Intrinsic Geometry, Topology, and Complexity of 3-Manifolds

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A mathematical object known as a 3-manifold resembles our physical space if one views a small piece of it. However on a large scale, 3-manifolds have properties that may be quite different from what we are used to seeing and experiencing. Such manifolds are common objects in science, appearing for example in physics, astronomy or data science, as well as in various fields of mathematics. Here the Principal Investigator will study intrinsic properties of 3-manifolds, and how they relate to various areas of mathematics. Since many properties lend themselves to computer study and yield interesting algorithms, the PI will also look at questions that encompass how "hard" 3-manifolds are computationally. The award provides funds for supporting a graduate student.Due to Geometrization (W. Thurston, Perelman), every 3-manifold can be canonically decomposed into pieces, and each piece has a certain geometric structure. Thus, on a global scale, one can match topological information for the manifold with the respective geometry. However, on a local scale, that is, intrinsically, the connection between the geometry and topology of a 3-manifold is not well understood. This is particularly so for hyperbolic 3-manifolds, though there are questions for other classes as well. The goal of the first part of this project is to obtain a deep insight into this, for 3-manifolds with finite hyperbolic or simplicial volume (since not all manifolds here are hyperbolic). There are projects on embedded surfaces and arcs in 3-manifolds, cusped or closed, addressing well-known conjectures in the field. While these questions are interesting a priori, the second part of the project is concerned with applications of the developed results and techniques, and aims to use the obtained insight for deepening connections with other areas. In particular, (a) to better understand the interplay between geometric topology and algebraic geometry of 3-manifolds, through the study representation variety of a 3-manifold; (b) to address the problems on the interface of theoretical computer science and low-dimensional topology, through an overlap with complexity theory and computable analysis.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.