In this article, we give a perspective on several results, old and new, concerning geometric structures of moduli spaces of Riemann surfaces with respect to the L2 metric (Weil-Petersson metric) on deformations of hyperbolic metrics. In doing so, we aim to demonstrate that the Weil-Petersson metric is suited to account for the geometry of moduli spaces while the topological type, genus in parti...

Suppose M is a compact connected odd-dimensional manifold with boundary, whose interior M comes with a complete hyperbolic metric of finite volume. We will show that the L2-topological torsion of M and the L2-analytic torsion of the Riemannian manifold M are equal. In particular, the L2-topological torsion of M is proportional to the hyperbolic volume of M , with a constant of proportionality w...

Recently, in [4] the author has proved that if B is an open unit ball in a Cartesian product l2× l2 furnished with the lp-norm ‖ · ‖ and kB is the Kobayashi distance on B, then the metric space (B,kB) is locally uniformly linearly convex. In this paper, we introduce this kind of local uniform convexity in bounded convex domains in complex reflexive Banach spaces and we apply this notion in the ...

The main purposes of this paper are (1) To survey the area of coarse embeddability of metric spaces into Banach spaces, and, in particular, coarse embeddability of different Banach spaces into each other; (2) To present new results on the problems: (a) Whether coarse non-embeddability into l2 implies presence of expander-like structures? (b) To what extent l2 is the most difficult space to embe...

Geometry and analysis of shape space. The core of shape space in one of its simplest forms is the orbit space of the action of the group of diffeomorphisms of the circle S1 (the reparametrization group) on the space of immersions of the circle into the plane R2. The aim is to find good Riemannian metrics which allow applications in pattern recognition and visualization. The contributions of the...

Consider the cotangent bundle of a closed Riemannian manifold and an almost complex structure close to the one induced by the Riemannian metric. For Hamiltonians which grow for instance quadratically in the fibers outside of a compact set, one can define Floer homology and show that it is naturally isomorphic to singular homology of the free loop space. We review the three isomorphisms construc...

Let X be an irreducible n-dimensional projective variety in CPN with arbitrary singular locus. We prove that the L2-∂-(p, 1)-cohomology groups (with respect to the Fubini-Study metric) of the regular part of X are finite dimensional.

This paper is about Gaussian regression with random design, where the observations are i.i.d., it is known from Le Cam (1973, 1975 and 1986) that the rate of convergence of optimal estimators is closely connected to the metric structure of the parameter space with respect to the Hellinger distance. In particular, this metric structure essentially determines the risk when the loss function is a ...

We consider the construction of adaptive controllers for minimum phase linear systems which achieve non-zero robustness margins in the sense of the (linear) L2[0,∞) gap metric. The gap perturbations may be more constrained for larger disturbances and for larger parametric uncertainty. Working within the framework of the nonlinear gap metric [3], universal adaptive controllers are first given ac...

We establish an upper bound of 13 + 8 √ 2 . 4.931 on the stretch factor of the Yao graph Y∞ 4 defined in the L∞-metric, improving upon the best previously known upper bound of 6.31. We also establish an upper bound of (11 + 7 √ 2) √ 4 + 2 √ 2 . 54.62 on the stretch factor of the Yao graph Y4 defined in the L2-metric, improving upon the best previously known upper bound of 662.16.