Discretizing Manifolds with the Help of Riesz Kernels

如果我们观察一个足球，我们会看到20个六边形和12个五边形。从数学上讲，这些六边形和五边形可以被视为它们中心的所谓“Voronoi细胞”。从数学的角度来看，这些细胞看起来非常好，特别是，它们几乎都是相同的宽度。在一个给定的形状上产生点，使其“Voronoi单元”看起来均匀，这是一个重要的和计算困难的问题。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。这意味着这些顶点足够好地表示球。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。这类问题在物理学、化学、统计学和数值积分等领域都有广泛的应用。更具体地说，假设许多粒子被放置在一个给定的流形（形状）上，并且已知这些粒子根据某种势相互排斥。从物理学上讲，已知这些粒子将围绕流形移动，试图使它们的势能最小化。一旦它们将能量最小化，它们就会停止运动。 人们想观察他们当时的位置。它被证明（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，关于限制措施、分离和覆盖特性）。该项目的一个目标是证明一个关于较弱（即，超调和）s-Riesz势和一类合适的流形。预计这将需要开发一种新的方法来超谐波势，因为目前的方法在这种情况下不工作。主要研究者将进一步考虑有点双重的“切比雪夫问题”，可以如下所示：一个人想在肿瘤中放置放射性种子，使得肿瘤的每个点都接受一定量的辐射。为了摧毁整个肿瘤，应该在哪些位置注射多少种子？第二个目标是证明当肿瘤是一个“d-可求长”集时，指定位置的均匀性属性（在极限测度方面）。由于缺乏光滑性，主要研究者打算使用几何测度理论的方法来近似d-可求长集的光滑集。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。在这里，它被建议与所谓的“决定点过程”，这是随机点的配置，从随机矩阵中产生。这个奖项反映了NSF的法定使命，并已被认为是值得的支持，通过评估使用基金会的智力价值和更广泛的影响审查标准。

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.