My main research interests are in nonlinear PDEs. The methods I use are derived from measure theory, PDEs, geometric analysis, and calculus of variations. I have published several advances on the Plateau problem, as well as a general approach to solving elliptic anisotropic minimization problems. I am also interested in fluids, optimal transport problems, and classical analysis.
The Plateau problem is to find a surface with smallest area which spans a given boundary. Many iterations of this problem have been solved over the course of the last century, but a number of important questions have remained open after a flurry of activity in the 1960s. Over the last five years, there has been a rebirth of interest in the problem and advances have been made by several independent groups. Perhaps most exciting has been the discovery of concise axioms that a given collection of surfaces may satisfy in order for there to be an area minimizer within that collection.
My research has focused on several problems: 1) The development of these axioms and the extension of the results of Reifenberg and Almgren to find minimizers among broad and natural classes of competitors; 2) The development of a theory for general ambient spaces, including manifolds with singularities and a large class of metric spaces; 3) The extension of the results of Reifenberg and Almgren to permit boundaries that may vary within a prescribed constraining set (a "sliding boundary" problem.)
I have written a series of three papers together with J. Harrison on the Plateau problem and related area minimization problems. The main contribution of the first paper, Existence and soap film regularity of solutions to Plateau's problem is a new spanning condition, by which a generalized surface (which is allowed to have triple junctions and other singularities) spans a given boundary wire if it intersects every Jordan curve which links the boundary once. We adapt the homological spanning lemmas in Adams's appendix to Reifenberg's 1960 paper, "Solution of the Plateau Problem for m-Dimensional Surfaces of Varying Topological Type," to this geometric spanning condition and show that the collection of generalized surfaces spanning a given boundary is closed under the operations necessary to prove existence and regularity of an area minimizer.
Our second paper, Solutions to the Reifenberg Plateau problem with cohomological spanning conditions, contains a modern solution to the soap-film Plateau problem, and is the first to satisfactorily take into account boundaries with multiple components. In this paper we generalize to higher codimension and minimize area weighted by a Hölder density function. We also greatly simplify the existence proof of Reifenberg using more modern techniques, such as Preiss's theorem and monotonicity formulae.
In our third and most recent paper, General Methods of Elliptic Minimization, we extend the results of the previous paper and minimize an anisotropic density function (that is, density which depends on tangent direction) within certain ambient metric spaces (including manifolds.) Our approach generalizes the results in Almgren's 1968 Annals paper and fixes some problems with his proof. We also do not use varifolds or currents in our proof in any way, only classical measure theory and set-theoretic constructions in ℝn, thus simplifying his approach.
In Topological aspects of differential chains, J. Harrison and I investigated the limit of a chain complex resolution of Whitney's space of sharp chains, a space of currents we called "differential chains" after their presentation as clouds of dipoles, quadrupoles and higher order k-poles. These discrete objects can be glued together to form larger structures, and are the basis of a discrete calculus with unique and powerful algebraic properties. See my undergraduate thesis and J. Harrison's paper Operator Calculus of Differential Chains and Differential Forms for an exposition.
In Differential Forms are Dual to a Differential Coalgebra, I develop the algebraic properties of these k-poles and show they have a differential coalgebra structure whose completion is quasi-isomorphic to singular chains.
It is known that every topological manifold admits a quasiconformal (i.e. bounded infinitesimal distortion) structure, except in dimension four (Sullivan, Hyperbolic geometry and homeomorphisms, 1978.) In dimension four, however, the quasiconformal category seems to behave more like the smooth category: there are topological 4-manifolds that do not admit a quasiconformal structure, and there are smooth 4-manifolds that are homeomorphic but not quasiconformally equivalent (Donaldson & Sullivan, Quasiconformal 4-manifolds, 1989.)
By work of Friedman and Morgan, there can be infinitely many non-diffeomorphic smooth 4-manifolds in a given homeomorphism class. For example, the homeomorphism class of the K3 surfaces admits infinitely many non-diffeomorphic smooth structures. However, by combining the work of Friedman-Morgan and Donaldson-Sullivan, I have shown that if one restricts to a quasiconformal class, there are only finitely many non-diffeomorphic Kähler smooth structures on the topological manifold underlying the K3 surface. This result holds for other complex surfaces, namely those that are simply connected with b2+>1. I am also working on the b2+=1 case, which is somewhat different due to Donaldson's polynomial invariant being defined only as a function on the hyperbolic subspace of H2 cut out by the intersection form ω=1. Thus (conjecturally in the b2+=1 case,) diffeomorphism class -> quasiconformal class is finite-to-one for simply connected complex surfaces.
I prove a generalization of Reifenberg's isoperimetric inequality. The main result of this paper is used to establish existence of a minimizer for an anisotropically-weighted area functional among a collection of surfaces that satisfies a set of axioms, namely being closed under certain deformations and Hausdorff limits. This problem is known as the axiomatic Plateau problem.
There is a fundamental asymmetry between algebras and their dual objects, coalgebras, namely that the dual of a coalgebra is an algebra, but the converse is only true in finite dimensions. We prove that there exists a differential graded coalgebra whose continuous dual is the differential graded algebra of differential forms. This coalgebra is constructed as an explicit subspace of de Rham currents. A general theory of topological duality between coalgebras and algebras is also developed.
I prove that under mild assumptions, a set is rectifiable if and only if its Hausdorff measure is lower semi-continuous under bounded Lipschitz perturbations.
We provide new general methods in the calculus of variations for the anisotropic Plateau problem in arbitrary dimension and codimension. A new direct proof of Almgren's 1968 existence result is presented; namely, we produce from a class of competing "surfaces," which span a given bounding set in some ambient space, one with minimal anisotropically weighted area. In particular, rectifiability of a candidate minimizer is proved without the assumption of quasiminimality. Our ambient spaces are a class of Lipschitz neighborhood retracts which includes manifolds with boundary and manifolds with certain singularities. Our competing surfaces are rectifiable sets which satisfy any combination of general homological, cohomological or homotopical spanning conditions. An axiomatic spanning criterion is also provided. Our boundaries are permitted to be arbitrary closed subsets of the ambient space, providing a good setting for surfaces with sliding boundaries.
This is a chapter of a book on outstanding open problems in mathematics that we were invited to contribute by John Forbes Nash, Jr. and Michael Rassias. We give a history of the Plateau problem over the past hundred years as well as a detailed synopsis of the current state of affairs, and outline several new avenues for future research on the problem and in related fields.
See the synopsis above for a general discussion. For experts in the field: We prove existence and regularity of minimizers for Hölder densities over general surfaces of arbitrary dimension and codimension in ℝn, satisfying a cohomological boundary condition, providing a natural dual to Reifenberg's Plateau problem. We generalize and extend methods of Reifenberg, Besicovitch, and Adams, in particular we generalize a particular type of minimizing sequence used by Reifenberg (whose limits have nice properties, including lower bounds on lower density and finite Hausdorff measure,) prove such minimizing sequences exist, and develop cohomological spanning conditions. Our cohomology lemmas are dual versions of the homology lemmas in the celebrated appendix by Adams found in Reifenberg's 1960 paper.
Plateau's soap film problem is to find a surface of least area spanning a given boundary. We begin with a compact orientable (n−2)-dimensional submanifold M of ℝn. If M is connected, we say a compact set X "spans" M if X intersects every Jordan curve whose linking number with M is 1. Picture a soap film that spans a loop of wire. Using (n−1)-dimensional Hausdorff spherical measure as the measure of the size of a compact set X in ℝn, we prove there exists a smallest compact set X0 that spans M. We also show that X0 is almost everywhere a real analytic (n−1)-dimensional minimal submanifold and if n=3, then X0 has the structure of a soap film as predicted by Plateau. We provide more details about the minimizer X0. Primarily, X0 is the support of a current S0 and M is the support of the algebraic boundary of S0. We also discuss the more general case where M has codimension >2.
In this paper we investigate the topological properties of the space of differential chains 'B(U) defined on an open subset U of a Riemannian manifold M. We show that 'B(U) is not generally reflexive, identifying a fundamental difference between currents and differential chains. We also give several new brief (though non-constructive) definitions of the space 'B(U), and prove that it is a separable ultrabornological (DF)-space. Differential chains are closed under dual versions of fundamental operators of the Cartan calculus on differential forms. The space has good properties some of which are not exhibited by currents B'(U) or D'(U). For example, chains supported in finitely many points are dense in 'B(U) for all open U in M, but not generally in the strong dual topology of B'(U).