IRFP: Topological Representations of Matroids and the Geometry of Phylogenetic Trees

Stamps Matthew T, Davis CA

Investigators

Abstract

The International Research Fellowship Program enables U.S. scientists and engineers to conduct nine to twenty-four months of research abroad. The program's awards provide opportunities for joint research, and the use of unique or complementary facilities, expertise and experimental conditions abroad. This award will support a twenty-four-month research fellowship by Dr. Matthew T. Stamps to work with Dr. Svante Linusson at the Royal Institute of Technology (KTH) in Stockholm, Sweden. A relatively new field of mathematics is topological combinatorics, which concerns, as its name suggests, the interactions between combinatorics and algebraic topology. The main idea is to translate a genuine combinatorial question into a topological problem whose solution is both well studied and answers the original question at hand. This approach has led to many significant breakthroughs over the past several decades; in fact, there are a number of deep theorems in combinatorics whose only known proofs require topological techniques. This project considers several applications of those techniques to matroid theory and evolutionary biology. Matroids are (discrete) mathematical objects that capture the notion of independence. They most frequently appear in combinatorics and optimization, but they also arise from arrangements of circles on spheres (along with higher dimensional analogs) that can be constructed from topological objects called homotopy colimits of a diagram of spaces. The primary aim of the PI is to establish an explicit role in which homotopy colimits can provide novel solutions to a collection of problems in matroid theory and, more generally, in algebraic combinatorics. A secondary component of the project explores the geometry of the edge-product space of phylogenetic trees, a probabilistic model from evolutionary biology aimed at allowing taxonomists to work with missing information (an important feature given that, in some areas, new species are discovered rather frequently). This line of research incorporates techniques from several disciplines in mathematics, including commutative algebra, combinatorics, discrete geometry, algebraic topology, and probability theory, while developing collaboration between the PI and research groups at the Royal Institute of Technology (KTH) in Stockholm, Sweden (Dr. Anders Björner and Dr. Svante Linusson), Aalto University in Helsinki, Finland (Dr. Alexander Engström), and the Freie Universität - Berlin in Germany (Dr. Günter Ziegler).