Fourier analysis and partial differential equations

摘要（Fefferman, 1265524）本文提出了以下几个问题：具有蜂窝晶格对称性的二维势的哈密顿算子的色散面一般表现出锥形奇点，在晶格的某些小变形下，这种奇点持续存在。特别是石墨烯是由这样一个哈密顿量控制的。此外，本文还研究了经典水波方程及其几种变体解在有限时间内奇点的形成。更准确地说，水与空气交界面的弦弧常数可以在有限时间内趋于零。研究的第三个主题是Sobolev空间中的函数插值数据。给定一个函数定义在欧几里德空间的一个大有限集合上，问题是在整个欧几里德空间上计算一个函数，该函数与有限集合上的给定函数一致，并且具有最小数量级的Sobolev范数。这种插值的有效算法现在已知为指定Sobolev空间的参数范围；但对于剩下的索博列夫空间来说，问题仍然存在。研究的第四个主题是从调和分析的观点出发，发展对由复分析引起的偏微分方程解的基本理解。石墨烯被广泛认为在物理学和技术领域具有非常重要的潜力，正是因为在这项资助下研究的不寻常的量子力学特性。流体及其之间的界面存在于自然界和工程应用中，但对其行为的基本数学理解仍然是一个重大挑战。与数据一致的光滑函数的计算是统计学中的一个重要课题，也是处理大数据的重要方法之一。复数变量方程的调和分析是数学中的一个基本问题，它与许多其他数学问题都有联系。

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.