CAREER: Isoperimetric and Minkowski Problems in Convex Geometric Analysis

职业：凸几何分析中的等周和闵可夫斯基问题

• 批准号: 2337630
• 负责人: Yiming Zhao
• 金额: $ 43.47万
• 依托单位: Syracuse University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2024
• 资助国家: 美国
• 起止时间: 2024-06-01 至 2029-05-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2337630&HistoricalAwards=false
• 关键词: CAREER; Isoperimetric; Minkowski; Problems; Convex

Isoperimetric problems and Minkowski problems are two central ingredients in Convex Geometric Analysis. The former compares geometric measurements (such as volume and surface area) while the latter recovers the shape of geometric figures using local versions of these measurements. The two types of problems are inherently connected. This project will exploit this connection to seek answers to either isoperimetric problems or Minkowski problems in various settings when answers to one exist while answers to the other remain elusive. Although these problems originate from a geometric background, their applications extend beyond mathematics into engineering and design, including areas like antenna reflector design and urban planning. The principal investigator will organize a series of events and workshops at local science museums, community centers, and schools, involving high school teachers and students as well as undergraduate and graduate students. These events and workshops aim to expose the fun and exploratory side of the principal investigator’s research and mathematics in general to students early in their educational careers, raise society's awareness and interest in mathematics, and promote mathematics among historically underrepresented populations.The existence of solutions to the dual Minkowski problem (that characterizes dual curvature measures) in the original symmetric case has been largely settled (by the principal investigator and his collaborators) through techniques from geometry and analysis. This naturally leads to conjectures involving isoperimetric problems connected to the dual Minkowski problem. Such conjectured isoperimetric inequalities are also connected to an intriguing question behind many other conjectures in convexity: how does certain symmetry improve estimates? The principal investigator will also study Minkowski problems and isoperimetric inequalities coming from affine geometry. Special cases of these isoperimetric inequalities are connected to an affine version of the sharp fractional Sobolev inequalities of Almgren-Lieb. The techniques involved in studying these questions are from Convex Geometric Analysis and PDE. In the last few decades (particularly the last two), there has been a community-wide effort to extend results in the theory of convex bodies to their counterparts in the space of log-concave functions. In this project, the principal investigator will also continue his past work to extend dual curvature measures, their Minkowski problems, and associated isoperimetric inequality to the space of log-concave functions.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

等周问题和闵可夫斯基问题是凸几何分析中的两个核心要素，前者比较几何测量（例如体积和表面积），而后者使用这些测量的局部版本来恢复几何图形的形状。这两类问题本质上是相同的。当一个问题的答案存在而另一个问题的答案仍然难以捉摸时，该项目将利用这种联系来寻找等周问题或明可夫斯基问题的答案。和设计，包括天线反射器设计和城市规划等领域。首席研究员将在当地科学博物馆、社区中心和学校组织一系列活动和研讨会，让高中教师和学生以及本科生和研究生参与这些活动。和研讨会的目的是在学生教育生涯的早期向他们展示主要研究者的研究和数学的有趣和探索性的一面，提高社会对数学的认识和兴趣，并在历史上代表性不足的人群中推广数学。闵可夫斯基问题（即原始对称情况下的特征（对偶曲率测量）已通过几何和分析技术在很大程度上得到解决（由主要研究者和他的合作者），这自然会导致涉及与对偶闵可夫斯基问题相关的等周不等式的猜想。也与许多其他凸性猜想背后的一个有趣的问题有关：某些对称性如何改善估计？首席研究员还将研究明可夫斯基问题和等周不等式。这些等周不等式的特殊情况与阿尔姆格伦-利布的尖锐分数索博列夫不等式的仿射版本有关，研究这些问题所涉及的技术来自于过去几十年（特别是最近几十年）。二），整个社区都在努力将凸体理论的结果扩展到对数凹函数空间中的盟友。在这个项目中，首席研究员。还将继续他过去的工作，以扩展双曲率测量、他们的明可夫斯基问题以及相关的等周测量，反映了对数凹函数空间的不平等。该奖项的法定使命，并通过使用基金会的智力价值和更广泛的评估进行评估，被认为值得支持影响审查标准。