Geometric Variational Problems and Rearrangement Inequalities

几何变分问题和重排不等式

• 批准号: RGPIN-2020-06826
• 负责人: Burchard, Almut
• 金额: $ 1.97万
• 依托单位: University of Toronto
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2022
• 资助国家: 加拿大
• 起止时间: 2022-01-01 至 2023-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=758674
• 关键词: Geometric; Variational; Problems; Rearrangement; Inequalities

This proposal describes a research agenda on non-local shape optimization problems, as they arise in potential theory, biological aggregation, and harmonic analysis. Each project involves at least one student or postdoc. In potential theory, non-local energy functionals result from integrating a singular interaction potential, such as the Newtonian repulsion, over pairs of charged particles. More complex pair interactions of attractive-repulsive type are used in biological models to describe how the collective behaviour of herds and swarms emerges from adding up interactions between individuals. I am interested in shape optimization problems for these energy functionals under suitable geometric constraints. The long-term objective is to characterize not just the optimal configurations but all critical points of the energy functionals, and understand how the energy landscape shapes the resulting classical or and quantum dynamics over long time scales. In harmonic analysis, convolution-type functionals are used to capture information about sum sets and incidence geometry. The general principle is that large values of these functionals are associated with frequent coincidences and small sumsets; they indicate that the densities appearing in the functional share some common additive structure. The long-term objective is to reliably detect the presence of such structure in a variety of situations, and characterize its implications. I am interested in sharp inequalities of isoperimetric type that can be used to quantify the distance of near-maximizers to intervals, convex sets, or Bohr sets in the continuum, and to arithmetic progressions in the discrete setting. These inequalities become more powerful with increasing dimension, giving rise to new concentration inequalities that are awaiting detailed study. I propose 8 problems in these areas. The first two, (P1) and (P2), are extremal Capacitor problems. (P3) concerns minimization of non-local interaction energies that decay at infinity, which is a toy model for biological aggregation. (P4) and (P5) concern rearrangement inequalities on non-Euclidean spaces, specifically Gauss space, spheres, and orthogonal groups. The next three problems (P6)-(P8) touch on related questions in additive combinatorics. Finally, I plan to continue a long-standing collaboration on problems in Computer Communication Networks.

该建议描述了关于非本地形状优化问题的研究议程，因为它们在潜在理论，生物聚集和谐波分析中产生。每个项目涉及至少一个学生或博士后。在潜在理论中，非本地能量函数是由于整合了成对的带电颗粒的奇异相互作用电位（例如牛顿排斥）而引起的。在生物学模型中使用了更复杂的对抑制类型的对相互作用，以描述牛群和群体的集体行为如何从增加个人之间的相互作用中出现。在适当的几何约束下，我对这些能量功能的形状优化问题感兴趣。长期目标不仅要表征最佳配置，还要表征能量功能的所有关键点，并了解能量景观如何在长期尺度上塑造所得的经典或量子动力学。在谐波分析中，卷积类型功能用于捕获有关和集合几何形状的信息。一般原则是，这些功能的巨大值与频繁的一致性和小总和有关。他们表明，功能中出现的密度共享一些常见的添加剂结构。长期目标是可靠地检测到各种情况下这种结构的存在，并表征其含义。我对等等类型的急剧不平等感兴趣，这些类型可用于量化连续体中近麦克西变量与间隔，凸集或BOHR集的距离，以及在离散设置中的算术进行。随着维度的增加，这些不平等变得更加强大，从而引起了等待详细研究的新集中不平等。我在这些领域提出了8个问题。前两个（P1）和（P2）是极端电容器问题。 （P3）涉及在Infinity衰减的非本地相互作用能量的最小化，这是生物聚集的玩具模型。 （P4）和（p5）涉及非欧盟人空间的重排不平等，特别是高斯空间，球体和正交组。接下来的三个问题（P6） - （P8）涉及添加剂组合学中的相关问题。最后，我计划继续在计算机通信网络中的问题上进行长期合作。