Combinatorics, Cohomology, and Matrix Spaces

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Combinatorics is the field of mathematics most concerned with concrete, finite objects. Over the past half century, it has grown from humble origins into a rich field with deep connections to and applications in disparate fields across mathematics and the sciences. A central mathematical framework that has spurred combinatorics' development from its inception is the challenge of understanding geometric spaces by their decomposition into constituent pieces. Recent breakthroughs of this type have led to deeper understanding of fundamental problems in theoretical physics, probability theory, algebraic geometry and many other fields. Since the 19th century, mathematicians have applied this framework to better understand collections of two-dimensional arrays of data called matrix spaces. This project will use combinatorial tools to determine properties of matrix spaces broken down into pieces determined by imposing redundancy conditions on the underlying data. In addition, funds will support undergraduate research, training graduate students and outreach efforts including work with a prison education program.To a mathematician, the spaces this project studies are quite simple: matrices, idempotent matrices, symmetric and skew-symmetric matrices. The pieces of matrix spaces considered in this project, called matrix Schubert varieties, are comprised of matrices satisfying rank restrictions on specified submatrices. Matrix Schubert varieties are invariant under certain row/column operations -- this extends to a natural group action. By taking this group action into account, geometric and topological properties of matrix Schubert varieties solve central questions in enumerative algebraic geometry and intersection theory. Using a collection of tools from combinatorics and commutative algebra known collectively as 'Grobner geometry', this project will describe polynomial representatives that encode the geometric properties of matrix Schubert varieties. A potential application is the first combinatorial description of structure constants in the Lagrangian Grassmannian.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.