Proceedings from an AMS Special Session examine the Singer-Hopf Conjecture and allied challenges in geometry and topology, highlighting research on normalized Betti numbers, L2-invariants, and aspherical manifolds. The articles bridge diverse viewpoints from differential geometry, complex algebraic geometry, and topology.
This volume contains the proceedings of the AMS Special Session on Singer-Hopf Conjecture in Geometry and Topology, held from March 18-19, 2023, at Georgia Institute of Technology, Atlanta, Georgia. It presents a multidisciplinary point of view on the Singer conjecture, the Hopf conjecture, the study on normalized Betti numbers, and several other intriguing questions on the fundamental group and cohomology of aspherical manifolds. This volume highlights many interesting research directions in the study of aspherical manifolds and covers a large collection of problems and conjectures about $L^2$-invariants of aspherical manifolds. It provides a snapshot of contemporary research in mathematics at the interface of geometry and topology, as well as algebraic geometry. The problems are presented from several distinct points of view, and the articles in this volume suggest possible generalizations and bridge a gap with closely related problems in differential geometry, complex algebraic geometry, and geometric topology. The volume can play a role in focusing the attention of the mathematical community on these fascinating problems which continue to resist the siege of geometers and topologists. It is our hope that this volume will become a valuable resource for early career mathematicians interested in these deep and important questions.