Computational Algebra and Applications

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Many real world problems have a significant mathematical component. This project uses computational algebra to attack such problems. The main themes are: (1) Data Analysis, (2) Graphics, visualization, geometric modeling, (3) Computer science and computational complexity. A main problem in Data Analysis is to extract meaning from a massive dataset: imagine a dense, cloud of moving points. One approach is to freeze the cloud at a moment in time, and then increase the size of the individual points until nearby groups coalesce. Topological Data Analysis uses sophisticated mathematical tools to study the problem, and has led to insights into visual cortex activity, cancer pathology, and viral evolution. Companies from Boeing to Pixar use computer graphics and geometric modeling in applications ranging from analyzing turbulent fluid flow (air passage over a plane's wing) to accurate simulations and virtual reality. Key mathematical tools are splines, which provide a way of assembling a coherent big picture from local pieces. A third theme of this project is the modeling of dynamic processes, which often comes down to multiplying (massive) matrices many times (a matrix is a rectangular array of numbers). Finding efficient ways to multiply matrices is the topic of a subfield of Computer Science known as complexity theory. The project builds on prior work of the PI using algebraic objects known as permanents to provide insight into the problem. While at his previous position at Iowa State, the PI partnered with the Iowa State Veterans Center to provide extra math tutoring and support for student veterans, including an intensive day long Math Boot Camp at the start of each semester. In this project, he will develop similar programs at Auburn, with the first boot camp planned for fall 2021.The proposal revolves around a few main themes: permanents, polytopal parameterizations, and persistent homology, approximation theory and splines. The first group of topics is unified by homological methods and simplicial complexes, and the second by the interplay of algebra and discrete geometry; the objects of investigation are central players in computational mathematics. The problems share three traits: they are of importance in real world applications; subtle geometric or combinatorial data is reflected algebraically; and they are computationally tractable. The project will use the power of computational algebra to discover hidden connections between important applied invariants and geometry, combinatorics, and algebra. There are elegant characterizations of key invariants for applications (e.g. dimension of the spline space for a polytopal subdivision, equations for Wachspress parametric surfaces, the support locus of persistent homology) in terms of algebra. The concrete and computational aspect of the problems means they are ideal for involving graduate students and postdocs, and makes prospects for progress excellent. In addition to scientific advances, the project will result in development of specialized software, to be coauthored with students and integrated into the NSF-sponsored Macaulay2 software package.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.