We consider the question of when an inequality between lengths of corresponding geodesics implies a corresponding inequality between volumes. We prove this in a number of cases for compact manifolds with and without boundary. In particular, we show that for two Riemannian metrics of negative curvature on a compact surface without boundary, an inequality between the marked length spectra implies the same inequality between the areas, with equality implying isometry.

Soit Γ un groupe Kleinien non élémentaire agissant sur une variété de Cartan-Hadamard $\tilde X$; on note Λ(Γ) l'ensemble non-errant du flot géodésique (φ_t) agissant sur le fibré unitaire tangent T¹($\tilde{X}$/Γ). Lorsque Γ est convexe cocompact (i.e. Λ(Γ) est compact), la restriction de (φ_t) à Λ(Γ) est un flot Axiom A : ainsi, d'après un théorème de Bowen-Ruelle, il existe sur Λ(Γ) une unique mesure de probabilité d'entropie maximale, invariante sous l'action de (φ_t). Dans cet article, nous nous affranchissons de l'hypothèse de compacité de Λ(Γ) et étudions le cas où Γ est quelconque. Nous montrons que la restriction de (φ_t) à Λ(Γ) admet une mesure de probabilité d'entropie maximale si et seulement si la mesure de Patterson-Sullivan est finie; de plus, lorsque la mesure de Patterson-Sullivan est finie, c'est l'unique mesure d'entropie maximale pour le flot géodésique.

Let Γ be a nonelementary Kleinian group acting on a Cartan-Hadamard manifold $\tilde{X}$; denote by Λ(Γ) the nonwandering set of the geodesic flow (φ_t) acting on the unit tangent bundle T¹($\tilde{X}$/Γ). When Γ is convex cocompact (i.e., Λ(Γ) is compact), the restriction of (φ_t) to Λ(Γ) is an Axiom A flow: therefore, by a theorem of Bowen and Ruelle, there exists a unique invariant measure on Λ(Γ) which has maximal entropy. In this paper, we study the case of an arbitrary Kleinian group Γ. We show that there exists a measure of maximal entropy for the restriction of(φ_t) to Λ(Γ) if and only if the Patterson-Sullivan measure is finite; furthermore when this measure is finite, it is the unique measure of maximal entropy.

By a theorem of Handel and Kitchens, the supremum of the measure-theoretic entropies equals the infimum of the entropies of the distances d on Λ(X); when Γ is geometrically finite, we show that this infimum is achieved by the Riemannian distance d on Λ(X).

The theta correspondence has been an important tool in studying cycles in locally symmetric spaces of orthogonal type. In this paper we establish for the orthogonal group O(p,2) an adjointness result between Borcherds's singular theta lift and the Kudla-Millson lift. We extend this result to arbitrary signature by introducing a new singular theta lift for O(p,q). On the geometric side, this lift can be interpreted as a differential character, in the sense of Cheeger and Simons, for the cycles under consideration.

We give a classification theorem for unital separable simple nuclear C^*-algebras with tracial topological rank zero which satisfy the universal coefficient theorem. Let A and B be two such C^*-algebras. We prove that A≌B if and only if

(K₀(A), K₀(A)₊, [1_A], K₁(A)) ≌ K₀(B), K₀(B)₊, [1_B], K₁(B)).

Modular compactifications of moduli spaces for polarized K3 surfaces are constructed using the tools of logarithmic geometry in the sense of Fontaine and Illusie. The relationship between these new moduli spaces and the classical minimal and toroidal compactifications of period spaces is discussed, and it is explained how the techniques of this paper yield models for the latter spaces over number fields. The paper also contains a discussion of Picard functors for log schemes and a logarithmic version of Artin's method for proving representability by an algebraic stack.

We correct a sign error in the published version of the paper.