Topology and Linear Algebra in Discrete Geometry

离散几何是研究几何对象的有限族的组合性质。这一领域的成果通常应用于搜索算法、数据分类和优化。用于回答离散几何问题的方法可能大不相同。有些是基于线性代数，其中空间的刚性结构很重要。有些是基于拓扑的，其结果支持所涉及对象的平滑变形。本研究计划旨在开发新的方法，在离散几何中使用拓扑和线性代数方法，以扩大组合学和其他领域之间的联系。驱动的动机是理解是什么使组合问题适合于拓扑技术，以及是什么使它适合于线性代数技术。该项目还包括可用于指导本科生进行研究的主题。该项目的重点是与凸集的有限族相关的问题。PI计划开发新的方法将拓扑方法应用于Tverberg型问题，这与Tverberg定理的长期“丰富多彩”版本有关。最近的进展，定量Helly型定理可以帮助建立一个桥梁之间的分析方面的凸性和线性规划，这是PI的目标形式化。该奖项反映了NSF的法定使命，并通过使用基金会的智力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 is the study of combinatorial properties of finite families of geometric objects. Results in this field often have applications to search algorithms, data classification, and optimization. The methods used to answer discrete geometry questions can be wildly different. Some are based on linear algebra, where the rigid structure of the space is important. Some are based on topology, and the results hold up to smooth deformations of the objects involved. This research project aims to develop new ways to use topological and linear algebraic methods in discrete geometry to expand the connections between combinatorics and other fields. The driving motivation is to understand what makes a combinatorial problem amenable to techniques from topology, and what makes it amenable to techniques from linear algebra. The project also includes topics that can be used to mentor undergraduate students in research. The project focuses on questions related to finite families of convex sets. The PI plans to develop new ways to apply topological methods to Tverberg-type problems, related to a longstanding "colorful" version of Tverberg’s theorem. The recent progress in quantitative Helly-type theorems can help build a bridge between the analytic side of convexity and linear programming, which the PI aims to formalize. Several of the questions under study aim to generalize classic results in extremal combinatorics to high-dimensional Euclidean spaces.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.