Cokernels of Random Matrices and the Geometry of Error-Correcting Codes

University Of California-Irvine, Irvine CA

Investigators

Abstract

This research project will investigate several questions about how objects from combinatorics and number theory vary in families. The kinds of objects that are the focus of this research play important roles in modern cryptography and coding theory. These questions lie at the boundary of several mathematical areas, and the principal investigator will use ideas from algebraic geometry and number theory to answer combinatorial questions and enhance our understanding of error-correcting codes, lattices, and sandpile groups of graphs. In many situations, these counting problems can be understood in terms of families of random matrices. A major idea is to study error-correcting codes from an algebraic perspective, focusing on the interplay between codes, polynomials over finite fields, and the algebraic varieties that they define. The principal investigator has extensive experience as a mentor for research by undergraduates and high school students and will lead student research projects in these areas. These projects will be supported by graduate student mentors will be who will gain valuable professional development experience. The research projects here combine ideas from multiple areas of pure and applied mathematics and are ideal for promoting collaboration across disciplines. The sandpile group of a connected graph is a finite abelian group whose order is equal to the number of spanning trees of the graph. The PI will study how these groups vary within certain families of graphs. These questions are motivated by connections to the Cohen-Lenstra heuristics from number theory. The PI will consider problems about families of random lattices, focusing on lattices with algebraic structure. An important tool here comes from the theory of zeta functions of groups and rings. In certain cases this additional algebraic structure will lead to very different behavior. Reed-Muller codes are multivariable analogues of Reed-Solomon codes and are some of the most fundamental examples in coding theory. A key idea is to use results from the theory of interpolation problems in algebraic geometry to prove new results about codes. The principal investigator will combine expertise in algebraic and arithmetic geometry, as well as in coding theory, to further our understanding of these important objects. 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.