Professor Jenny Harrison

Much of my research has been in dimension theory, a field that did not really exist when I was a graduate student. In both my Denjoy Conjecture and Seifert Conjecture works, there appeared a duality between dimension of domains and differentiability of functions. This duality also showed up in the famous counterexamples of Denjoy and in Sard's theorem. It seemed there ought to be a "metatheorem" to help explain this. The quest for the basis and extent of this duality has led to the development of chainlet geometry, a blend of geometric measure theory and geometric integration theory. The duality of dimension and differentiability has evolved into a broader duality of geometry and analysis. After building a theoretical basis for over a decade, I am now working on applications.

The application which interests me the most at this time is a new approach to discrete exterior calculus.

The continuous and the discrete are thought by many to be two broad and antithetical domains of mathematical thought. Newton considered calculus primarily from the viewpoint of the continuous and Leibniz, the discrete. The advent of the computer has made discrete techniques all the more important, but rigor has often been lacking. The question of convergence of the discrete to the continuous has been an open problem. I have been using chainlet geometry to develop a rigorous theory of discrete exterior calculus. The idea is to replace domains, differential forms, operators and relations with simple discrete versions that converge to the smooth continuum. I am interested a number of applications, including combinatorial differential topology, electromagnetism, quantum theory, string theory, stochastic calculus, PDE's, and vision modeling.

Chainlets help elucidate the seach for area minimizing soap filims spanning a given Jordan curve.

A soap film can be viewed as a thin solid fluid bounded by two surfaces of opposite orientation. It is natural to model the film using one polyhedron for each side. Two challenges are to position the polyhedra for both sides in the same place without cancelling each other out and to model triple junctions without introducing extra boundary components. In Cartan's Magic Formula and Soap Film Structures I used chainlet geometry to create dipole cells and mass cells which accomplish these goals and model faithfully all observable soap films. The models can deal with films that have been problematic with other models, including nonorientable films, films with singularities, films that deformation retract onto their boundary and films that do not touch the entire knotted wire.

A number of people have asked me how my work on Plateau's problem relates to other results. In my paper, On Plateau's Problem for Soap Films with Bounded Energy, I assume that energy, the sum of the area of a surface and length of its singular curves, is bounded by some large constant over the space of candidate spanning surfaces which I call dipolyhedra. One can simply take a cone over the given curve to find a spanning dipolyhedron with finite area so the assumption of finite area is not a problem. The second part of the assumption, that the length of the singular, triple branching curves is uniformly bounded is also natural, coming from physics. Real life soap films do not exhibit an unbounded amount of branching for a bounded amount of surface area spanning a fixed Jordan curve. For that matter, we could assume the energy is bounded above by the number of atoms in the universe and be confident that our solution to Plateau's problem has area bounded above by the soap film spanning the given curve with smallest area. Mathematicians should and do make assumptions in modeling justified by physics. For example, Almgren made the following assumption about films: Roughly put, if you replace a bit of soap film by another surface with the same boundary you will increase area. Both his assumption and mine are supported by physics Thus, it is natural to assume that energy, the sum of the area and the length of the singular curves is uniformaly bounded in our space of dipolyhedra.

I have tried to further clarify how my work compares to others when it comes to modeling soap films. Consider the following categories for modeling soap films:

If mathematicians accept the assumption of an energy bound then 7 beats out all the competition for an existence theorem, as it is the only one that contains all naturally arising soap film solutions for a given curve. Even without the assumption, it contains all solutions produced by 1,2,3,4, and 6. It is strictly larger than these categories as can be seen by considering simple examples such as the Moebius strip and a mildly complicated knotted curve that forces branching on any soap film solution. Douglas was awarded one of the first Field's medals for his solution to Plateau's problem and Federer and Fleming won the Steele prize for their seminal work.