【学术会议】2024年凸几何与几何分析研讨会

告题目及摘要

Measure Equations, Optimization Problems and PDEs in Convex Geometry

张高勇

美国纽约大学

Finding unknown convex geometric objects from their prescribed measurements went back to Minkowski in the 1890s and Aleksandrov and Fenchel in the 1930s. The geometric problems are called Minkowski problems, which require to solve equations of convex geometric measures. The smooth cases of such measure equations are fully nonlinear partial differential equations. Convex geometric optimization problems and variational methods are designed to study these measure equations. There have been revolutionary studies in recent years and a theory of convex geometric measures is emerging. This talk explains the measure equations, optimization problems and PDEs.

张高勇教授简介：国际著名几何学家，美国纽约大学库朗(Courant)数学研究所终身教授，美国数学会首届会士，长江学者讲座教授(08-10)，研究方向为凸几何、几何分析、积分几何等，在几何不等式和几何测度论等方面做出了一系列杰出的工作。张高勇教授的研究成果丰硕，他解决了四维空间中著名的 Busemann-Petty问题，被称为“几何学此领域中最有意义的结果之一”。证明了已成经典的“Zhang projection inequality"和affine Sobolev-Zhang inequality”。张高勇教授与其合作者在对数 Minkowski问题、对偶Minkowski问题、积分几何中的Minkowski问题等方面做出了杰出贡献，在凸几何分析、偏微分方程、几何测度论等领域产生了深远影响。他们提出的log-Brunn-Minkowski猜想现在已成为凸儿何领域最困难的且广受关注的问题之一。张商勇教授与国际著名几何学家Erwin Lutwak, Deane Yang近三十年在仿射等周问题方面取得的研究成果，极大地推动了仿射几何与凸几何的发展。张高勇教授的部分工作发表在Acta Math、Ann. Math、JAMS、CPAM、Duke Math.等顶级数学期刊上。

Gelfand and Helly inequality and Von Neumann invariant subspace problem

蒋春澜

河北师范大学

In the 1920s, Von Neumann proposed the famous invariant subspace problem: Does every bounded linear operator acting on the infinite dimensional Hilbert space have a non trivial invariant subspace? Gelfand and Helly gave the following result in the 1950s: For operators with a single point spectrum, If the n power of such operators is polynomial growth, the answer to the Von Neumann invariant subspace problem is affirmative. In this report, we extend the depth of Gelfand and Helly inequality, thereby proving that a larger class of operators have non trivial invariant subspaces. The spectrum of this type of operator that may be infinitely set.

蒋春澜教授简介：河北师范大学教授，博士生导师，河北省燕赵学者。曾任河北师范大学校长、中国科学院访问教授、美国波多黎各大学客座教授，长期从事算子代数可约性与强不可约性研究。在无穷维希尔伯特空间算子理论中作出了享有国际声誉的贡献。在PNAS、Adv. Math、J.Funct Anal、Trans. Amer. Math. Soc、IMRN等杂志发表学术论文70多篇。主持完成《国家重点基础研究发展规划》（973计划）项目、教育部重大课题、国家自然科学基金重点项目和国家自然科学基金面上项目十余项。曾获国家教委科技进步奖二等奖、上海市科技进步奖二等奖等。2000年获国务院政府特殊津贴，2003年获河北省自然科学奖一等奖，2013年获教育部自然科学奖二等奖。

 John ellipsoid, its generalizations and applications in convex geometry

熊革

同济大学

It is known that there exists a unique ellipsoid of maximal volume inside a convex body (a compact convex set with non-empty interiors) in  . This ellipsoid is called John ellipsoid (named after mathematician Fritz John), and has many applications in convex geometry, functional analysis, and optimizations. In this talk, I will present our work on John ellipsoid, including its generalizations and applications in convex geometry. Our research on this topic has been ongoing for over 10 years.

熊革教授简介：同济大学长聘教授，博士生导师。熊革教授解决了凸体几何中的几个公开问题，包括Lutwak-Yang-Zhang关于锥体积泛函极值问题的2,3维情形；由截面确定凸体的Baker-Larman问题的2维情形；最早提出并解决了静电容量的Lp Minkowski问题，完全解决了纽约大学 张高勇教授关于凸体的John椭球与对偶惯性椭球一致性的问题。熊革教授在国际数学的重要期刊JDG, Math. Ann., Adv. Math., IUMJ，IMRN, CVPDE, JFA, CAG, Israel Journal of Mathematics, Discrete and Computational Geometry等上发表论文30余篇。

The reverse log-Brunn-Minkowski inequality

席东盟

上海大学

We established an equivalent form of the log-Brunn-Minkowski conjecture. As an application, we proved the log-Minkowski inequality in the case where one convex body is a zonoid (the inequality part was first proved by van Handel), and we also provided the equality characterization, which turned out to be new.

席东盟教授简介：上海大学理学院数学系教授，上海大学伟长学者。主要研究凸几何与积分几何中的分析问题，尤其是在等周问题与几何测度的Minkowski问题取得了若干成果，包括：在积分几何中引入了并解决了一族平移不变几何测度的Minkowski问题，对应了一族收敛到sigma_k的Monge-Ampere型算子；解决了2维Dar猜想；建立了非对称凸体的log-Brunn-Minkowski不等式。相关成果发表在Comm. Pure Appl. Math.、J. Differential Geom.、Adv. Math.、Math. Ann.、Trans. Amer. Math. Soc.、J. Funct. Anal.等期刊上。2023年获得国家自然科学基金优秀青年基金资助。