CAREER: Moduli spaces of surfaces

Regents Of The University Of Michigan - Ann Arbor, Ann Arbor MI

Investigators

Abstract

Moduli spaces of surfaces are of central importance in mathematics and theoretical physics and are a meeting ground for researchers working in different fields. They parametrize different geometries on surfaces, so that a point in the moduli space encodes a shape which a surface can assume. Classically, one considers the evenly curved shapes, with richly different points of views arising from hyperbolic geometry, complex analysis, algebraic geometry, and matrix groups. More recently, singular flat geometries have gained prominence because of their relationship to the classical moduli space as well as their connections to important examples in the theory of dynamical systems (systems that evolve over time). This project will advance the study of moduli spaces in five interrelated research programs tied together by shared techniques and analogies. Progress on these research programs will advance the understanding of the geometry of surfaces and higher dimensional spaces and will unlock applications to dynamical systems. The nature of the topics makes their simultaneous investigation synergistic and allows for their integration into educational activities, including the training and mentoring of graduate students, vertically integrated research with undergraduate students, and the development of a new course for the bridge to PhD program. Undergraduate participants will be encouraged to participate in the MathCorp summer outreach program, and the more senior participants will receive training on mentoring. The five research programs that will be undertaken are as follows. First, the PI will build quasi-convex co-bounded planes in Teichmüller spaces, and eventually convex cocompact surface subgroups of mapping class groups, by gluing together components obtained via dynamics. Second, the PI will determine if there is a non-trivial orbit closure of translation surfaces of rank at least 3, by classifying special constructions and investigating low genus moduli spaces. Third, the PI will show typical high genus surfaces are good spectral expanders, using the trace formula and work of Mirzakhani. Fourth, the PI will develop Patterson-Sullivan theory for mapping class groups of non-orientable surfaces. Fifth, the PI will develop a criterion for unique ergodicity in the context of dilation surfaces. 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.