Normalized Betti Numbers, Non-Positive Curvature, and the Singer Conjecture

Differential geometry is a broad and active subject that plays a crucial role in modern mathematics, theoretical physics, and computer science. It studies spaces called differentiable manifolds that when zoomed in look like pieces of the familiar Euclidean space, and that strikingly are the correct model for many of our physical theories such as General Relativity and String Theory.

In modern differential geometry, the so-called Singer conjecture predicts a fascinating connection between the geometry and topology of such spaces in the presence of non-positive curvature. A principal objective of the funded research will be to build a multidisciplinary study of such a problem, and to explore its connections with other important problems in the field such as Yau's question on normalized Betti numbers and the Hopf problem.

The goal of this proposal will be not only to build a comprehensive program towards the solution of these problems, but also to propose extensions of such questions, to clarify their interdependence, and to bridge a gap with closely related problems in differential geometry, complex algebraic geometry, and geometric topology. Moreover, progress in these areas could have significant repercussions outside of geometry.

Indeed, these questions are intimately connected to problems in partial differential equations, geometric group theory, as well to areas of mathematical theoretical physics. Undergraduate and graduate students will be trained through their participation in the proposed activities, and the PI will continue to organize seminars and conferences and to participate in outreach efforts targeting students who have experienced reduced access to education.

More specifically, this project addresses the study of normalized Betti numbers and L2-Betti numbers on non-positively curved spaces with geometric analysis techniques and Hodge theory. In 2017, the PI together with Mark Stern developed the theory of Price inequalities for harmonic forms on Riemannian manifolds. The PI will explore and elucidate the connections between the theory of Price inequalities for harmonic forms and the Singer conjecture for compact manifolds.

The PI will also study non-compact finite volume negatively curved spaces with Price inequalities, and he will apply these techniques to higher dimensional aspherical Dehn filled manifolds, higher graph manifolds, and non-positively curved toroidal compactifications. Also, he will study the cohomology of sequences of negatively curved Riemannian manifolds which converge, in the sense of Benjamini and Schramm, to their Riemannian universal cover.

Finally, in the Kaehler setting the PI will focus his research on smooth irregular varieties. This will yield extensions of the original conjecture of Singer outside the class of aspherical manifolds, and it will also open new avenues of research related to Yau's question on normalized Betti numbers.

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.