Fourier analysis and partial differential equations

翻译后摘要（February man，1265524）这个建议包括几个主题，包括以下内容：分散表面的哈密顿二维潜力具有对称性的蜂窝晶格一般表现出圆锥形奇点，坚持在一定的小变形的晶格。特别地，石墨烯由这样的哈密顿量控制。 此外，该提案研究了在有限时间内的经典水波方程和几个变种的解决方案的奇点的形成。更准确地说，水和空气之间的界面的弦弧常数可以在有限时间内趋于零。第三个主题调查是插值数据的功能在Sobolev空间。给定一个定义在欧氏空间中一个大的有限集上的函数，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，并且具有最小可能数量级的Sobolev范数。这种插值的有效算法现在已知的参数范围内指定的索伯列夫空间，但其余的索伯列夫空间的问题仍然是开放的。研究的第四个主题是从调和分析的角度对由复分析产生的偏微分方程的解进行基本的理解。石墨烯被广泛认为在物理和技术方面非常重要，正是因为在该资助下研究的不寻常的量子力学性质。流体和它们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。与数据相符的光滑函数的计算是统计学的一个重要课题，也是处理大数据的重要方法之一。多复变量方程的调和分析是数学的一个基本问题，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.