离散限制性问题及其在数论与PDEs中的应用

The restriction problem originated in the 1970s, and its dual version is the boundedness problem of extension operator on hyper-surface or smooth manifold. Its special case corresponds to Strichartz estimates of solutions for the linear dispersion equations, which have constituted the basic research framework and research tools for nonlinear dispersion equations. At the same time, the variable-coefficient expansion operator corresponds to the famous Hormander oscillation integral operator. In order to establish Strichartz estimate of the solution of periodic dispersion equation, Bourgain developed the number theory approach and Fourier analytic method, and reduced Strichartz estimate of the periodic solutions of dispersion equations, the number of integer solutions of diophantine equations, the exponential sum estimates, the Lp - estimate of eigen-function of Laplacian operator on torus, the Vinogradov mean theorem for curve with torsion to discrete restriction problems. Through BCT's multilinear restriction estimates, decoupling theory and algebraic geometry methods, a series of open problems in different disciplines such as harmonic analysis, analytical number theory, partial differential equation and geometric measure theory are solved, and a bridge for cross research among different mathematical fields is built. The purpose of this project is to provide some challenging problems for young mathematicians in China, understand the research progress in this field through Tianyuan mathematics advanced seminar, strengthen exchanges and cooperation with international mathematicans, cultivate a number of young mathematicians in cross fields such as harmonic analysis, PDEs and analytic number theory in China, and make substantive contributions to the development and progress of mathematical research in China.

限制性问题起源于上个世纪七十年代,对偶版本是超曲面或光滑流形上扩张算子有界性问题,其特殊情形对应着色散方程的Strichartz估计，这类估计构成了研究非线性色散方程的基本研究工具,与此同时，扩张算子的变系数版本对应着著名的Hörmander振荡积分算子。为研究周期色散方程解的时空估计，Bourgain开创了数论方法与傅里叶解析方法，将周期色散方程解的时空估计、丢番图方程整数解的个数、指数求和估计、具非平凡挠率的曲线上Vinogradov平均值问题等归结为离散限制性问题。通过多线性限制性估计与分离性理论等，解决了调和分析、解析数论、PDEs等不同学科中一系列公开问题，搭建了不同数学学科交叉研究的桥梁。本项目旨在为国内年轻数学家提供一些挑战性问题，加强与国际同行之间的交流与合作，在国内培养一批调和分析、PDEs、解析数论等交叉领域青年数学家,为中国数学研究发展与进步做出实质性贡献。

猜想限制性问题起源于上个世纪七十年代, 对偶版本是超曲面或光滑流形上扩张算子有界性问题,其特殊情形对应着线性色散方程的Strichartz估计，这类估计构成了研究非线性色散方程的基本研究工具,与此同时，扩张算子的变系数版本对应着著名的Hörmander振荡积分算子。 为研究周期色散方程解的Strichartz估计，Bourgain开创了数论方法与Fourier解析方法，将周期色散方程解的Strichartz估计、丢番图方程整数解的个数估计、指数求和估计、平坦环上Laplace算子特征函数的Lp-估计、具非平凡挠率的曲线上Vinogradov平均值问题等归结为离散限制性问题。通过多线性限制性估计、分离性理论、代数几何方法，解决了调和分析、解析数论、偏微分方程、几何测度论等不同学科中一系列公开问题，搭建了不同数学学科交叉研究的桥梁。本项目旨在为国内年轻数学家提供一些挑战性问题，通过该天元数学高级研讨班了解该领域的研究进展，加强与国际同行之间的交流与合作，在国内培养一批调和分析、PDEs、解析数论等交叉领域青年数学家,为中国数学研究发展与进步做出实质性贡献。

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