Research in Harmonic Analysis with applications to Geometric Measure Theory and PDE's

摘要Dr.埃尔多安将进行调和分析，几何测度理论和偏微分方程的基础研究，重点是围绕勒贝格范数不等式为中心的欧几里得空间中的调和分析问题。正在进行的研究的一个主题是限制估计的傅立叶变换。最近，PI在某些情况下得到了令人满意的结果，这意味着在几何测度论中一个长期存在的公开问题--法尔科纳距离集猜想的方向上得到了新的部分结果.这些结果也意味着波和薛定谔方程的新的Schehartz型估计。正在进行的研究的另一个例子是广义Radon变换（GRT）的映射特性-一个巨大的类平均算子的低维子流形的欧几里得空间。通过应用技术布尔甘，沃尔夫和其他开发的Kakeya问题，他获得（部分联合基督）有趣的结果在某些情况下。近年来，PI研究了薛定谔演化的动力学性质（与Killip和Schlag联合）。他将继续他的调查各种问题，从数学物理，并连接与谐波分析。调和分析在自然科学和工程中有着广泛的应用。它是目前广泛应用于应用程序中的一系列强大而多样的工具的基础，并为未来的进一步应用提供了希望。拟议的研究涉及基础问题，这可能最终有助于支持未来的应用。广义相对论映射性质的研究在工程中有着广泛的应用。例如，应用于患者身体的密度函数的X射线变换（其是特定的GRT）本质上是通过磁共振成像获得的数据。 GRT的研究也是分析多维傅立叶级数和积分、更一般的振荡积分和广泛的一类偏微分方程的可和性的重要组成部分。拟议的研究将有助于对这些问题的普遍理解。计划中的研究还涉及组合学和数论中某些感兴趣的离散问题，这些问题在大多数情况下仍然是开放的。

ABSTRACTDr. Erdogan will conduct basic research on harmonic analysis, geometric measure theory and partial differential equations, focusing on problems in harmonic analysis in Euclidean spaces centered around Lebesgue norm inequalities. One subject of ongoing research is the restriction estimates for the Fourier transform. Recently, the PI has obtained satisfactory results in some cases, which imply new partial results in the direction of Falconer's distance set conjecture -- a long standing open problem in geometric measure theory. These results also imply new Strichartz type estimates for the wave and Schrodinger equations. Another example of ongoing research is the mapping properties of Generalized Radon transforms (GRT) -- a huge class of averaging operators over lower dimensional sub-manifolds of Euclidean spaces. By applying the techniques of Bourgain, Wolff and others developed for Kakeya problems, he obtained (part of it jointly with Christ) interesting results in some cases. In recent years, the PI studied the dynamical properties of Schrodinger evolution (joint with Killip and Schlag). He will continue his investigations on various problems from mathematical physics, and the connections with harmonic analysis. Harmonic analysis has always found wide applications in natural sciences and engineering. It underlies a powerful and diverse array of tools currently widely used in applications, and offers the promise of further applications in the future. The proposed research deals with foundational issues, which may ultimately help to underpin such future applications. The study of the mapping properties of GRT has various applications in engineering. For example, the X-ray transform (which is a particular GRT) applied to the density function of a patients body is essentially the data obtained by magnetic resonance imaging. The study of GRT is also an important ingredient in the analysis of sumability of multi-dimensional Fourier series and integrals, of more general oscillatory integrals, and of a wide class of partial differential equations. The proposed research would make a contribution to the general understanding of these problems. The planned research is also related to certain discrete problems of interest in combinatorics and number theory, which in most cases remain wide open.