Discretizing Manifolds with the Help of Riesz Kernels

如果我们观察一个足球，我们会看到二十个六边形和十二个五边形。从数学上讲，这些六边形和五边形可以被看作是它们中心的所谓“Voronoi细胞”。从数学的角度来看，这些细胞看起来非常漂亮，特别是，它们几乎都有相同的宽度。在给定的形状上产生“Voronoi细胞”看起来均匀的点是一个重要且计算困难的问题。如果我们现在只向一个陌生人展示上面提到的六边形和五边形的顶点，这个陌生人可能会想到一个球。这意味着这些顶点可以很好地表示一个球。这个项目的目标是证明某些具体的算法能够在给定的形状上产生许多点，这样如果我们只看这些点，我们就可以很好地恢复形状。这些问题引起了人们极大的兴趣，并在物理、化学、统计和数值积分等领域都有应用。更具体地说，假设许多粒子被放置在给定的流形（形状）上，并且已知这些粒子根据某些势相互排斥。从物理学上讲，我们知道这些粒子会在流形周围运动，试图使它们的势能最小化。一旦能量最小化，它们就会停止运动。我们想观察一下它们当时的位置。据推测（并在某些情况下证明），对于所谓的s-Riesz势，这些粒子将在许多意义上均匀地填充流形（例如，关于极限测度、分离和覆盖性质）。这个项目的一个目标是证明一个关于较弱（即超谐波）s-Riesz势和合适的流形类的分离距离的猜想。预计这将需要开发一种新的超谐波电位方法，因为目前的方法在这种情况下不起作用。首席研究员将进一步考虑有点双重的“切比雪夫问题”，它可以被看作如下：一个人想要在肿瘤中放置放射性种子，以便肿瘤的每个点都能接受一定数量的辐射。为了摧毁整个肿瘤，需要在哪些部位注射多少种子？第二个目标是证明指定位置的均匀性（根据极限测度），当肿瘤是“d可整流”集时。由于缺乏平滑性，主要研究者打算利用几何测量理论中的方法用光滑集近似d-可整流集。最后，该项目的第三个目标是比较上述确定性配置的分布与随机配置的分布。对于均匀和独立分布的点，这已经由主要研究者完成了。在这里，我们建议使用所谓的“决定点过程”，它是由随机矩阵产生的随机点配置。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

If we observe a soccer ball, we will see twenty hexagons and twelve pentagons. Mathematically speaking, these hexagons and pentagons can be viewed as the so-called "Voronoi cells" of their centers. From the mathematical perspective, these cells look very nice, in particular, they are all at almost the same width. It is an important and a computationally hard problem to produce points on a given shape whose "Voronoi cells" look that uniform. If we now show to a stranger only the vertices of the mentioned hexagons and pentagons, this stranger will likely think about a ball. This means that these vertices represent a ball well enough. The objective of this project is to prove that certain concrete algorithms are able to produce many points on a given shape so that if we look only at these points, we can recover the shape well enough. Such problems are of great interest and have applications to physics and chemistry, statistics and numerical integration.More specifically, assume many particles are placed on a given manifold (shape), and it is known that these particles repel each other according to some potential. From physics, it is known that these particles will move around the manifold trying to minimize their potential energy. As soon as they minimize the energy, they will stop moving. One would like to observe their locations at that time. It is conjectured (and in some cases proved) that for so-called s-Riesz potentials, these particles will fill the manifold uniformly in many senses (e.g., with respect to the limiting measure, separation and covering properties). One goal of this project is to prove a conjecture on separation distance for weaker (i.e., superharmonic) s-Riesz potentials and a suitable class of manifolds. It is anticipated that this will require developing a new approach to superharmonic potentials, as the current approaches do not work in this case. The principal investigator will further consider the somewhat dual "Chebyshev problem", which can be viewed as follows: one wants to place radioactive seeds in a tumor so that every point of the tumor receives some required amount of radiation. How many seeds, and at which locations, should they be injected in order to destroy the whole tumor? The second goal is to prove the uniformness properties (in terms of the limiting measure) for the specified locations, when the tumor is a "d-rectifiable" set. Since there is a lack of smoothness, the principal investigator intends to use methods from the geometric measure theory to approximate d-rectifiable sets by smooth sets. Finally, the third goal of the project is to compare the distributions of the deterministic configurations mentioned above to the distributions of random configurations. For uniformly and independently distributed points this has already been done by the principal investigator. Here it is proposed to work with so-called "determinantal point processes", which are random point configurations that arise from random matrices.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.