LEAPS-MPS: Geometry and Topology of Dual Artin Groups

A group is a mathematical structure which can be understood either geometrically (encoding the symmetries of a particular object) or algebraically (using symbolic manipulation according to certain abstract rules). Groups are of fundamental importance in mathematics and have been applied in a wide variety of areas, including high energy physics, crystallography, robotics, and cryptography. One area of particular interest is the class of Artin groups, which are straightforward to describe algebraically but difficult to understand geometrically. This project will examine the topology and geometry of Artin groups by developing the relationship between these groups and the closely-related analogue known as "dual Artin groups." In addition, the project will support undergraduate research and enhance student programming (such as the mathematics colloquium series) at Lafayette College.

The goals of this project fall into four categories. First, the PI will examine the relationship between Artin groups and dual Artin groups. These two definitions are conjectured to produce isomorphic groups in general, but this has only been demonstrated in select cases. One goal of the project is to expand the classes of Artin groups for which this isomorphism is known to exist (e.g. to the class of "even" Artin groups). Next, the PI will solidify connections between the dual Artin groups corresponding to braid groups and other areas such as hyperplane arrangements, tree complexes, and the work of Thurston on complex dynamics. In each of these areas, prior and ongoing work of the PI has identified topological connections with the dual braid group; this project will produce a more comprehensive understanding of how and why these connections arise. Third, a major component of this project concerns recent work of the PI with McCammond on the "branched annulus complex," which is a finite piecewise-Euclidean cell structure on a new compactification for the space of monic complex polynomials with distinct centered roots. In collaboration with McCammond, the PI will introduce a generalization of this complex for each dual Artin group and examine its topological and geometric properties. Finally, this project will support undergraduate research on the related combinatorial topic of noncrossing partition lattices. In particular, the PI will guide undergraduates in studying the lattices of noncrossing partitions arising from planar configurations and identifying when useful combinatorial properties appear (e.g. gradedness, rank-symmetry, shellability).

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.

Award Number: 2532608
Principal Investigator: Michael Dougherty
Funds Obligated: $250,000
State: PA