Fourier积分算子及相应局部光滑性猜想

The main purpose of the Fourier integral operator is to study the singularity propagation of solutions to wave equation under the framework of cotangent bundles, the local smoothness associated with to the half-wave operators, and the construction of parametrix of strong hyperbolic equations. It originated from the study of potential theory and pseudo-differential operators which are one of powerful tools for elliptic equations. As a classical model of Fourier integral operators, local smoothness conjecture of the half-wave operator in Euclidean space is an advanced form of many well-known mathematical conjectures, which implies many conjecture such as Bochner-Riesz conjecture, restriction conjecture, Kakeya conjecture, etc. This fact fully reflects the inherent relationship between PDEs and different mathematical fields such as harmonic analysis and geometric measure theory. With the foundational work of Hörmander and the development of mathematicians such as Stein, Bourgain, the Fourier integral operator has become the core of variable coefficient or non-flat harmonic analysis. In particular, the development of the variable coefficient square function estimates and variable coefficient decoupling theory have provided a powerful tool for studying a series of mathematical conjectures，including local smoothness conjectures. Based on the developing status of this field in China, we will invite Delort, Guth, Sogge and young mathematicians such as Hickman and Demeter to give a series of lectures or min-course.In meantime, Tianyuan Advanced Mathematics Seminar will provides for a platform of academic exchanges between the experts and young mathematicans in China. We hope that young domestic mathematicians can master relevant theories and methods as soon as possible, so that they can make a great progress in this highly competitive field of mathematics, laying the foundation for solving open problems in related research fields.

Fourier积分算子的主旨是在余切丛框架下研究波动方程解的奇性传播、半波算子对应的局部光滑性及构造强双曲方程的拟基本解。作为Fourier积分算子典型范例，欧几里得空间的半波算子局部光滑性猜想是许多著名数学猜想的高级形式，它意味着调和分析与几何测度论中Bochner-Riesz猜想、限制性猜想、Kakeya猜想等,体现了PDEs与调和分析、几何测度论等不同数学领域之间的内在联系。经过Hörmander奠基性工作，Fourier积分算子已经成为变系数或非平坦调和分析的核心。特别是变系数分离性方法的发展，为研究一系列数学猜想提供有力工具。基于国内该研究领域的发展状况，该项目拟邀请Delort、Guth、Sogge及青年数学家Hickman、Demeter来华，通过天元数学高级研讨班的平台进行学术交流与短期课程、让国内年轻数学家尽快掌握相关理论与方法，在这一竞争激烈的数学领域取得突破奠定基础。

Fourier积分算子的主旨是在余切丛框架下研究波动方程解的奇性传播、半波算子对应的局部光滑性及构造强双曲方程的拟基本解。作为Fourier积分算子典型范例，欧几里得空间的半波算子局部光滑性猜想是许多著名数学猜想的高级形式，它意味着调和分析与几何测度论中Bochner-Riesz猜想、限制性猜想、Kakeya猜想等,体现了PDEs与调和分析、几何测度论等不同数学领域之间的内在联系。经过Hörmander奠基性工作，Fourier积分算子已经成为变系数或非平坦调和分析的核心。特别是变系数分离性方法的发展，为研究一系列数学猜想提供了有力工具。基于国内该研究领域的发展状况，该项目邀请了Sogge、Killip、Visan、Murphy及青年数学家金龙、席亚昆等专家学者，通过天元数学高级研讨班的平台进行学术交流与短期课程、让国内年轻数学家尽快掌握相关的理论与方法，在这一竞争激烈的数学领域取得突破奠定基础。