代数多项式方法在调和分析、PDEs与几何测度论中的应用

The starting point of polynomial method stems from “Ham Sandwich” thorem in Algebraic topology. Dvir utilized the polynomial method to solve the finite field Kakeya conjecture, and it promoted the development of other subjects in mathematics. Inspired by this work, Guth and Katz solved the Erdös distance conjecture. Bôcher Prize winner Guth combines the polynomial method with multi-scale techniques developed by Bourgain-Guth to promote the research of several conjectures in geometry, analysis and PDEs.And he solved the famous Carleson conjecture, and made substantial progress in Kakeya conjecture, restricted conjecture, distance set conjecture, etc. The idea of the polynomial method is to divide the spatial region Rn into some connected regions and small neighborhoods of n-1 dimensional manifolds through the hypersurface determined by the zero points of the polynomial, so as to realize dimensionality induction and scale induction in physical space.Based on the above reasons, the Tianyuan Advanced Seminar intends to invite a few of mathematicanssuch as Guth, Hickman, Ruixiang Zhang, Hong Wang to give a series of lectures or min-course on polynomial method, and to enable domestic young mathematics workers and doctoral students to master relevant theories and methods as soon as possible. Meanwhile, based on the Tianyuan Advanced Workshop, a platform for cooperation and exchanges for young scholars in this field will be built to promote the development of domestic research fields such as harmonic analysis, PDEs, number theory, and geometric measure theory.

多项式方法出发点是代数拓扑中“三明治定理” 。 Dvir用多项式方法解决了有限域上Kakeya猜想,Guth-Katz解决了沉浸多年的Erdös距离集问题, 在数学的许多领域引发了一场革命.Bôcher奖得主Guth将多项式方法与Bourgain-Guth的多尺度技术相结合，推动了几何、分析与PDEs中的若干猜想的研究，在 Kakeya猜想、限制性猜想、距离集猜想等取得实质进展。多项式方法的理念是通过多项式的零点决定的超曲面将空间区域R^n划分为一些连通区域与n-1维流形的小邻域，从而在相空间中实现维数归纳与尺度归纳。基于上述理由，拟邀请Guth、Hickman及在普林斯顿张瑞祥、王虹等数学家讲解代数多项式方法，使国内年轻数学工作者与博士研究生尽快掌握相关的理论。与此同时，以天元高级讲习班为基础，为从事这一领域青年学者搭建一个合作交流的平台，促进国内调和分析、PDEs、几何测度论的发展。

多项式方法出发点是代数拓扑中“三明治定理”。 Dvir用多项式方法解决了有限域上Kakeya猜想, Guth-Katz解决了沉浸多年的Erdös距离集问题, 在数学的许多领域引发了一场革命。Bôcher奖得主Guth将多项式方法与Bourgain-Guth的多尺度技术相结合，推动了几何、分析与PDEs中的若干猜想的研究，在 Kakeya猜想、限制性猜想、距离集猜想等取得实质进展。多项式方法的理念是通过多项式的零点决定的超曲面将空间区域R^n划分为一些连通区域与n-1维流形的小邻域，从而在相空间中实现维数归纳与尺度归纳。基于国内该研究领域的发展状况，该项目邀请了Sogge、Katz及在加州大学洛杉矶分校王虹等数学家讲解代数多项式方法，使国内年轻数学工作者与博士研究生尽快掌握相关的理论。与此同时，以天元高级讲习班为基础，为从事这一领域青年学者建一个合作交流的平台，促进国内调和分析、PDEs、几何测度论的发展。