Combinatorial Properties of Convex Sets and Measures in Euclidean spaces

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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.