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.

该提案描述了非局部形状优化问题的研究议程，这些问题出现在势论、生物聚合和调和分析中。每个项目至少涉及一名学生或博士后。在势论中，非局域能量泛函是由带电粒子对上的奇异相互作用势（例如牛顿斥力）积分产生的。生物模型中使用更复杂的吸引-排斥类型的配对相互作用来描述群体和群体的集体行为如何通过个体之间的相互作用而产生。我对这些能量泛函在适当的几何约束下的形状优化问题感兴趣。长期目标不仅是表征最佳配置，而且是能量泛函的所有关键点，并了解能量景观如何在长时间尺度上塑造最终的经典或量子动力学。在调和分析中，卷积型泛函用于捕获有关和集和关联几何的信息。一般原则是，这些泛函的大值与频繁的重合和小的总和相关。它们表明泛函中出现的密度具有一些共同的加性结构。长期目标是在各种情况下可靠地检测这种结构的存在，并描述其含义。我对等周类型的尖锐不等式感兴趣，它可用于量化连续体中接近最大化者与区间、凸集或玻尔集的距离，以及离散设置中的算术级数。随着维度的增加，这些不平等变得更加强大，从而产生新的集中不平等，有待详细研究。我在这些方面提出了8个问题。前两个（P1）和（P2）是极值电容器问题。 （P3）涉及无穷大衰减的非局部相互作用能量的最小化，这是生物聚集的玩具模型。 (P4) 和 (P5) 涉及非欧几里得空间上的重排不等式，特别是高斯空间、球体和正交群。接下来的三个问题（P6）-（P8）涉及加法组合学中的相关问题。最后，我计划继续就计算机通信网络问题进行长期合作。