Fourier analysis and partial differential equations

摘要(Fefferman，1265524)这一建议包括以下几个主题：具有蜂窝晶格对称性的二维势的哈密顿量的色散面一般表现为锥形奇点，这种奇点在晶格的某些小变形下仍然存在。特别是，石墨烯受这样的哈密顿量支配。此外，该方案还研究了经典水波方程及其几种变种的解在有限时间内的奇性形成问题。更准确地说，水和空气界面的弦弧常数可以在有限的时间内趋于零。研究的第三个主题是Sobolev空间中的函数对数据的插补。给出一个定义在欧氏空间中的大型有限集上的函数，问题是计算整个欧氏空间上的一个函数，该函数与有限集上的给定函数一致，并且具有最小可能量级的Soblev范数。对于指定Sobolev空间的一系列参数，现在已知用于这种内插的有效算法；但对于剩余的Soblev空间，这个问题仍然悬而未决。第四个研究课题是从调和分析的角度对复变分析产生的偏微分方程解的基本理解。石墨烯被广泛认为在物理和技术上具有潜在的非常重要的意义，正是因为在这项授权下研究的不寻常的量子力学性质。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。计算与数据一致的光滑函数是统计学中的一个重要主题，也是处理大数据这一重要问题的一种方法。多复变方程的调和分析是一个基本的数学问题，与许多其他数学问题有着密切的关系。

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.