Combinatorial Properties of Convex Sets and Measures in Euclidean spaces

离散几何和拓扑组合是数学的重要领域。他们以新颖的方式强调数学不同领域之间的联系。这些学科的研究在计算机科学中有应用，如搜索算法和优化。该奖项将支持PI开发新的数学方法来解决组合学和拓扑学边界问题的研究。离散几何问题往往很容易表述，是一个很好的数学推广课题。PI的目标是让本科生参与他的研究。这个项目的研究将集中在三种主要类型的开放问题上。这些类型的问题突出了离散几何的不同方面，是多学科的；包括组合学，拓扑学和线性代数。由于它们密切相关，一个领域的进展将有利于其他领域的研究。第一类问题，测度的划分，是等变拓扑和离散几何相互作用的基础。PI将侧重于附加几何约束对该领域经典问题的影响。第二个是Tverberg理论，它侧重于从组合的角度理解有限点集的凸包。PI扩展了用于该领域的线性代数方法，旨在研究最近结果的拓扑版本。第三类研究欧几里德空间中凸集有限族的交结构。这项工作将集中于继续发展经典结果的定量版本的最新趋势，例如helly型定理。这些问题中的一些在计算机科学和其他数学领域有额外的动机应用。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。

Discrete geometry and topological combinatorics are important areas of mathematics. They highlight connections between different areas of mathematics in novel ways. The research in these subjects has applications in computer science, such as in search algorithms and optimization. This award will support the PI's research to develop new mathematical methods to solve problems on the boundary of combinatorics and topology. Problems in discrete geometry are often easy to state, and a good subject for popularization of mathematics. The PI will aim at involving undergraduate students in his research. The research of this project will focus on three main types of open problems. These type of problems highlight different aspects of discrete geometry and are multi-disciplinary; involving combinatorics, topology and linear algebra. As they are closely related, progress in one area will benefit the research in the others. The first kind of problems, partitions of measures, is a cornerstone of the interaction of equivariant topology and discrete geometry. The PI will focus on the effect of additional geometric constraints to classic problems in the field. The second is Tverberg theory, which focuses on understanding the convex hulls of finite sets of points from a combinatorial point of view. The PI has extended the linear-algebraic methods used for this area, and aims at working on topological versions of recent results. The third kind focuses on understanding the intersection structure of finite families of convex sets in Euclidean spaces. The work will focus on continuing a recent trend of developing quantitative versions of classic results, such as Helly-type theorems. Several of these problems have as additional motivation applications in computer science and other areas of mathematics.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.