Reidemeister torsion was the first topological invariant that could distinguish between spaces which were homotopy equivalent but not homeomorphic. The Cheeger-Müller theorem established that the Reidemeister torsion of a closed manifold can be computed analytically. I will report on joint work with Frédéric Rochon and David Sher on finding a topological expression for the analytic torsion of a manifold with fibered cusp ends. Examples of these manifolds include most locally symmetric spaces of rank one. We establish our theorem by controlling the behavior of analytic torsion as a space degenerates to form hyperbolic cusp ends.[-]

An oriented manifold possesses an L-homology fundamental class which is an integral refinement of its Hirzebruch L-class and assembles to the symmetric signature. In joint work with Gerd Laures and James McClure, we give a construction of such an L-homology fundamental class for those oriented singular spaces, which are integral intersection homology Poincaré spaces. Our approach constructs a morphism of ad theories from intersection Poincaré bordism to L-theory. We shall indicate an application to the stratified Novikov conjecture. The latter has been treated analytically by Albin, Leichtnam, Mazzeo and Piazza.[-]

The intersection cohomology of a complex projective variety $X$ agrees with the usual cohomology if $X$ is smooth and satisfies Poincare duality even if $X$ is singular. It has been proven in various contexts (and conjectured in more) that the intersection cohomology may be represented by the $L^2$- cohomology of a Kähler metric defined on the smooth locus of $X$. The various proofs, though different, often depend on a notion of weight which manifests itself either through representation theory, Hodge theory, or metrical decay. In this talk we discuss the relations between these notions of weight and report on new work in this direction.[-]

This is joint work with Jasmin Matz. The goal is to introduce a regularized version of the analytic torsion for locally symmetric spaces of finite volume and higher rank. Currently we are able to treat quotients of the symmetric space $SL(n,\mathbb{R})/SO(n)$ by congruence subgroups of $SL(n,\mathbb{Z})$. The definition of the analytic torsion is based on the study of the renormalized trace of the corresponding heat operators. The main tool is the Arthur trace formula. I will also discuss problems related to potential applications to the cohomology of arithmetic groups.[-]

Hermitian complex spaces are a large class of singular spaces that include for instance projective varieties endowed with the metric induced by the Fubini-Study metric. Many of the problems raised by Cheeger, Goresky and MacPherson in the case of complex projective varieties admit a natural extension also in this setting. The aim of this talk is to report about some recent results concerning the Hodge-Kodaira Laplacian acting on the canonical bundle of a compact Hermitian complex space. More precisely let $(X,h)$ be a compact and irreducible Hermitian complex space of complex dimension $m$. Consider the Dolbeault operator $\bar{\partial}_{m,0}$ : $L^2 \Omega^{m,0}(reg(X),h) \to L^2\Omega^{m,1}(reg(X),h)$ with domain $\Omega{_c^{m,0}}(reg(X))$ and let $\bar{\mathfrak{d}}_{m,0} : L^2 \Omega^{m,0}(reg(X),h)\to L^2\Omega^{m,1}(reg(X),h)$ be any of its closed extension. Now consider the associated Hodge-Kodaira Laplacian $\bar{\mathfrak{d}^*} \circ\bar{\mathfrak{d}}_{m,0}$ : $L^2 \Omega^{m,0}(reg(X),h)\to L^2\Omega^{m,0}(reg(X),h)$. We will show that the latter operator is discrete and we will provide an estimate for the growth of its eigenvalues. Finally we will prove some discreteness results for the Hodge-Dolbeault operator in the setting of both isolated singularities and complex projective surfaces (without assumptions on the singularities in the latter case).[-]

In the first part of this talk we will show how classical tools of Riemannian geometry can be used in the setting of stratfied spaces in order to obtain a lower bound for the spectrum of the Laplacian, under an appropriate assumption of positive curvature. Such assumption involves the Ricci tensor on the regular set and the angle along the stratum of codimension 2. We then show that a rigidity result holds when the lower bound for the spectrum is attained. These results, restricted to compact smooth manifolds, give a well-known theorem by M. Obata and A. Lichnerowicz.
Finally, we will explain some consequences of the previous theorems on the existence of a conformal metric with constant scalar curvature on a stratified space.[-]