conjugacy classes of g 2 and spectral rigidity the free group 2 is generated by a 1 2 0 1 and b 10 21 and setting x a exp 2 i x b exp 2 i defines a unitary character on 2 for 0 1 a program is devised to compute tr conj class summed over all primitive conjugacy classes of 2 of trace tr combined with a luo sarnak theorem this yields lower bounds for the spectral variance for a large sampling of characters in the range 0 1 the results indicate that the berry conjecture for spectral rigidity does not hold for this set of classically chaotic systems the program is also used to compute x ln n summed over all primitive conjugacy classes of 2 of norm n x the function 0 x is asymptotic to x and the remainder can be written as x x x the values of x are computed for all traces between 3202 and 4802 here x tr 2 2 the s cluster around 0 6 attaining a maximum of 2 3 finally it is proved that the remainder 0 x x has a negative bias by showing that the mean normalized remainder converges to a negative limit

the second moment of gl 3 times gl 2 l functions at special points for a fixed sl 3 mathbb z maass form phi we consider the family of l functions l phi times u j s where u j runs over the family of hecke maass cusp forms on sl 2 mathbb z we obtain an estimate for the second moment of this family of l functions at the special points frac 1 2 it j consistent with the lindelof hypothesis we also obtain a similar upper bound on the sixth moment of the family of hecke maass cusp forms at these special points this is apparently the first occurrence of a lindelof consistent estimate for a sixth power moment of a family of gl 2 l functions

prime geodesic theorem for the picard manifold abstract let psl 2 z i be the picard group and h 3 be the three dimensional hyperbolic space we study the prime geodesic theorem for the quotient h 3 called the picard manifold obtaining an error term of size o x 3 2 2 where denotes a subconvexity exponent for quadratic dirichlet l functions defined over gaussian integers

stochastic calculus of variations abstract a theory of stochastic calculus of variations is presented which generalizes the ordinary calculus of variations to stochastic processes generalizations of the euler equation and noether s theorem are obtained and several conservation laws are discussed an application to nelson s probabilistic framework of quantum mechanics is also given

extension of chronological calculus for dynamical systems on manifolds abstract we propose an extension of the chronological calculus developed by agrachev and gamkrelidze for the case of c smooth dynamical systems on finite dimensional c smooth manifolds to the case of c m smooth dynamical systems and infinite dimensional c m manifolds due to a relaxation in the underlying structure of the calculus this extension provides a powerful computational tool without recourse to the theory of calculus in frechet spaces required by the classical chronological calculus in addition this extension accounts for flows of vector fields which are merely measurable in time to demonstrate the utility of this extension we prove a variant of chow rashevskii theorem for infinite dimensional manifolds

extremal properties of the determinant of the laplacian in the bergman metric on the moduli space of genus two riemann surfaces we study extremal properties of the determinant of the laplacian in the bergman metric on the moduli space of compact genus two riemann surfaces by a combination of analytical and numerical methods we identify four non degenerate critical points of this function and compute the signature of the hessian at these points the curve with the maximal number of automorphisms the burnside curve turns out to be the point of the absolute maximum our results agree with the mass formula for virtual euler characteristics of the moduli space a similar analysis is performed for bolza s strata of symmetric riemann surfaces of genus two

simple geodesics and a series constant over teichmuller space we investigate the birman series set in a neighborhood of a cusp on a punctured surface showing that it is homeomorphic to a cantor set union countably many isolated points cross a line the local topology of the cantor set is shown to be related in a simple way to the global behavior of simple geodesics from this we deduce that a certain series is constant across the teichmuller space

the quantum unique ergodicity conjecture for thin sets abstract we consider some analogs of the quantum unique ergodicity conjecture for geodesics and shrinking families of sets in particular we prove the analog of the que conjecture for eisenstein series restricted to the infinite geodesic connecting 0 and inside the modular surface

a sum formula for a pair of closed geodesics on a hyperbolic surface we consider an arbitrary pair of closed geodesics and the corresponding period integrals for the eigenfunctions of the laplacian on a compact hyperbolic surface a summation formula that relates geometric information about the geodesics namely the angles of intersection and lengths of common perpendiculars between them to the period integrals is proved as a corollary an asymptotic is obtained for the second moment of the period integrals for a fixed geodesic as an average over the eigenvalue with an error term that can be interpreted in terms of the geometric data

rankin selberg method for real analytic cusp forms of arbitrary real weight 1 2 indeed rankin and selberg applied their method to the more general situation of holomorphic cusp forms of type f r v where f is of finite index in f r 0 is the weight and v is a compatible multiplier system let f z be such a cusp form f f m and m1 mm be representatives of distinct right cosets then the vector valued function f z clz d l f m l z cruz d rf mmz t means transpose c dl is the second row of m is of type f r e where e is a unitary m dimensional multiplier system so the concept of vector valued automorphic forms for f is intimately related to the investigation of automorphic forms for subgroups of finite index in v cf 13

lagrangians of stochastic mechanics the stochastic variational principle which applies to diffusions and extremises the mean value of the classical action is generalised to include non markovian diffusions a whole family of different stochastic actions labelled by a time dependent parameter is obtained and all of them are shown to lead to the shrodinger equation as a result the well known degeneracy of the class of stochastic processes which correspond to a single quantum state is recovered it is also related to an arbitrariness in the choice of the noise which can occur in two places in the definition of the noise through the time dependent parameter and in the auto correlations of the noise which although constant during the variational method remain undetermined the whole scheme is given a covariant form

a variational principle for the navier stokes equation abstract motivated from arnold s variational characterization of the euler equation in terms of geodesic families of diffeomorphisms a variational principle for the motion of incompressible viscous fluids is presented a volume preserving diffusion process with drift velocity field subject to the navier stokes equation is shown to extremize the energy functional of the fluid under a certain class of stochastic variations

quantum mechanics and stochastic control theory a time symmetric stochastic control theory is proposed as one of the representatives of quantum mechanics the main idea is based on nelson s probability theoretical approach to quantum mechanics his approach is reformulated as a time symmetric stochastic control problem several different control constraints equivalent to nelson s are obtained one of them has a close connection to the lagrangian formalism of classical mechanics this suggests to us the use of stochastic calculus of variations within the realm of this time symmetric stochastic control theory it is shown why schrodinger s original variational method of quantization was successful several advantageous points of the stochastic control theoretical approach to quantum mechanics including the analysis of the classical limit are also discussed

local average in hyperbolic lattice point counting with an appendix by niko laaksonen the hyperbolic lattice point problem asks to estimate the size of the orbit gamma z inside a hyperbolic disk of radius cosh 1 x 2 for gamma a discrete subgroup of hbox psl 2 mathbb r selberg proved the estimate o x 2 3 for the error term for cofinite or cocompact groups this has not been improved for any group and any center in this paper local averaging over the center is investigated for hbox psl 2 mathbb z the result is that the error term can be improved to o x 7 12 varepsilon the proof uses surprisingly strong input e g results on the quantum ergodicity of maas cusp forms and estimates on spectral exponential sums we also prove omega results for this averaging consistent with the conjectural best error bound o x 1 2 varepsilon in the appendix the relevant exponential sum over the spectral parameters is investigated

genericity results for singular curves let m be a smooth manifold and cal d m m geq 2 be the set of rank m distributions on m endowed with the whitney c infty topology we show the existence of an open set o m dense in cal d m so that every nontrivial singular curve of a distribution d of o m is of minimal order and of corank one in particular for m geq 3 every distribution of o m does not admit nontrivial rigid curves as a consequence for generic sub riemannian structures of rank greater than or equal to three there does not exist nontrivial minimizing singular curves

growth of weil petersson volumes and random hyperbolic surfaces of large genus in this paper we investigate the geometric properties of random hyperbolic surfaces of large genus we describe the relationship between the behavior of lengths of simple closed geodesics on a hyperbolic surface and properties of the moduli space of such surfaces first we study the asymptotic behavior of weil petersson volume vg n of the moduli spaces of hyperbolic surfaces of genus g with n punctures as g then we discuss basic geometric properties of a random hyperbolic surface of genus g with respect to the weil petersson measure as g

number variance for arithmetic hyperbolic surfaces we prove that the number variance for the spectrum of an arithmetic surface is highly nonrigid in part of the universal range in fact it is close to having a poisson behavior this fact was discovered numerically by schmit bogomolny georgeot and giannoni it has its origin in the high degeneracy of the length spectrum first observed by selberg

extremals of determinants of laplacians on etudie le determinant associe au laplacien en fonction de la metrique sur une surface donnee et en particulier ses valeurs extremes quand la metrique est bien restreinte

generalizations of mcshane s identity to hyperbolic cone surfaces we generalize mcshane s identity for the length series of simple closed geodesics on a cusped hyperbolic surface to a general identity for hyperbolic cone surfaces with all cone angles possibly with cusps and or geodesic boundary the general identity is obtained by studying gaps formed by simple normal geodesics emanating from a distinguished cone point cusp or boundary geodesic in particular by applying the generalized identity to the quotient orbifolds of a hyperbolic one cone one hole torus by its elliptic involution and of a hyperbolic closed genus two surface by its hyperelliptic involution we obtain general weierstrass identities for the one cone one hole torus and an identity for the genus two surface which are also obtained by mcshane using different methods we also give an interpretation of the general identity in terms of complex lengths of the cone points cusps and geodesic boundary components

from classical periodic orbits to the quantization of chaos we present a quantitative analysis of selberg s trace formula viewed as an exact version of gutzwiller s semiclassical periodic orbit theory for the quantization of classically chaotic systems two main applications of the trace formula are discussed in detail i the periodic orbit sum rules giving a smoothing of the quantal energy level density ii the selberg zeta function as a prototype of a dynamical zeta function defined as an euler product over the classical periodic orbits and its analytic continuation across the entropy barrier by means of a dirichlet series it is shown how the long periodic orbits can be effectively taken into account by a universal remainder term which is explicitly given as an integral over an orbit selection function numerical results are presented for the free motion of a point particle on compact riemann surfaces hadamard gutzwiller model which is the primary testing ground for our ideas relating quantum mechanics and classical mechanics in the case of strong chaos our results demonstrate clearly the crucial role played by the long periodic orbits an exact rule for quantizing chaos is derived for such systems where the dirichlet series representing the selberg zeta function converges on the critical line explicit formulae are given for the computation of the abscissae of absolute and conditional convergence respectively of these dynamical dirichlet series for the two riemann surfaces considered it turns out that one can cross the entropy barrier but that the critical line cannot be reached by a convergent dirichlet series it would seem that this is the main reason why the riemann siegel lookalike formula recently conjectured by m v berry and j p keating fails in generating the lower lying quantal energies for these strongly chaotic systems

noether s theorem in symmetric stochastic calculus of variations quite recently a symmetric stochastic calculus of variations was proposed to formulate canonical stochastic dynamics which is an extension of nelson s stochastic mechanics in this article a noether s theorem is formulated within this calculus of variations conservation laws of momentum angular momentum and energy are proved which are related with the same laws in quantum mechanics

prime geodesic theorem in the 3 dimensional hyperbolic space for a cofinite kleinian group acting on h3 we study the prime geodesic theorem on m h3 which asks about the asymptotic behavior of lengths of primitive closed geodesics prime geodesics on m let e x be the error in the counting of prime geodesics with length at most log x for the picard manifold psl 2 z i we improve the classical bound of sarnak e x o x5 3 e to e x o x13 8 e in the process we obtain a mean subconvexity estimate for the rankin selberg l function attached to maass hecke cusp forms we also investigate the second moment of e x for a general cofinite group and we show that it is bounded by o x16 5 e

perturbative series and the moduli space of riemann surfaces on utilise des techniques de la theorie quantique des champs pour calculer des quantites reliees aux groupes de symetrie de paires f g ou f est une surface et g une epine de f

the hexagonal versus the square lattice schmutz schaller s conjecture regarding the lengths of the hexagonal versus the lengths of the square lattice is shown to be true the proof makes use of results from computational prime number theory using an identity due to selberg it is shown that in principle the conjecture can be resolved without using computational prime number theory by our approach however this would require a huge amount of computation

bounds on faltings s delta function through covers let x be a compact riemann surface of genus gx 1 in 1984 g faltings introduced a new invariant fal x associated to x in this paper we give explicit bounds for fal x in terms of fundamental dierential geometric invariants arising from x when gx 1 as an application we are able to give bounds for faltings s delta function for the family of modular curves x0 n in terms of the genus only in combination with work of a abbes p michel and e ullmo this leads to an asymptotic formula for the faltings height of the jacobian j0 n associated to x0 n

maupertuis principle of least action in stochastic calculus of variations within the framework of stochastic calculus of variations for time symmetric semimartingales x t we consider two different stochastic versions of maupertuis least action principle in lagrangian and hamiltonian terms the general results are applied to classical statistical mechanics where they coincide with those of classical calculus of variations and to nelson s stochastic mechanics an approach to quantum mechanics where a time symmetric semimartingale represents the position of a particle and the dynamics is expressed by a stochastic version of hamilton s principle of least action some historic examples of old quantum theory are discussed

on the set of points of smoothness for the value function of affine optimal control problems we study the regularity properties of the value function associated with an affine optimal control problem with quadratic cost plus a potential for a fixed final time and initial point without assuming any condition on singular minimizers we prove that the value function is continuous on an open and dense subset of the interior of the attainable set as a byproduct we obtain that it is actually smooth on a possibly smaller set still open and dense

moduli space heights and isospectral sets of plane domains let q be a plane domain of finite connectivity n with smooth boundary and choose a fixed domain 2 of the same type then there exists a flat metric g on 2 such that q is isometric with 29g in what follows we do not distinguish between isometric domains by the spectrum of 29 we mean the spectrum of the laplace beltrami operator ag on 29 with dirichlet boundary conditions the height h eg log det ag is a spectral invariant and plays a central role in this paper among all suitably normalized flat metrics on 2 conformal to a given metric g there is a unique flat metric for which the height is a minimum this metric is characterized by the fact that d 2 has constant geodesic curvature we call such a metric uniform and denote it by u the set of all such metrics is denoted by u 2 we can therefore identify with the moduli space 2 of conformal structures on e for n 3 we introduce a special parametrization for mu 2 by means of which we show that h u oo as u approaches the boundary of fu 2 using this along with the heat invariants for the laplacian we then show that any isospectral set of plane domains is compact in the c topology similar results hold for n 1 and 2

chebyshev s bias for composite numbers with restricted prime divisors let x d a denote the number of primes p x with p a modd chebyshev s bias is the phenomenon for which more often x d n x d r than the other way around where n is a quadratic nonresidue mod d and r is a quadratic residue mod d if x d n x d r for every x up to some large number then one expects that n x d n n x d r for every x here n x d a denotes the number of integers n x such that every prime divisor p of n satisfies p a mod d in this paper we develop some tools to deal with this type of problem and apply them to show that for example n x 4 3 n x 4 1 for every x in the process we express the so called second order landau ramanujan constant as an infinite series and show that the same type of formula holds for a much larger class of constants

the sard conjecture on martinet surfaces given a totally nonholonomic distribution of rank two on a three dimensional manifold we investigate the size of the set of points that can be reached by singular horizontal paths starting from a same point in this setting the sard conjecture states that that set should be a subset of the so called martinet surface of 2 dimensional hausdorff measure zero we prove that the conjecture holds in the case where the martinet surface is smooth moreover we address the case of singular real analytic martinet surfaces and show that the result holds true under an assumption of non transversality of the distribution on the singular set of the martinet surface our methods rely on the control of the divergence of vector fields generating the trace of the distribution on the martinet surface and some techniques of resolution of singularities

singular trajectories of control affine systems when applying methods of optimal control to motion planning or stabilization problems we see that some theoretical or numerical difficulties may arise due to the presence of specific trajectories namely minimizing singular trajectories of the underlying optimal control problem in this article we provide characterizations for singular trajectories of control affine systems we prove that under generic assumptions such trajectories share nice properties related to computational aspects more precisely we show that for a generic system with respect to the whitney topology all nontrivial singular trajectories are of minimal order and of corank one these results established both for driftless and for control affine systems extend results of y chitour f jean and e trelat comptes rendus math 337 2003 pp 49 52 in french y chitour f jean and e trelat j differential geom 73 2006 pp 45 73 as a consequence for generic control affine systems with or without drift defined by more than two vector fields and for a fixed cost there do not exist minimizing singular trajectories besides we prove that given a control affine system satisfying the lie algebra rank condition larc singular trajectories are strictly abnormal generically with respect to the cost we then show how these results can be used to derive regularity results for the value function and in the theory of hamilton jacobi equations which in turn have applications for stabilization and motion planning from both theoretical and implementational points of view

canonical stochastic dynamical systems a canonical stochastic dynamical system for time symmetric semimartingales is formulated by the stochastic least action principle in a new stochastic calculus of variations a certain class of stochastic dynamical systems gives a hamiltonian formalism of nelson s stochastic mechanics in a manner analogous to classical mechanics the notions of a stochastic poisson bracket and canonical transformation are introduced to the stochastic dynamical systems it is shown that the phase factor of the wave function plays the role of a generating function of the canonical transformation

the euler characteristic of the moduli space of curves let fg 1 g 1 be the mapping class group consisting of all isotopy classes of base point and orientation preserving homeomorphisms of a closed oriented surface f of genus g let 1 be its euler characteristic in the sense of wall that is z f i fgl f l z e f where f is any torsion free subgroup of finite index in f 1 and e is a contractible space on which f acts freely and properly discontinuously an example of such a space is the teichmiiller space 1 and g f can be interpreted as the orbifold euler characteristic of f i join the moduli space of curves of genus g with base point the purpose of this paper is to prove the following formula for f i

a geometric characterization of arithmetic fuchsian groups the trace set of a fuchsian group encodes the set of lengths of closed geodesics in the surface h luo and sarnak showed that the trace set of a cofinite arithmetic fuchsian group satisfies the bounded clustering property sarnak then conjectured that the b c property actually characterizes arithmetic fuchsian groups schmutz stated the even stronger conjecture that a cofinite fuchsian group is arithmetic if its trace set has linear growth he proposed a proof of this conjecture in the case when the group contains at least one parabolic element but unfortunately this proof contains a gap in the present paper we point out this gap and we prove sarnak s conjecture under the assumption that the fuchsian group contains parabolic elements

a dynamical approach to the sard problem in carnot groups abstract we introduce a dynamical systems approach for the study of the sard problem in sub riemannian carnot groups we show that singular curves can be obtained by concatenating trajectories of suitable dynamical systems as an application we positively answer the sard problem in some classes of carnot groups

morse sard type results in sub riemannian geometry let m g be a sub riemannian manifold and x0 m assuming that chow s condition holds and that m endowed with the sub riemannian distance is complete we prove that there exists a dense subset n1 of m such that for every point x of n1 there is a unique minimizing path steering x0 to x this trajectory admitting a normal extremal lift if the distribution is everywhere of corank one we prove the existence of a subset n2 of m of full lebesgue measure such that for every point x of n2 there exists a minimizing path steering x0 to x which admits a normal extremal lift is nonsingular and the point x is not conjugate to x0 in particular the image of the sub riemannian exponential mapping is dense in m and in the case of corank one is of full lebesgue measure in m

injectivity radius and gonality of a compact riemann surface we obtain a sharp lower bound for the volumes of purely 1 dimensional complex analytic subvarieties in a geodesic tubular neighborhood of the diagonal of the cartesian product of a compact riemann surface with itself this leads to a lower bound of the seshadri number of the canonical line bundle of the cartesian product with respect to the diagonal as a consequence we obtain an upper bound for the hyperbolic injectivity radii of compact riemann surfaces of a fixed gonality in particular we obtain the limiting behavior of the gonalities of a tower of compact riemann surfaces we also give an application of our results to an invariant related to the ample cone of the symmetric product of a riemann surface

stochastic embedding of dynamical systems most physical systems are modeled by an ordinary or a partial differential equation like the n body problem in celestial mechanics in some cases for example when studying the long term behavior of the solar system or for complex systems there exist elements which can influence the dynamics of the system which are not well modeled or even known one way to take these problems into account consists of looking at the dynamics of the system on a larger class of objects that are eventually stochastic in this paper we develop a theory for the stochastic embedding of ordinary differential equations we apply this method to lagrangian systems in this particular case we extend many results of classical mechanics namely the least action principle the euler lagrange equations and noether s theorem we also obtain a hamiltonian formulation for our stochastic lagrangian systems many applications are discussed at the end of the paper