Geometric Variational Problems and Rearrangement Inequalities

该提案描述了非局部形状优化问题的研究议程，因为它们出现在潜在的理论，生物聚集和谐波分析。每个项目至少涉及一名学生或博士后。在势理论中，非定域能量泛函是由对带电粒子对的奇异相互作用势（如牛顿斥力）积分而得。在生物模型中，吸引-排斥类型的更复杂的对相互作用被用来描述群体和群的集体行为是如何从个体之间的相互作用中产生的。我感兴趣的形状优化问题，这些能量泛函下适当的几何约束。长期目标是不仅要表征最佳配置，还要表征能量泛函的所有临界点，并了解能量景观如何在长时间尺度上塑造所产生的经典或量子动力学。在调和分析中，卷积型泛函用于捕获关于和集和关联几何的信息。一般的原则是，这些泛函的大值与频繁的重合和小的和集;它们表明，出现在泛函中的密度共享一些共同的加法结构。长期目标是在各种情况下可靠地检测这种结构的存在，并描述其影响。我感兴趣的是等周型的尖锐不等式，这些不等式可以用来量化近最大化者到连续统中的区间、凸集或玻尔集以及到离散设置中的算术级数的距离。这些不等式随着维数的增加而变得更加强大，引起了新的浓度不等式，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)关注在无穷远处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）涉及非欧几里德空间上的重排不等式，特别是高斯空间、球面和正交群。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，我计划继续在计算机通信网络问题上的长期合作。

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.