15 jun 2017 -- 15:00
Sala Riunioni, Dipartimento di Matematica, Pisa
Seminari dei Baby-Geometri
Abstract.
Gromov raised the question whether there is a universal bound for the $L^2$-Betti numbers of an aspherical manifold by its simplicial volume. A positive answer would yield, in combination with Gromov's main inequality, an upper bound of $L^2$-Betti numbers of an aspherical manifold by its Riemannian volume provided a lower Ricci curvature bound. While the above conjecture remains open, the implication was shown by Sauer using so-called randomization techniques.
After a short introduction to $L^2$-Betti numbers, I will address the randomization techniques and new developments around curvature-free versions of the main inequality for $L^2$-Betti numbers.