CAREER: Geometry of Mapping Class Groups and Surface Bundles

The mathematics in this research project centers around questions in geometry and topology, which are broadly concerned with understanding various notions of shape. This project focuses on 2-dimensional spaces called surfaces, which are fundamental in many areas of mathematics. Surfaces can be flat, like a piece of paper, or curved, like the outside of a ball, a donut, or a saddle, and the various shapes they take often strongly constrain the shapes of the higher dimensional spaces in which they live.

The educational portion of this project involves a variety of activities aimed at recruiting and supporting marginalized students into mathematics. The first part continues a series of workshops featuring mini-courses by early career speakers from underrepresented groups on their cutting-edge research aimed at graduate students. The second part establishes a series of undergraduate research and recruitment events connecting undergraduate mathematics researchers in Southern California with graduate recruiters from programs in the region.

The third part is a Topical Pedagogy Seminar which will provide graduate students and postdocs training in incorporating topical material into foundational mathematics courses.

A remarkable and ubiquitous example of this mathematical phenomenon is a surface bundle, which just like a donut, can be sliced so that the cross-sections are surfaces. Unlike a donut, however, as one moves through most surface bundles, the surface cross-sections can twist and deform in complicated ways. This twisting–which essentially determines the bundle–is encoded in the mapping class group, which, among other things, is the collection of all symmetries of the space of shapes that a surface can take, also known as Teichmüller space.

The first part of the research program aims to develop a powerful fleet of combinatorial techniques for studying the geometry of mapping class groups and Teichmüller spaces, as well as their structure at infinity. The second part focuses on the geometry of surface bundles arising naturally from dynamics.

In more detail, this project will investigate the coarse geometry of the mapping class group, Teichmüller space, and surface bundles using the tools of geometric group theory. The first part aims to address a family of results showing that mapping class groups can be coherently locally modeled by CAT(0) cube complexes, allowing for the construction of new metrics with a variety of applications, including about their geometry and topology at infinity.

The second part studies the geometry of surface bundles of Veech surfaces and their combinations, as well as developing a Sullivan-like notion of structural stability for the subgroups associated to a variety of surface bundles with nice curvature properties.

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.