The proposal focuses on weakly differentiable mappings with first-order weak derivatives. This class includes, but is not limited to, quasiconformal and quasiregular mappings. A differential inclusion restricts the essential range of the derivative to a certain set of matrices. Main sources of differential inclusions are differential geometry and the theory of nonlinear partial differential equations. Typically, various notions of convexity and connectedness in the matrix space drive the analysis of existence and regularity of solutions. One of basic questions is which differential inclusions imply local invertibility. Invertibility is often established in two steps: first it is shown that the mapping is discrete and open (that is, a branched cover); the second step is to prove that the branch set is empty. Implementation of each step encounters problems that continue to challenge the available methods of analysis and topology. The proposal also brings the tools of geometric function theory into the field of ordinary differential equations. In the presence of canonical coordinates on the Euclidean space the driving vector field in an autonomous system of differential equations can be identified with a mapping, and the geometry of this mapping turns out to be related to the uniqueness of solution. Thirdly, the techniques introduced in quasiconformal analysis are also effective in the studies of smooth (e.g., harmonic) mappings, which in turn find applications in the theory of minimal surfaces
Minimal surfaces are mathematical models of thin films, for instance soap bubbles. Our understanding of their shapes develops through the solution of extremal problems. For example, how far apart can one move two boundary curves of a minimal surface before the surface breaks down? The principal investigator will apply the techniques of geometric function theory to such extremal problems. This approach is not limited to models of thin films and is also relevant in the studies of elastic deformation of solid materials. Another part of the proposal addresses uniqueness and stability of solutions of ordinary differential equations, which are commonplace in physics and engineering. They appear as equations of motion for one or several particles, with the number of particles affecting the dimensionality of the problem and the geometry of vector fields involved. Geometric function theory allows one to establish the uniqueness of a solution in situations where the standard results of the theory of ordinary differential equations do not apply. The PI works with post-docs and graduate students and organizes the Syracuse Analysis Study group.
A key concept in mathematics is an invertible transformation. It can be visualized as a change of shape in an elastic body, which can be perfectly undone. The property of being invertible is captured at the small scale by the derivatives of the transformation. However, derivatives do not rule out large-scale overlaps, or interpenetration of matter. The search for precise conditions under which interpenetration does not occur is a recurring theme in theoretical mechanics, in particular nonlinear elasticity. The project contributed to this search by identifying specific conditions under which a transformation, subjected to a physically natural principle of minimization of energy, is guaranteed to be invertible. Not every transformation can be adequately described with the classical calculus tools (i.e., derivatives). For example, sharp bends, like in folded of a paper, preclude the existence of derivatives. A known way around this difficulty is to introduce wider classes of transformation, called Sobolev spaces. They provide a robust framework which allows for formation of rough features such as sharp edges, yet excludes pathological objects such as nowhere differentiable functions of Weierstrass. Another important feature of Sobolev spaces is that every element of the space can be approximated, with arbitrary high precision, by smooth transformations. The existence of such approximation provides an indirect way of doing calculus with transformations that do not possess classical derivatives. However, the key property of invertibility may be lost in the process of approximation: smoothing out an invertible transformation to remove a sharp bend may accidentally create a region of interpenetration. One of achievements of this project is the proof of the existence of a smooth, invertible approximations to an invertible transformation in a Sobolev space. The proof applies to two-dimensional objects (mathematical models of thin plates and elastic films). The three-dimensional analog of this problem remains open. An elastic string or film, stretched too far, will eventually break. An easily reproduced instance of this phenomenon is the behavior of soap film formed between two circular wires. As the wires are moved away from each other, the film assumes the shape of a catenoid, until the moment when it disintegrates. The precise moment when this happens has been a subject of studies in the theory of minimal surfaces (which are a mathematical model of thin films). In 1962 Johannes C. C. Nitsche made a conjecture regarding the range of geometric parameters in which such a surface can exist. This conjecture, phrased in terms of the existence of invertible harmonic transformations, was open for half a century before being settled in a paper based on the results of this project ("The Nitsche conjecture", 2011). The project also contributed to the development of future generations of mathematicians. During the course of this project, the principal investigator supervised the research of two Ph.D. students and one undergraduate senior, all of whom will graduate in near future.
