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.

等级问题和Minkowski问题是凸几何分析中的两个核心因素。前者比较了几何测量值（例如体积和表面积），而后者则使用这些测量值的局部版本恢复了几何形状。两种类型的问题固有地连接。该项目将利用这种联系，以寻求各种环境中的等值问题或Minkowski问题的答案，尽管这些问题源于几何背景，但它们的应用超越了数学对工程和设计的范围，包括天线反射器设计和城市规划等领域。首席研究人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讲习班，涉及高中教师和学生以及本科生和毕业生。学生。这些事件和研讨会旨在揭示主要研究者的研究和数学的乐趣和探索者方面，总的来说，学生在他们的教育事业中提高了学生的教育事业，提高社会对数学的认识和兴趣，并在历史上促进数学不足的人群中的数学。首席研究员及其合作者）通过几何和分析的技术。这自然会导致涉及与双重Minkowski问题相关的等等问题问题的猜想。这种猜想的等距不平等也与许多其他猜想背后的有趣问题有关：某些对称性改进估计如何？首席研究者还将研究Minkowski问题和仿射几何形状的等含量不平等。这些等距不平等的特殊情况与Almgren-Lieb的敏锐分数Sobolev不平等的仿射版有关。研究这些问题所涉及的技术来自凸几何分析和PDE。在过去的几十年中（部分是最后两个），在社区范围内努力将凸体理论的结果扩展到对数凸函数的空间中的同行。在这个项目中，首席研究员还将继续他的过去工作，以扩展双重曲率措施，其Minkowski问题以及与Log-Concave功能空间相关的等速度不平等。这项奖项反映了NSF的法定任务，并通过使用该基金会的知识绩效和广泛的影响来评估NSF的法定任务，并被认为是珍贵的支持。