Fourier analysis and partial differential equations

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Abstract (Fefferman, 1265524)This proposal includes several topics, including the following: The dispersion surface of Hamiltonians for two-dimensional potentials having the symmetry of the honeycomb lattice generically exhibit conical singularities, which persist under certain small deformations of the lattice. In particular, graphene is governed by such a Hamiltonian. In addition, the proposal studies the formation of singularities in finite time for solutions of the classical water wave equation and several variants. More precisely, the chord-arc constant for the interface between water and air can tend to zero in finite time. A third topic investigated is the interpolation of data by functions in Sobolev spaces. Given a function defined on a large finite set in Euclidean space, the problem is to compute a function on the whole Euclidean space that agrees with the given function on the finite set, and has Sobolev norm of the least possible order of magnitude. Efficient algorithms for such interpolation are now known for a range of the parameters specifying the Sobolev space; but for the remaining Sobolev spaces the problem remains open. A fourth topic of investigation is to develop a fundamental understanding of the solutions of partial differential equations arising from complex analysis, from the viewpoint of harmonic analysis.Graphene is widely regarded as potentially very important in physics and technology, precisely because of the unusual quantum-mechanical properties studied under this grant. Fluids and the interfaces between them occur in nature and in engineering applications, yet the fundamental mathematical understanding of their behavior remains a major challenge. The computation of smooth functions agreeing with data is a major theme in statistics, and is one approach to the important problem of coping with big data. The harmonic analysis of the equations of several complex variables is a fundamental problem of mathematics, with relations to many other mathematical problems.