调和分析中四大猜想及其应用

The restriction problem originated by studying the Fourier transform of L^p functions, and the subject of restriction estimates is the relation between the decay on Fourier transform of the surface measure at infinity and the geometrical properties on its support. It is closely related to Kakeya conjecture, Bocher-Riesz conjecture, local smoothness conjecture，which gradually formed the famous four conjecture among Harmonic Analysis. The restriction estimates in the PDE is known as the STRCHARTZ estimates, and has become the basic tool to study nonlinear dispersive equation. The methods to study the four well-known conjectures have become effective arguments and tools to deal with other important issues in the fields of Harmonic Analysis, Analytic Number Theory, PDEs, Geometric Measure Theory, Incidence Geometry, Mathematical Physics etc. Bourgain's genius discoveries have brought together many seemingly unrelated research fields, including Harmonic Analysis, Addition number theory, Incidence Geometry, Arithmetic Combinatorics, PDEs and other mathematical fields. Through the Tianyuan Advanced Seminar, we hope to introduce some challenging questions to young mathematicians, and cultivate a generation of first mathematicians working in this intersecting research field among Modern Analysis, Number Theory and Geometrical Measure Theory, making important contribution to the progress of Chinese mathematics.

限制性问题源于L^p函数Fourier变换的研究，主题是关于曲面测度的Fourier变换在无穷远处的衰减和支集的几何性质之间的联系，它与Kakeya猜想、Bocher-Riesz猜想、局部光滑性猜想等紧密相关，这就逐步形成调和分析领域著名的四大猜想。限制性估计在PDE中的表现形式就是著名的STRCHARTZ估计，已经成为研究非线性色散方程解的基本工具.研究四大猜想的各种方法已经成为解决调和分析、解析数论、偏微分方程、几何测度论、关联几何、数学物理等数学领域公开问题的有效方法与工具。经过Bourgain的天才发现，将许多看似不相关的研究领域有机联系在一起，研究方法涉及的领域包括调和分析、堆垒数论、关联几何学、算术组合学等数学领域。希望通过该天元数学高级研讨班，给年轻数学家提供一些挑战性问题，培养一批从事现代分析、PDE、数论、几何测度论等交叉领域一流数学家,为中国数学的进步做出重要贡献。

限制性问题源于L^p函数Fourier变换的研究，主题是关于曲面测度的Fourier变换在无穷远处的衰减和支集的几何性质之间的联系，它与Kakeya猜想、Bocher-Riesz猜想、局部光滑性猜想等紧密相关，这就逐步形成调和分析领域著名的四大猜想。限制性估计在PDE中的表现形式就是著名的STRCHARTZ估计，已经成为研究非线性色散方程解的基本工具.研究四大猜想的各种方法已经成为解决调和分析、解析数论、偏微分方程、几何测度论、关联几何、数学物理等数学领域公开问题的有效方法与工具。经过Bourgain的天才发现，将许多看似不相关的研究领域有机联系在一起，研究方法涉及的领域包括调和分析、堆垒数论、关联几何学、算术组合学等数学领域。通过该天元数学高级研讨班，给年轻数学家提供一些挑战性问题，培养一批从事现代分析、PDE、数论、几何测度论等交叉领域一流数学家,为中国数学的进步做出重要推动作用。