their product spaces and on fractafolds constructed therefrom).  At the same time I am investigating the behavior of the metric in harmonic coordinates, existence of embeddings via eigenfunction coordinates (which is related to some questions in applied mathematics) and the nature of the maps between the resistance metric and harmonic coordinate metric (when it exists).  The overall goal here is to understand the extent to which analysis of this type is similar to the study of Sturm-Liouville problems associated to singular measures.  Finally I am working on some methods for the construction of Dirichlet forms on sets which do not fit within the existing theory.

For completeness, I should return to a point mentioned earlier, which is that there are other approaches to analysis on metric measure spaces. A good place to learn about one important and well known one is the book “Analysis on metric spaces” by Juha Heinonen.  In this approach one assumes existence of a large class of rectifiable curves, from which it follows that the Lipschitz functions are a rich and interesting collection.  Pursuing this idea leads to a sort of first order calculus on the space.  In general this is a different kind of theory than that in analysis on fractals, where there are often no rectifiable curves and the Lipschitz functions are not the natural class to consider.  However there are some results in the intersection between these two theories, and I believe that many more natural connections will become clear over time.

Some results

Sobolev Spaces and other spaces of “smooth” functions

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.

Analysis on fractals

Much of my work in this area is joint with Bob Strichartz (Cornell), and different projects have involved a number of other people, especially Kasso Okoudjou (University of Maryland), Erin Pearse (Cal Poly), Huojun Ruan ( Zhejiang U.), Alexander Teplyaev (U. of Connecticut) and Marius Ionescu (U.S. Naval Acad.).

One problem 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.  So far we have quite complete results for the existence of smooth bump functions on metric measure spaces, 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, as well as study pseudo-differential operators of various kinds. The tools involved in the proofs are both analytic and probabilistic.

A related paper I wrote with Coulhon uses the failure of multiplication to preserve smoothness in the fractal setting to give examples where Sobolev spaces are not algebras. The methods build on an approach of Ben-Gal, Strichartz and Teplyaev, which in turn is related to older work of Kusuoka on the singularity of energy measures (and hence the natural candidates for gradients) to the usual self-similar reference measure on a fractal.

A very useful property of the Kigami Laplacian on a suitable self-similar set is that one can give a concrete description of the Green kernel. A group of us found an analogous constructive approach to the resolvent kernel for the Laplacian on p.c.f. self-similar sets. We were able to give a series description of this kernel in which the terms are rescaled and localized copies of functions that satisfy an eigenfunction equation on the interior of the fractal.  I subsequently proved some estimates which allow this approach to be applied to infinite blowups of these fractals and which generalize known bounds for the resolvent to the complex plane with the negative real axis removed.  This work is related to earlier results I obtained with (undergraduate student) Jessie DeGrado and with Bob Strichartz, which allow the computation of the harmonic gradients introduced by Teplyaev in the case of the Sierpinski Gasket. Quite recently, I have returned to some related questions in a paper with REU students Luke Brown, Giovanni Ferrer and Karuna Sangam, and PhD student Gamal Mograby.

A separate set of projects I have worked on is related to describing certain quantum particle models on fractal substrates. The foundation for this work is a functional analytic approach to constructing differential one-forms on fractals that admit a resistance form in the sense of Kigami. This was inspired by work of Sauvageot and Cipriani on derivations and Dirichlet forms; what we add to their approach in the following paper is a concrete description of these structures in the case of resistance forms.

Hinz and Teplyaev used some of these ideas in as sequence of papers treating vector fields and Schrodinger equations for Dirichlet forms on metric measure spaces in the case where the energy measures are absolutely continuous with respect to the reference measure. Hinz and I then generalized some features of this to the case of resistance forms under mild conditions which permit them to be applied to the case of magnetic Schrodinger operators on certain specific fractals. Magnetic operators on some fractals of this type were previously studied in the physics literature, in particular by Bellissard and collaborators, using renormalization-group methods. Two of my papers have graduate and undergraduate co-authors from my REU.

An interesting feature of analysis on the Sierpinski gasket, which is one of the best-studied examples in analysis on fractals, is that Laplacian eigenfunctions are precisely localized. This a common phenomenon on highly symmetric fractals (as was proven by Barlow and Kigami) and has many interesting consequences. For example, there is an elegant symmetry-breaking argument of Teplyaev that proves the Laplacian on an infinite blowup of the Sierpinski gasket has pure-point spectrum. We make substantial use of the localization of Laplacian eigenfunctions to prove versions of the classical Szegö theorem on the Sierpinski Gasket. This is a very different approach than that used in the classical setting, and although our methods should apply with suitable modifications to many highly symmetrical fractals with resistance form we do not know how to prove analogous results outside this class.

A well-known class of fractals are the Julia sets obtained from the dynamics of quadratic polynomials on the complex plane. Despite the simplicity of their construction they contain a wealth of interesting mathematics. It is then natural to wonder whether they support a non-trivial intrinsic differential structure. One approach to obtaining such a structure in the case of quasicircles has been proposed by Connes as an application of methods from his theory of non-commutative geometry. Methods from analysis on fractals are applicable to more complicated Julia sets, but the results are not directly comparable to those of Connes. For example, Teplyaev and I showed how to construct a Laplacian in the Kigami sense on the Basilica Julia set. Significant progress regarding the spectra of a wider classes of Julia sets has been made by Strichartz and collaborators by viewing them as quotients of the circle in a manner that respects the dynamics; this approach uses celebrated results of Douady and Hubbard. Finally, there is recent progress some related questions for the Schreier graphs of the Basilica group; this began with the PhD thesis of Brzoska (whom I co-advised, with Teplyaev) and was continued by one of my REU groups. We were able to prove that there are infinitely many gaps in the spectrum and, using Teplyaev’s symmetry breaking argument for the infinite Sierpinksi gasket, that the spectra of generic blowups is pure point.

In a collaboration with physicists Akkermans and Dunne, as well as my frequent collaborator Teplyaev and a number of graduate and REU students, I have continued various investigations in the mathematical physics of fractal structures.  One involves a generalization of an idea of Feynman regarding AC impedence on infinite networks to the fractal setting.  Another is connected to the work of Hinz, Lancia, Teplyaev and Vernole towards a magnetostatic theory for Euclidean domains with fractal boundary.

Also as part of a large collaboration, I have been working on developing a theory of functions of bounded variation on metric-measure-Dirichlet spaces that incorporates the case of fractals.  There is an extensive existing theory, built substantially by Ambrosio and his students and collaborators, especially Miranda, Pallara and Savare, which connects to the theory of analysis on metric spaces in the sense that the foundational assumption is that there is a sufficiently rich class of rectifiable curves that (either in the sense of curve modulus or using notions from optimal transportation).  However, some fractals admit no rectifiable curves in certain natural metrics, and for these a different approach is required.  Our starting point is that if one has a Dirichlet form on the fractal then the domain of the form is typically a Besov space (this is due to Jonsson and generalized by Pietruska-Paluba); reversing this one finds that a heat operator (which is equivalent to the Dirichlet form) provides a large class of Besov spaces, and that in classical settings some of these correspond to Sobolev spaces and to the space of bounded variation.  We work directly with these Besov spaces in order to develop an abstract version of the BV space arising from the Dirichlet form and establish some its properties, particularly those in the circle of ideas around sets of finite perimeter, coarea formula and embedding theorems.


Modulation Spaces, multipliers and PDE

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.