Variations of a conjecture of Singer

Abstract: The Chern-Hopf conjecture predicts that the sign of the Euler characteristic of a closed, nonpositively curved manifold is determined by its dimension. Singer and Dodziuk observed that it would follow if one could show that the universal cover of such a manifold has no L^2-harmonic forms outside the middle dimension, observed that this is true in many examples and suggested it might be true quite generally. Since then, this question has been extended to aspherical manifolds, termed the Singer conjecture, and studied by combinatorial, algebraic, dynamical as well as analytic means. In this talk I will discuss what is known and unknown about the Singer conjecture, its connection to homology growth and embedding theory, and recent constructions showing that it does not have rational or mod p analogues (for odd primes p). Joint work with Boris Okun and Kevin Schreve.