Geometric counts on surfaces

Cornell University, Ithaca NY

Investigators

Abstract

Surfaces are geometric shapes that are everywhere two dimensional. Such geometric objects play an important role in pure mathematics, as well as in various areas of science and engineering. This project focuses on understanding metric and geometric properties of surfaces, i.e. notions of length and angle. Deforming the metric structure of a surface gives rise to so-called `moduli spaces’; each point in the moduli space represents a particular geometry on the underlying surface. The PI will continue to investigate two natural classes of surface geometry: (i) singular flat metrics, and (ii) hyperbolic (negatively curved) metrics. Improved understanding of flat metrics has led, and will continue to lead, to new results in dynamical systems. Hyperbolic metrics are, in a certain sense, the most natural metrics to put on these surfaces. This project will also investigate various structures on the moduli spaces of surfaces, notably measures, which provide for a well-defined notion of `random surface’. The PI will use and further develop analogies between the two types of metrics, as well as connections to graph theory, probability theory, spectral geometry, and algebraic geometry. The broader impact of this project includes the generation of questions and topics suitable for graduate students. The PI will maintain an involvement with programs aimed at bringing together early-career researchers to work on problems in concentrated and collaborative settings. In the flat metric setting, the PI has used the new multi-scale compactification of strata of translation surfaces to prove strong regularity of ergodic SL_2-invariant probability measures, verifying a natural heuristic about affine invariant manifolds. This result, and the techniques in its proof, have already been applied by others to study random Teichmuller geodesics and counts of pairs of saddle connections. In another joint work, the PI has proved restrictions on the type of equations that can define an affine invariant manifold near the multi-scale boundary, thereby providing a new proof of Wright's Cylinder Deformation Theorem and generalizing it to meromorphic strata. The PI will use the above results and techniques to study, via degeneration, the centrally important classification problem for affine invariant manifolds, as well as problems about the less studied k-differentials for k>2. The PI has also developed a research program concerning random hyperbolic surfaces, a young and active field. Together with Sapir, the PI has proved a conjecture concerning the relative number of simple versus non-simple closed geodesics for various notions of random surfaces, as genus tends to infinity. There is a fruitful analogy between random hyperbolic surfaces and random regular graphs; the PI has solved, and continues to investigate, problems on both sides of the coin. This research leads to further questions about the shape of a typical geodesic on a generic surface, as well as to certain questions about geodesics on every hyperbolic surface. 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.