PRIMES: The Topology of Knots and Replication as a Vehicle for Student Research

California State University-Long Beach Foundation, Long Beach CA

Investigators

Abstract

This project will build a partnership between the American Institute of Mathematics (AIM) and California State University Long Beach (CSULB) that will invigorate mathematics research at CSULB by creating innovative student research activities and expanding a research culture for students and faculty. CSULB is an urban comprehensive university and Hispanic Serving Institution with nearly 60% of students identifying as members of underrepresented groups and 60% identifying as female. Approximately 51% of CSULB students receive Pell grants. This project will improve representation in mathematics by providing high-quality research opportunities for students and scaling these opportunities up in size to reach a broad group of CSULB students. These projects will empower students at all levels to answer previously unsolved mathematical questions and engage in innovative research. The PI will leverage a leave at AIM to design and implement research projects in the areas of knot theory and low-dimensional topology for students at CSULB, establish a faculty learning community to support research mentors, and help create a sustainable model for ongoing student research engagement and mentorship. The PI will pursue two research directions, chosen based on their accessibility for student research projects. The first direction is knot theory, the mathematical study of loops in 3D space. The Meridional Rank Conjecture states that the meridional rank and bridge index of a knot are equal. The conjecture has been verified for an array of infinite classes of knots, but the general case is open. The PI will adopt a novel perspective to this conjecture which makes use quotients of the knot group and new definitions of bridge number. The PI has assembled a group of top researchers who are committed to attending an AIM SQuaRE led by the PI to investigate the Meridional Rank Conjecture. The second direction is the topology of self-replication, which seeks to make questions like ``Which shapes can self-replicate?'' mathematically rigorous. Topological models of self-replication include manifolds that can be decomposed along a surface into two homeomorphic copies of the original (these are known as idempotents in a topological category) and n-manifolds that embed in n-dimensional real space and can be decomposed in to a collection of isometric shapes each of which is a scaled version of the original (these are known as rep-tiles). The PI will investigate novel classification theorems for rep-tiles and idempotents, strengthen existing classification theorems for these objects, and pursue applications of these theories to the field of 4-manifold topology. 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.