quantum monodromy and nonconcentration near a closed semi hyperbolic orbit for a large class of semiclassical operators p h z which includes schrodinger operators on manifolds with boundary we construct the quantum monodromy operator m z associated to a periodic orbit of the classical flow using estimates relating m z and p h z we prove semiclassical estimates for small complex perturbations of p h z in the case is semi hyperbolic as our main application we give logarithmic lower bounds on the mass of eigenfunctions away from semi hyperbolic orbits of the associated classical flow as a second application of the monodromy operator construction we prove if is an elliptic orbit then p h admits quasimodes which are well localized near

the principal series for a reductive symmetric space ii eisenstein integrals in this paper we develop a theory of eisenstein integrals related to the principal series for a reductive symmetric space g h here g is a real reductive group of harish chandra s class an involution of g and h an open subgroup of the group g of xed points for the group g itself is a symmetric space for the left right action of g g we refer to this setting as the group case up to a normalization our eisenstein integrals generalize those of harish chandra 18 associated with a minimal parabolic subgroup in the group case

a class of polynomial planar vector fields with polynomial first integral abstract we give an algorithm for deciding whether a planar polynomial differential system has a first integral which factorizes as a product of defining polynomials of curves with only one place at infinity in the affirmative case our algorithm computes a minimal first integral in addition we solve the poincare problem for the class of systems which admit a polynomial first integral as above in the sense that the degree of the minimal first integral can be computed from the reduction of singularities of the corresponding vector field

asymptotic expansions and boundary values of eigenfunctions on riemannian symmetric spaces soit x g k un espace de riemann symetrique de type non compacte a bord b k m et soit d x l algebre de tous les operateurs differentiels g invariants sur x on obtient des developpements asymptotiques pour une grande classe de fonctions jointives f sur x

geometric bounds on the density of resonances for semiclassical problems in this paper we shall give upper bounds on the number of resonances in certain regions in the complex plane close to the real axis for semiclassical operators like h 2 v x when h is small

cones of curves and of line bundles at infinity abstract we consider pencils at infinity v f z d in the projective plane p 2 there exists a minimal composition of point blowing ups x v p 2 eliminating the indeterminacies of the rational map p 2 p 1 induced by v we systematically study the mordell weil group of v the cone of curves ne x v q of x v and their relationship in theorem 3 we explicitly compute both objects when the projective curve defined by f is a union of curves with one place at infinity we furthermore ask two questions on pencils in p 2 our question 2 is answered in the affirmative in a particular case theorem 4 which can be viewed as a version at infinity but for pencils of any degree of a classical result on cubic pencils

the total coordinate ring of a smooth projective surface in 5 cox introduced the homogeneous coordinate ring of a toric variety which is a polynomial ring that allows to show that such a variety behaves like a projective space in many ways an analogous to that ring can also be defined for a smooth projective variety x over an algebraically closed field k such that linear and numerical equivalence coincide for divisors on x condition which is assumed for all varieties considered in this paper indeed let us fix li i 1 a z basis of pic x and set n n1 n2 nr z and dn ri 1 nili by regarding the vector spaces h 0 x ox dn f k divx f dn 0 0 as k subvector spaces of the function field k of x the total coordinate ring of x or the cox ring of x is defined as the graded k subalgebra of k

cones of curves and of line bundles on surfaces associated with curves having one place at infinity let v be a pencil of curves in bf p 2 with one place at infinity and x longrightarrow bf p 2 the minimal composition of point blow ups eliminating its base locus we study the cone of curves and the cones of numerically effective and globally generated line bundles on x it is proved that all of these cones are regular in particular this result provides a new class of rational projective surfaces with a rational polyhedral cone of curves the surfaces in this class have non numerically effective anticanonical sheaf if the pencil is neither rational nor elliptic an application is a global version on x of zariski s unique factorization theorem for complete ideals we also define invariants of the semigroup of globally generated line bundles on x depending only on the topology of v at infinity 2000 mathematical subject classification primary 14c20 secondary 14e05

fourier inversion on a reductive symmetric space let x be a semisimple symmetric space in previous papers 8 and 9 we have dened an explicit fourier transform for x and shown that this transform is injective on the space c 1 c x ofcompactly supported smooth functions on x in the present paper which is a continuation of these papers we establish an inversion formula for this transform

the plancherel decomposition for a reductive symmetric space i spherical functions we prove the plancherel formula for spherical schwartz functions on a reductive symmetric space our starting point is an inversion formula for spherical smooth compactly supported functions the latter formula was earlier obtained from the most continuous part of the plancherel formula by means of a residue calculus in the course of the present paper we also obtain new proofs of the uniform tempered estimates for normalized eisenstein integrals and of the maass selberg relations satisfied by the associated c functions

local boundary data of eigenfunctions on a riemannian symmetric space let x g k be a riemannian symmetric space of non compact type and let d x be the algebra of invariant differential operators on x in a previous paper 1 we developed a theory of asymptotic expansions for joint eigenfunctions of d x of at most exponential growth in the present paper we show that local asymptotic data determine the eigenfunctions completely we also develop a theory of asymptotic expansions along walls both results are of importance for the theory of the discrete series of a semisimple symmetric space let a be a maximal abelian split subspace of the lie algebra 9 of g and denote by a its complexified dual let e be the restricted root system of a in g and w the associated weyl group then harish chandra s isomorphism d x s a w determines a bijection 2 z from a w to the set d x of algebra homomorphisms d x c given 2ca let ga x denote the joint eigenspace of functions f e c o x satisfying

trace formulae for three dimensional hyperbolic lattices and application to a strongly chaotic tetrahedral billiard abstract this paper is devoted to the quantum chaology of three dimensional systems a trace formula is derived for compact polyhedral billiards which tessellate the three dimensional hyperbolic space of constant negative curvature the exact trace formula is compared with gutzwiller s semiclassical periodic orbit theory in three dimensions and applied to a tetrahedral billiard being strongly chaotic geometric properties as well as the conjugacy classes of the defining group are discussed the length spectrum and the quantal level spectrum are numerically computed allowing the evaluation of the trace formula as is demonstrated in the case of the spectral staircase n e which in turn is successfully applied in a quantization condition

the plancherel decomposition for a reductive symmetric space ii representation theory we obtain the plancherel decomposition for a reductive symmetric space in the sense of representation theory our starting point is the plancherel formula for spherical schwartz functions obtained in part i the formula for schwartz functions involves eisenstein integrals obtained by a residual calculus in the present paper we identify these integrals as matrix coefficients of the generalized principal series

on the computation of darboux first integrals of a class of planar polynomial vector fields abstract we study the class of planar polynomial vector fields admitting darboux first integrals of the type i 1 r f i i where the i s are positive real numbers and the f i s are polynomials defining curves with only one place at infinity we show that these vector fields have an extended reduction procedure and give an algorithm which from a part of the extended reduction of the vector field computes a darboux first integral for generic exponents

meromorphic plane curves let f x y be a reduced polynomial of c x y and consider the family f c in c x y where for any c f f is a reduced polynomial of c x y let fx fy be the derivatives of f with respect to x and y and assume that f has only one place at infinity i e the projective curve associated to f has only one point at the line at infinity and it is analytically irreducible at this point in 6 s s abhyankar and t t moh have proved the following

plongements de series discretes pour un espace symetrique reductif resume using the realization of a discrete series a of a semi simple symmetric space g h as a space of boundary values in the dual space gd kd we construct an explicit embedding of a in a principal series ind 1 associated to an open subgroup q1 of a minimal stable parabolic subgoup q of g the construction of q1 and uses the properties of what we call a fundamental system of roots

the cone of curves and the cox ring of rational surfaces given by divisorial valuations abstract we consider surfaces x defined by plane divisorial valuations of the quotient field of the local ring r at a closed point p of the projective plane p 2 over an arbitrary algebraically closed field k and centered at r we prove that the regularity of the cone of curves of x is equivalent to the fact that is non positive on o p 2 p 2 l where l is a certain line containing p under these conditions we characterize when the characteristic cone of x is closed and its cox ring finitely generated equivalent conditions to the fact that is negative on o p 2 p 2 l k are also given

support d asymptotiques et croissance des fonctions propres sur un espace symetrique semi simple abstract we associate several distribution boundary values to an eigenfunction with moderate growth on a riemannian symmetric space g k the associated character of the algebra d g k of invariant differential operators is allowed to be non regular we prove results on the support of these boundary values these allow us to recover the theorems of matsuki oshima and oshima on the equivalence between growth of an eigenfunction and limitations on the supports of its boundary values our approach is based on an asymptotic analysis that makes no use of hyperfunction theory

local smoothing for the schrodinger equation with a prescribed loss we consider a family of surfaces of revolution each with a single periodic geodesic which is degenerately unstable we prove a local smoothing estimate for solutions to the linear schrodinger equation with a loss that depends on the degeneracy and we construct explicit examples to show our estimate is saturated on a weak semiclassical time scale as a byproduct of our proof we obtain a cutoff resolvent estimate with a sharp polynomial loss