LEAPS-MPS: Braids and Mapping Class Groups: Investigating Left-orders, Twisting, and Positivity

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This award is funded in whole or in part under the American Rescue Plan Act of 2021 (Public Law 117-2).This project concerns several research questions in topology, which is the study of shapes. Topology can help us investigate difficult and interesting questions, such as what the shape of our universe is, how we can extract meaningful information from noisy data sets, or how human DNA, which unravels to be several feet long, can fold up to fit inside a cell. The project investigates mathematical properties of knots, which are knotted up loops in three-dimensional space, and three-manifolds, which are objects that locally look like the three-dimensional world we live in. The central theme is to investigate how information about knots and three-manifolds is dictated by two-dimensional information, namely, maps on surfaces. Turning a three-dimensional question into a two-dimensional one is a powerful tool throughout topology and geometry. In this project, knots can be represented by objects called braids, and three-manifolds can be represented by objects called open book decompositions, both of which give ways of associating a map on a surface to a knot or three-manifold, respectively. One can ask how properties of these maps dictate properties of the corresponding knot or three-manifold. The investigator also plans to hold career preparation events, math circles in areas of broad interest such as public health, and a distinguished speaker series at her institution, with the goal of increasing the participation of underrepresented groups in STEM and enhancing the research environment at her institution. The investigator will undertake three projects that investigate how properties of three-manifolds and knots are dictated by two-dimensional information. The first project explores whether results about braids relating to left-orders and twisting can be extended to more general maps on surfaces, with potential applications to the study of three-manifolds and the L-Space Conjecture. The second project asks whether the Euler characteristics of open book decompositions for a given fixed three-manifold are bounded, with potential applications to an open question in contact geometry. The third project is an investigation of the notion of braid positivity and knot cables, with potential applications to an open question of Yasui.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.