Complex Monge-Amp\`ere equations, Grauert tube and estimate of Betti numbers.

In this talk I shall explain a geometric construction motivated by the study of complex Monge-Amp\`ere equations, namely the so-called Grauert tube, and its application in obtaining an effective estimate on the  Betti numbers of the loop space of a compact Riemannian manifold whose sectional curvature is bounded from below.