Geometric Variational Problems and Rearrangement Inequalities

这项建议描述了一个关于非局部形状优化问题的研究议程，因为它们出现在势能理论、生物聚集和调和分析中。每个项目至少涉及一名学生或博士后。在势论中，非局域能量泛函是在带电粒子对上积分奇异相互作用势的结果，例如牛顿排斥。生物模型中使用了更复杂的吸引-排斥类型的配对相互作用，以描述群体和群体的集体行为是如何通过将个体之间的相互作用相加而出现的。我对这些能量泛函在适当几何约束下的形状优化问题很感兴趣。长期目标是不仅描述能量泛函的最优构型，而且刻画所有临界点，并了解能量图景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，卷积型泛函被用来捕捉关于和集和和几何的信息。一般原理是，这些泛函的大值与频繁重合和小和集有关；它们表明泛函中出现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到这种结构的存在，并描述其影响。我对等周型的尖锐不等式感兴趣，它可以用来量化接近最大化者到连续统中的区间、凸集或玻尔集的距离，以及到离散环境中的算术级数的距离。随着规模的增加，这些不平等变得更加强大，导致了新的集中度不平等，有待详细研究。我在这些方面提出了8个问题。前两个问题(P1)和(P2)是极值电容问题。(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，特别是Gauss空间、球面和正交群上的重排不等式。接下来的三个问题(P6)-(P8)涉及到加性组合数学中的相关问题。最后，我计划继续就计算机通信网络中的问题进行长期合作。

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.