Given a smooth projective curve X, we give effective very ampleness bounds for generalized theta divisors on the moduli spaces SU_X(r,d) and U_X(r,d) of semistable vector bundles of rank r and degree d on X with fixed, respectively, arbitrary, determinant.

We study the boundary rigidity problem for domains in Rⁿ: Is a Riemannian metric uniquely determined, up to an action of diffeomorphism fixing the boundary, by the distance function ρ_g(x,y) known for all boundary points x and y? It was conjectured by Michel [M] that this was true for simple metrics. In this paper, we first study the linearized problem that consists of determining a symmetric 2-tensor, up to a potential term, from its geodesic X-ray integral transform I_g. We prove that the normal operator N_g=I^*_gI_g is a pseudodifferential operator (ΨDO) provided that g is simple, find its principal symbol, identify its kernel, and construct a microlocal parametrix. We prove a hypoelliptic type of stability estimate related to the linear problem. Next, we apply this estimate to show that unique solvability of the linear problem for a given simple metric g, up to potential terms, implies local uniqueness for the nonlinear boundary rigidity problem near that g.

We study some local invariants attached via multiplier ideals to an effective divisor or ideal sheaf on a smooth complex variety. These jumping coefficients consist of an increasing sequence of positive rational numbers beginning with the log-canonical threshold of the divisor or ideal in question. They encode interesting geometric and algebraic information, and we see that they arise naturally in several different contexts.

It is well known that on any given hyperbolic surface of finite area, a closed horocycle of length ℓ becomes asymptotically equidistributed as ℓ→∞. In this paper we prove that any subsegment of length greater than ℓ^{1/2 + ε} of such a closed horocycle also becomes equidistributed as ℓ→∞. The exponent 1/2 + ε is the best possible and improves upon a recent result by Hejhal [He3]. We give two proofs of the above result; our second proof leads to explicit information on the rate of convergence. We also prove a result on the asymptotic joint equidistribution of a finite number of distinct subsegments having equal length proportional to ℓ.

Given a Lie groupoid G over a manifold M, we show that multiplicative 2-forms on G relatively closed with respect to a closed 3-form ϕ; on M correspond to maps from the Lie algebroid of G into T^*M satisfying an algebraic condition and a differential condition with respect to the ϕ-twisted Courant bracket. This correspondence describes, as a special case, the global objects associated to ϕ-twisted Dirac structures. As applications, we relate our results to equivariant cohomology and foliation theory, and we give a new description of quasi-Hamiltonian spaces and group-valued momentum maps.

In this paper, we study the problem of restricting a square integrable representation of a connected semisimple Lie group to a reductive subgroup. Using a geometric method of restricting sections of a vector bundle to a submanifold, we obtain information about both the discrete and the continuous spectrum. We also show the (L²,L²)-continuity of the associated Berezin transform and that, under suitable general conditions, the Berezin transform is (L²,L²)-continuous for 1≤p≤∞