Combinatorial Connections with Algebra, Geometry, Probability and Applications

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Combinatorics is an area of mathematics which is quintessential for applications in computer science, biology, physics, chemistry, and industry. Optimized algorithms based on combinatorics have revolutionized business decisions in our lifetime. Current research is uncovering connections to all areas of mathematics and science. This project focuses specifically on interdisciplinary applications of combinatorics in connection with problems in algebra,geometry, probability, and computer science. Combinatorial connections have been at the core of the investigator's prior work and continue to inspire innovation and collaboration. This project describes three main themes for research. The first relates to the classical study of the coinvariant algebra and its representation theory. The problem is to study the asymptotics of the underlying decomposition into irreducible symmetric group modules. The methods of attack include tools from combinatorics, algebra and probability theory. The second studies a newly proposed family of symmetric functions related to the matroid of 0-1 vectors in n-dimensional space. Conjectures and theorems in this direction use tools from algebraic geometry. This research has connections to physics and economics. The third topic, which is a mixture of combinatorics, theoretical computer science and discrete geometry, pertains to placements of circles in the plane with a wrapping condition. This area of research was inspired by the general discrete geometry problem of finding an appropriate polygonization of a region in the plane which appears in graphics and optimization.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.