Computing harmonic 1-forms on hyperbolic 3-manifolds

I will discuss numerical methods for approximating harmonic representatives of cohomology classes on closed hyperbolic 3-manifolds $M$.  This leads to a practical numerical method for computing harmonic (i.e. $L^2$) norms on $H^1(M)$, cohomology regulators, and the Ray-Singer analytic torsion for such manifolds.  I will show experimental data comparing these to known topological bounds coming from the Thurston norm, and explore what they can tell us about open questions related to torsion growth.  The method itself is a variant of the method of particular solutions, which goes back to the work of Moler-Payne in the 1960s, and has been used by Strohmaier-Uski to compute very precise eigenvalues of the Laplacian on functions for hyperbolic surfaces.  No prior knowledge of numerical methods will be assumed and I will focus mostly on background and applications.  Finally, I will show pictures at the end of some links in the 3-sphere with 1,000+ crossings whose complements are congruence arithmetic. This is mainly joint work with Anil Hirani, but also Malik Obeidin and Cameron Rudd for the links.