Geodesics of Random Riemannian Metrics

We analyze the disordered Riemannian geometry resulting from random perturbations of the Euclidean metric. We focus on geodesics, the paths traced out by a particle traveling in this quenched random environment. By taking the point of the view of the particle, we show that the law of its observed environment is absolutely continuous with respect to the law of the random metric, and we provide an explicit form for its Radon-Nikodym derivative. We use this result to prove a "local Markov property" along an unbounded geodesic, demonstrating that it eventually encounters any type of geometric phenomenon. We also develop in this paper some general results on conditional Gaussian measures. Our Main Theorem states that a geodesic chosen with random initial conditions (chosen independently of the metric) is almost surely not minimizing. To demonstrate this, we show that a minimizing geodesic is guaranteed to eventually pass over a certain "bump surface," which locally has constant positive curvature. By using Jacobi fields, we show that this is sufficient to destabilize the minimizing property.Comment: 55 pages. Supplementary material at arXiv:1206.494

On Schwartz Groups

In this paper we introduce a notion of a Schwartz group, which turns out to be coherent with the well known concept of a Schwartz topo- logical vector space. We establish several basic properties of Schwartz groups and show that free topological Abelian groups, as well as free locally convex spaces, over a hemicompact k{space are Schwartz groups. We also prove that every hemicompact k{space topological group, in particular the Pontryagin dual of a metrizable topological group, is a Schwartz group

Completeness properties of locally quasi-convex groups

It is natural to extend the Grothendieck Theorem on completeness, valid for locally convex topological vector spaces, to abelian topological groups. The adequate framework to do it seems to be the class of locally quasi-convex groups. However, in this paper we present examples of metrizable locally quasi-convex groups for which the analogue to Grothendieck Theorem does not hold. By means of the continuous convergence structure on the dual of a topological group, we also state some weaker forms of Grothendieck Theorem valid for the class of locally quasi-convex groups. Finally, we prove that for the smaller class of nuclear groups, BB-reflexivity is equivalent to completeness

On the Set of Locally Convex Topologies Compatible with a Given Topology on a Vector Space: Cardinality Aspects

For a topological vector space (X, τ ), we consider the family LCT (X, τ ) of all locally convex topologies defined on X, which give rise to the same continuous linear functionals as the original topology τ . We prove that for an infinite-dimensional reflexive Banach space (X, τ ), the cardinality of LCT (X, τ ) is at least c

Absorption adjunctable semigroups

The paper studies the topological semigroups that admit the adjunction of a non-isolated absorbing element and the structure and permanence properties of the class AA of topological semigroups admitting this type of adjunctions. No precompact topological group can belong to the class AA. More generally, a subsemigroup X of a compact Hausdorff semigroup K belongs to AA iff X misses the Sushkevich kernel of the closure cl(X) in K. Every non-torsion abelian group belongs to AA when equipped with the discrete topology. The interest in the class AA stems from the question of quasi-uniformizability of semigroups (every topological AA-group, after the adjunction of a non-isolated absorbing element, gives rise to a non-quasi-uniformizable semigroup