A current research statement is available here. Here are some of the major areas I have focused on throughout my career:
I. Duality between the differentiability class of functions and the dimension of domains
My 1975 thesis Unsmoothable diffeomorphisms, settled a conjecture of Denjoy who had proposed that every diffeomorphism of class \( C^r \) is topologically conjugate to a diffeomorphism of class \( C^{r+1} \) for sufficiently large \( r \). Denjoy had proved this for diffeomorphisms of the circle, but the problem was unknown in higher dimensions. My counterexample was influenced by Denjoy's well known diffeomorphism of the circle which was \( C^1 \) but not topologically conjugate to a \( C^2 \) diffeomorphism, and was published in the Annals. A similar curious link between dimension and differentiability, not yet recognized as any kind of formal duality, arose again in \( C^2 \text{ Counterexamples to the Seifert Conjecture} \), published in 1988-89. It appeared again in the thesis of my then-student Alec Norton who proved a fractal version of Sard's theorem. It seemed there ought to be some kind of meta-theorem underlying these patterns and relations. Were dimension and differentiability class dual to each other in some formal way?
This question led to the development of the theory of differential chains which builds upon work of George Mackey, Hassler Whitney and Alexander Grothendieck. In recent years, the theory evolved from a rather ad hoc collection of methods on a chain complex of Banach spaces of polyhedral chains to a beautiful and canonical theory based in a single topological vector space determined by three simple axioms, as established with H. Pugh in Topological Aspects of Differential Chains. A duality of dimension and differentiability indeed exists. Roughly speaking, the smoother the integrand, the rougher can be the domain of integration. Developing a coherent and useful theory took several stages:
II. Stokes' theorem for nonsmooth domains
I vividly recall telling my students in a large vector calculus class at Berkeley in 1990 the standard method of integrating a smooth vector field along an embedded curve. For this method to work, the curve itself needed to be smooth. In mid-sentence, it occurred to me that although the method certainly required this assumption, there was no a priori reason to assume it. I started playing with fractal curves in the plane and proposed a norm on polyhedral chains using area and the boundary operator. Within the year Norton and I had proved Stokes' theorem for fractal curves in the plane. We later learned that our result was a Hölder version of Whitney's result for flat chains. It was discovered independently by B.A. Kats, and periodically by others. Codimension one followed easily. We did not prove either the divergence or curl theorems, just Stokes' theorem. (The title of our paper, The Gauss-Green Theorem for Fractal Boundaries was a misnomer.)
Stokes' theorem for nonsmooth curves in \( \mathbb{R}^3 \) was another problem altogether and required new ideas. Whitney's flat theory was not sufficient because volume needed to be considered for norms on curves in \( \mathbb{R} \), not just area. While on sabbatical at Yale, I found a family of norms on curves in \( \mathbb{R}^n \) which took into account \( r \)-dimensional volume for all \( r \le n \). This extends to norms on polyhedral \( k \)-chains in \( \mathbb{R}^n \) for \( 0 \le k \le n \). The boundary operator was used in a recursive fashion. Stokes' theorem held for limits of polyhedral \( k \)-chains which we called "chainlets" for a number of years. The name was later changed to "differential chains" when the theory matured, so I will use that term now. I announced this result in Stokes' Theorem for nonsmooth chains.
The divergence and curl operators needed another operator besides boundary, and this was a geometric dual to the Hodge star operator on differential forms. It was not obvious how to do this from the starting point of norms on polyhedral chains, but it was the subject of Flux across nonsmooth boundaries and fractal Gauss/Green/Stokes theorems and Geometric Hodge * operator with applications to theorems of Gauss and Green. The next step was to classify cochains. This was accomplished at the level of the chain complex of differential chains in Isomorphisms of differential forms and cochains. This was again not obvious from the viewpoint of polyhedral chains. Although I regularly posted updates on the arxiv, I began to publish less at this point. The theory was in a state of evolution and I did not want to freeze it in a formal publication until I knew what form it should take. Polyhedral chains were not so easy to work with, and I wanted something simpler. It was also important to find a major application to justify what was becoming a new approach to calculus.
III. Unification of discrete and smooth continuum
A discrete theory began to evolve when I discovered that point masses were differential chains. It was fun to apply the operators we knew about at the time to point masses to see what happened at the infinitesimal level. Boundary was intriguing, for one could take the boundary of a vector, say, and get a nonzero chain that behaved like a geometric version of a dipole. I then discovered that chains of point masses were dense in the Banach spaces of differential chains. Jerry Marsden invited me to talk about my early thoughts on the discrete theory at Caltech and there I met Robert Kotiuga and Alain Bossavit. They got me interested in applications to electrical engineering, and Lauri Kettunen invited me to give five lectures in Tampere. Antonio De Carlo, Paolo Podio-Guidugli and Gianfranco Capriz also began to show interest in the theory for its applications to continuum mechanics and invited me to give five lectures in Ravello. But it was still too difficult to use the theory from its initial starting point of norms on polyhedral chains. I knew that the theory was not ready for applications and asked everyone to wait.
I called Morris Hirsch in Wisconsin and we are now amused at how shocked he was when I proposed starting the theory fresh from the viewpoint of what we now call "Dirac chains." I next discarded boundary from the definition of the norms, hoping to simplify it. This required starting everything from scratch and checking what seemed like thousands of details to make sure everything still worked. It was hardly clear that it would all work out. Something could have gone wrong in a minute detail and the entire effort would have been for nought. Miraculously, everything worked as it should. The new definition greatly simplified everything. Polyhedral chains became secondary, appearing as a dense subspace. Dirac chains acted a lot like tangent spaces, even for nonsmooth chains. It was often simple to define operators on Dirac chains and prove they were bounded. For example, the divergence and curl theorems now followed easily from Stokes' theorem because the geometric Hodge operator on Dirac chains was trivial. The theory was becoming less piecemeal and ad hoc, but there was more to come.
I soon discovered the Koszul complex was deeply involved which showed an interplay between algebraic properties and geometric properties of the operators. See Operator Calculus of Differential Chains and Differential Forms, as well as Berkeley Colloquium Lecture (slides) 11/2003 for early thoughts about the discrete theory. The most up-to-date reference is Operator Calculus of Differential Chains and Differential Forms, to appear in the Journal of Geometric Analysis.
IV. Topology of the inductive limit
Using linking maps, the chain complex of differential chains began to take shape as an inductive limit of Banach spaces with the inductive limit topology. But this limiting space was far from easy to analyze, and I was often stuck. Harrison Pugh started to work on the theory at that time. Grothendieck's book "Topological Vector Spaces" held the keys for us. We learned from Grothendieck that simple results for projective limits of Banach spaces, as with smooth differential forms were often very difficult, if not impossible, to prove for inductive limits of Banach spaces. After reading numerous books and papers in topological vector spaces, we solved a number of fundamental questions and published our paper Topological Aspects of Differential Chains in the Journal of Geometric Analysis 2012. The paper is short, but dense, for it relies on some nontrivial results from Grothendieck and others. This paper establishes the important topological isomorphism theorem of differential forms of type \( \mathcal{B} \) and cochains. (These are differential forms each with a uniform bound on each derivative.) At last, we had found the long sought formal duality.
We began to denote the inductive limit endowed with the Mackey topology by \( '\mathcal{B} \) since its topological dual was \( \mathcal{B} \). There was now a single rich space to hold the operator algebra. It was now legitimate to compare properties of our theory with other theories of generalized functions, including de Rham currents and Schwartz distributions \( \mathcal{D}' \). The space \( '\mathcal{B} \) has good properties some of which are not exhibited by \( \mathcal{B}' \) or currents \( \mathcal{D}' \). For example, Dirac chains, which are supported in finitely many points, are dense in \( '\mathcal{B} \), but not in the strong dual topology of \( \mathcal{B}' \) or \( \mathcal{D}' \). They form a proper subspace of currents \( \mathcal{B}' \) and their topology is stronger than the weak topology.
V. Plateau's problem -- the problem of existence
The theory needed a major application for credence. The main applications of geometric measure theory, as described by Federer in the introduction to his book, were a generalization of the Gauss-Green divergence theorem and Plateau's problem for bounded, orientable surfaces. A general version of the Gauss-Green divergence theorem was now featured as an application of differential chains. I hoped that the more geometrical approach afforded by differential chains would lead to a simpler proof of Plateau's problem. I would have been happy with a simpler proof, for I had heard rumor that the proof using varifolds, attributed to Almgren, was very complicated. But I could not find any proof in print. I asked for a reference from experts who maintained that Almgren had proved this, but no one could provide one. I did not realize at the time that the problem for soap films was wide open, and only special cases had been solved. (Brakke provided models using double covers for a handful of simple soap films with triple junctions, and did not prove any type of existence theorem.)
My first papers on Plateau's problem, Cartan's Magic Formula and Soap Film Structures and On Plateau's Problem for Soap Films with Bounded Energy introduced new models of soap films using differential chains. "Dipole cells" were used to model triple junctions and non-orientable manifolds. In order to prove existence of a solution, the total length of triple junctions was assumed to have a uniform bound in the collection of all competing films.
I revisited Plateau's problem from time to time as the theory of differential chains became enriched as a way to test the strength of the theory. Finally in 2010, I saw how to avoid the extra hypothesis and prove an existence theorem, Soap Film Solutions to Plateau's Problem, which will appear in the Journal of Geometric Analysis. But I was not able to prove regularity. I was also not happy with the lack of a simple and concise statement of the problem. Many believed that my approach would never yield soap film regularity, but I felt that the simple characterization of differential chains given above, the nice models of triple junctions afforded by dipole cells, and the compactness theorem were strong hints that this would be a good space to work with.
VI. Plateau's problem -- the problem of existence and regularity
H. Pugh and I posted our paper, "Existence and Soap Film Regularity of Solutions to Plateau's Problem", on October 1, 2013. (After helpful feedback, we revised our title, abstract and introduction. It is now "Plateau's Problem: Existence and Soap Film Regularity of Size-Minimizing Current Solutions.") This paper provides an elegant statement of the problem using our new definition of a spanning set, similar to one I had used in Soap Film Solutions to Plateau's Problem. It establishes for the first time both existence and soap film regularity of a solution with minimal Hausdorff measure taken from the collection of all closed spanning sets of a prescribed Jordan curve. The solution supports a de Rham current and the prescribed curve is the support of the boundary of this current. We use a blend of methods from differential chains, Reifenberg's celebrated 1960 paper, and a theorem of Whitney equating positively oriented n-dimensional sharp chains with Borel measures.
VII. Further Applications
A number of unrelated applications are under development. These include a proof of the existence of a continuous coproduct on differential chains which commutes with boundary and dualizes to wedge product of forms; Hodge theory for differential chains and a mathematical derivation proving existence of the Coulomb field of a charged particle; and a generalization of the Reynold's transport theorem to evolving pairs of differential chains and differential forms where the chains represent evolving bodies which might be highly irregular.