Geometry and dynamics in moduli spaces of surfaces

University Of Wisconsin-Madison, Madison WI

Investigators

Abstract

Moduli spaces pervade mathematics. Given a mathematical object the corresponding moduli space parameterize the shapes that the object can have. For example, following a path in the moduli space of triangles corresponds to watching a movie of one triangle deforming into another. In this way moduli spaces helps us understanding how manifestations of a mathematical object can be deformed one to the other. The PI will investigate moduli spaces of surfaces with some additional geometric structure and use this to make advances in solving a suite of long-standing conjectures about their geometry and topology. The moduli spaces in question admit an action, i.e. a way of “mixing-up” the space, that is intimately connected to understanding the ``physics” governing the space. The PI and his collaborators have recently developed techniques for studying this action, which the PI will use to solve a series of problems. The PI will integrate his research with efforts to engage students from a diverse pool of backgrounds and mathematical talent. This will include devising computational research projects for undergraduate students, including those with minimal mathematical background, inviting graduate students to act as research mentors. Students will also be recruited to help produce computational and educational materials, which will be made available to the public. The project will make advances in PI's ongoing investigation of the GL(2, R) action on the Hodge bundle and applications to the study of associated moduli spaces. These questions connect to various problems in dynamics, low-dimensional topology, and algebraic geometry and use new techniques developed by the PI and collaborators. Recent groundbreaking work has shown that each GL(2, R) orbit closure of a point in a stratum of the Hodge bundle is locally linear in period coordinates, but as yet no classification of these orbit closures exists. The PI will make progress in classifying all GL(2, R) orbit closures in hyperelliptic loci of strata that are “sufficiently big”, and will use the theory of Hurwitz spaces to build a purely combinatorial mechanism for producing new orbit closures. In addition, using recent work on geminal orbit closures, the PI will uncover properties of totally geodesic submanifolds in the moduli space of Riemann surfaces. This inquiry will lead to a deeper understanding of when complex geodesics in Teichmuller space are holomorphic retracts. The PI will also study the moduli spaces of complex affine structures to make progress on a conjecture that strata of the Hodge bundle are aspherical in the orbifold sense; and will work on a program to determine the Hausdorff dimension of the set of divergent Teichmuller geodesic rays. Put together, these projects will resolve open questions about the geometry of moduli space, while shedding fresh light on the study of rational billiards. 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.