the graph of the generating d tuples of a finite soluble group and the swap conjecture abstract for a d generated group g we consider the graph d g in which the vertices are the ordered generating d tuples and in which two vertices x 1 x d and y 1 y d are adjacent if and only if there exists i 1 d such that i d 2 and x i y i for each i i we prove that if g is a finite soluble then d g is connected we consider also the swap graph d g in which two generating d tuples are adjacent if they differ only by one entry proving the following if g is an arbitrary finite group and d d g 1 then d g is connected

on the period spectrum of a symplectic mapping let m be a compact riemannian manifold it has been known for a long time that the singularities of the wave trace trace cos t are located at the periods of the closed geodesics do these singularities also contain information about the geometry of m in the neighborhood of a closed geodesic we prove that the birkhoff canonical form of the poincare map can be determined from the singularities of the wave trace

second variation and stabilities of minimal lagrangian submanifolds in kahler manifolds first we derive a new second variation formula which holds for any minimal lagrangian submanifold in k ihler manifolds using this we prove that any minimal lagrangian submanifold in kfihler manifolds with nonpositive chern class is stable we also prove that in kfihler manifolds with positive chern class any stable minimal lagrangian submanifold l satisfies that h l 0 next we introduce the notion of hamiltonian stability either local or global of minimal lagrangian submanifolds in einstein kfihler manifolds using our second variation formula we find a general criterion for the local hamiltonian stability for minimal lagrangian submanifolds in einstein k ihler manifolds this criterion is satisfied for all real forms which are einstein of irreducible hermitian symmetric spaces e g the cannonical rp c cp it is also satisfied by the clifford torus t c cp

profinite groups in which the probabilistic zeta function coincides with the subgroup zeta function abstract we examine finitely generated profinite groups in which two formal dirichlet series the subgroup zeta function and the probabilistic zeta function coincide we call these groups reversible in the class of prosolvable groups of finite rank we show some sufficient conditions for this property to hold and we produce a structural characterisation of torsion free prosolvable groups of rank two which are reversible for pro p groups several results support the conjecture that reversibility is equivalent to the property that every open subgroup has the same minimal number of generators of the group itself in particular this holds for powerful pro p groups

on boundedly generated subgroups of profinite groups abstract in this paper we investigate the following general problem let g be a group and let i g be a property of g is there an integer d such that g contains a d generated subgroup h with i h i g here we consider the case where g is a profinite group and h is a closed subgroup extending earlier work of lucchini and others on finite groups for example we prove that d 3 if i g is the prime graph of g which is best possible and we show that d 2 if i g is the exponent of a finitely generated prosupersolvable group g

minimal and random generation of permutation and matrix groups we prove explicit bounds on the numbers of elements needed to generate various types of finite permutation groups and finite completely reducible matrix groups and present examples to show that they are sharp in all cases the bounds are linear in the degree of the permutation or matrix group in general and logarithmic when the group is primitive they can be combined with results of lubotzky to produce explicit bounds on the number of random elements required to generate these groups with a specified probability these results have important applications to computational group theory our proofs are inductive and largely theoretical but we use computer calculations to establish the bounds in a number of specific small cases

angle variable holonomy in adiabatic excursion of an integrable hamiltonian if an integrable classical hamiltonian h describing bound motion depends on parameters which are changed very slowly then the adiabatic theorem states that the action variables i of the motion are conserved here the fate of the angle variables is analysed because of the unavoidable arbitrariness in their definition angle variables belonging to distinct initial and final hamiltonians cannot generally be compared however they can be compared if the hamiltonian is taken on a closed excursion in parameter space so that initial and final hamiltonians are the same the result shows that the angle variable change arising from such an excursion is not merely the time integral of the instantaneous frequency omega dh di but differs from it by a definite extra angle which depends only on the circuit in parameter space not on the duration of the process the 2 form which describes this angle variable holonomy is calculated

coadjoint orbits vortices and clebsch variables for incompressible fluids this paper is a study of incompressible fluids especially their clebsch variables and vortices using symplectic geometry and the lie poisson structure on the dual of a lie algebra following ideas of arnold and others it is shown that euler s equations are lie poisson equations associated to the group of volume preserving diffeomorphisms the dual of the lie algebra is seen to be the space of vortices and kelvin s circulation theorem is interpreted as preservation of coadjoint orbits in this context clebsch variables can be understood as momentum maps the motion of n point vortices is shown to be identifiable with the dynamics on a special coadjoint orbit and the standard canonical variables for them are a special kind of clebsch variables point vortices with cores vortex patches and vortex filaments can be understood in a similar way this leads to an explanation of the geometry behind the hald beale majda convergence theorems for vorticity algorithms symplectic structures on the coadjoint orbits of a vortex patch and filament are computed and shown to be closely related to those commonly used for the kdv and the schrodinger equations respectively

classical adiabatic angles and quantal adiabatic phase a semiclassical connection is established between quantal and classical properties of a system whose hamiltonian is slowly cycled by varying its parameters round a circuit the quantal property is a geometrical phase shift gamma n associated with an eigenstate with quantum numbers n nl the classical property is a shift delta theta l i in the lth angle variable for motion round a phase space torus with actions i il the connection is delta theta l delta gamma delta nl two applications are worked out in detail the generalised harmonic oscillator with quadratic hamiltonian whose parameters are the coefficients of q2 qp and p2 and the rotated rotator consisting of a particle sliding freely round a non circular hoop slowly turned round once in its own plane

equivariant asymptotics for bohr sommerfeld lagrangian submanifolds suppose given a complex projective manifold m with a fixed hodge form the bohr sommerfeld lagrangian submanifolds of m are the geometric counterpart to semi classical physical states and their geometric quantization has been extensively studied here we revisit this theory in the equivariant context in the presence of a compatible hamiltonian action of a connected compact lie group

connections of berry and hannay type for moving lagrangian submanifolds abstract berry s phase is the holonomy of the natural connection on the canonical circle bundle over projectivized quantum hilbert space given a symplectic manifold p a classical limit of this phase is constructed as the holonomy of a berry connection over a finite codimensional isodrastic foliation defined by constancy of action integrals on the space of lagrangian submanifolds in p equipped with smooth densities of total measure 1 if the densities are determined by a kahler metric compatible with the symplectic structure the curvature of the berry connection at the lagrangian submanifold l is given by a simple formula involving curvature of l in particular the berry connection defines a homotopy invariant for isodrastic loops of minimal lagrangian submanifolds in a simply connected kahler manifold a similar invariant constructed by the author for loops of symplectomorphisms is also a special case of the classical berry phase a classical analogue of berry s phase was discovered by hannay for moving families of completely integrable systems following berry and hannay we interpret hannay s angles as the holonomy of a hannay connection on a bundle of tori over the isodrastic foliation on the space of lagrangian toral layers consisting of lagrangian tori with a flat affine structure and an extension of this structure to the first infinitesimal neighborhood finally we show that the hannay angles are the derivatives with respect to action variables of the classical berry phase

the local structure of poisson manifolds varietes de poisson et applications decomposition structures de poisson lineaires approximation lineaire systemes hamiltoniens le probleme de linearisation groupes de fonction realisations et applications impulsion paires duales et groupes de jauge existence des realisations unicite des realisations les problemes des 3 corps restreints et autres exemples

the coset poset and probabilistic zeta function of a finite group abstract we investigate the topological properties of the poset of proper cosets xh in a finite group g of particular interest is the reduced euler characteristic which is closely related to the value at 1 of the probabilistic zeta function of g our main result gives divisibility properties of this reduced euler characteristic

the diameter of the generating graph of a finite soluble group abstract let g be a finite 2 generated soluble group and suppose that a 1 b 1 a 2 b 2 g then there exist c 1 c 2 such that a 1 c 1 c 1 c 2 c 2 a 2 g equivalently the subgraph g of the generating graph of a 2 generated finite soluble group g obtained by removing the isolated vertices has diameter at most 3 we construct a 2 generated group g of order 2 10 3 2 for which this bound is sharp however a stronger result holds if g has odd order or g is nilpotent in this case there exists b g with a 1 b a 2 b g

deformation of singular lagrangian subvarieties we investigate deformations of lagrangian manifolds with singularities we introduce a complex similar to the de rham complex whose cohomology calculates deformation spaces this cohomology turns out to be constructible in many cases examples of singular lagrangian varieties are presented and deformations are calculated explicitly

counting homomorphisms onto finite solvable groups we present a method for computing the number of epimorphisms from a finitely presented group g to a finite solvable group which generalizes a formula of

classical non adiabatic angles if a family of tori in phase space is driven by a time dependent hamiltonian flow in such a way as to return after some time to the original family there generally results a shift in the angle variables one realisation of this process is in the cyclic adiabatic change of a classical hamiltonian and the angle change has previously been shown to separate naturally into a dynamical part and a geometrical part here the same geometrical angle change is extracted when the return is achieved non adiabatically and the dynamical remainder calculated two examples are given the precession of a spin and the rotation of phase space ellipses

cohomology of symplectomorphism groups and critical values of hamiltonians let p co be a compact simply connected symplectic manifold sym p co its group of automorphisms the period group f p co c r is defined to be the image of co x h2 p z under the integration pairing h 2 p r x h2 p z r since co 4 0 the quotient g p co r f p co is either a circle or the quotient of a circle by a countable dense subgroup the purpose of this paper is to define and study a natural homomorphism a from 7h sym p co to g p co simply stated a measures the average over p with respect to the symplectic measure of the action integral around the trajectories of a loop of symplectomorphisms in terms of hamiltonians we give a formula for a which becomes especially simple when the loop is a 1 parameter subgroup in which case we can make some concrete computations these are related to the work of guillemin sternberg 9 and duistermaat heckmann 7 on momentum mappings we also give an interpretation of a in terms of lifting symplectomorphisms from p to a prequantization i e a g p co bundle over p with a connection whose curvature form is co here we extend the theory from the usual case where g p co is a circle by using the concept of diffeological structure introduced by souriau 16 in a sequel to this paper 19 taking as a starting point the observation that the graphs of symplectomorphisms are lagrangian submanifolds in a product manifold we shall study the possible extension of a to an invariant on loops in the space of lagrangian submanifolds in a symplectic manifold it turns out that this invariant should be defined as the holonomy of a connection whose definition involves choosing a probability measure on each lagrangian submanifold the connection is flat only for certain ways of choosing these measures and the curvature in the general case is closely related to the geometric phase introduced by berry 4 which is a subject of current interest among theoretical and experimental physicists as well as mathematicians 1 6 i0 11 13 173

on the notion of phase in mechanics the notion of phase plays an essential role in both semiclassical and quantum mechanics but what is exactly a phase and how does it change with time it turns out that the most universal definition of a phase can be given in terms of lagrangian manifolds by exploiting the properties of the poincare cartan form such a phase is defined not in configuration space but rather in phase space and is thus insensitive to the appearance of caustics surprisingly enough this approach allows us to recover the heisenberg weyl formalism without invoking commutation relations for observables

bohr sommerfeld conditions for integrable systems with critical manifolds of focus focus type we present a detailed study in the semi classical regime h of microlocal properties of systems of two commuting h pseudo dierential operators ph ph such that the joint principal symbol p p p has a special kind of singularity called a focusfocus singularity typical examples include the quantum spherical pendulum or the quantum champagne bottle in the spirit of colin de verdiere and parisse we show that such systems have a universal behavior described by singular quantization conditions of bohr sommerfeld type these conditions are used to give a precise description of the joint spectrum of such systems including the phenomenon of quantum monodromy and dierent formulations of the counting function for the joint eigenvalues close to the singularity in which a logarithm of the semi classical constant h appears thanks to numerical computations done by ms child for the case of the champagne bottle we are able to accurately illustrate our statements

on geometric phases for soliton equations this paper develops a new complex hamiltonian structure forn soliton solutions for a class of integrable equations such as the nonlinear schrodinger sine gordon and korteweg de vries hierarchies of equations that yields amongst other things geometric phases in the sense of hannay and berry for example one of the possible soliton geometric phases is manifested by the well known phase shift that occurs for interacting solitons the main new tools are complex angle representations that linearize the corresponding hamiltonian flows on associated noncompact jacobi varieties this new structure is obtained by taking appropriate limits of the differential equations describing the class of quasi periodic solutions a method of asymptotic reduction of the angle representations is introduced for investigating soliton geometric phases that are related to the presence of monodromy at singularities in the space of parameters in particular the phase shift of interacting solitons can be expressed as an integral over a cycle on an associated riemann surface in this setting soliton geometric asymptotics are constructed for studying geometric phases in the quantum case the general approach is worked out in detail for the three specific hierarchies of equations mentioned some links with functions the braid group and geometric quantization are pointed out as well

d wise generation of prosolvable groups abstract let g be a topological group for 2 d n denote by d g the largest m for which there exists an m tuple of elements of g such that any of its d entries generate g topologically we obtain a lower bound for d g in the case when g is a prosolvable group our result implies in particular that if g is d generated then the difference d g d tends to infinity when the smallest prime divisor of the order of g tends to infinity one of the aim of the paper is to draw the attention to an intriguing question in linear algebra whose solution would allow to improve our bounds and determine the precise value for d g in several relevant cases for example when d 2 and g is a prosolvable group

the connection whose holonomy is the classical adiabatic angles of hannay and berry and its generalization to the non integrable case we show how averaging defines an ehresmann connection whose holonomy is the classical adiabatic angles which hannay defined for families of completely integrable systems the averaging formula we obtain for the connection only requires that the family of hamiltonians has a continuous symmetry group this allows us to extend the notion of the hannay angles to families of non integrable systems with symmetry we state three geometric axioms satisfied by the connection these axioms uniquely determine the connection thus enabling us to find new formulas for the connection and its curvature two examples are given

new class of random matrix ensembles with multifractal eigenvectors department of physics university of florida gainesville florida 32611 february 1 2008 three recently suggested random matrix ensembles rme are linked together to represent a class of rme with multifrac tal eigenfunction statistics the generic form of the two levelcorrelation function for the case of weak and extremely strongmultifractality is suggested pacs numbers 71 25 s 72 15 rn 05 45 b

crowns and factorization of the probabilistic zeta function of a finite group abstract we employ the concept of crown to study the generalized euler factors of the probabilistic zeta function of finite groups

quantal phase factors accompanying adiabatic changes a quantal system in an eigenstate slowly transported round a circuit c by varying parameters r in its hamiltonian r will acquire a geometrical phase factor exp i c in addition to the familiar dynamical phase factor an explicit general formula for c is derived in terms of the spectrum and eigenstates of r over a surface spanning c if c lies near a degeneracy of c takes a simple form which includes as a special case the sign change of eigenfunctions of real symmetric matrices round a degeneracy as an illustration c is calculated for spinning particles in slowly changing magnetic fields although the sign reversal of spinors on rotation is a special case the effect is predicted to occur for bosons as well as fermions and a method for observing it is proposed it is shown that the aharonov bohm effect can be interpreted as a geometrical phase factor

symplectic groupoids and poisson manifolds on definit la notion de groupoide symplectique on donne la variete de poisson de ce groupoide symplectique on presente une construction

the calabi homomorphism lagrangian paths and special lagrangians let mathcal o be an orbit of the group of hamiltonian symplectomorphisms acting on the space of lagrangian submanifolds of a symplectic manifold x omega we define a functional mathcal c mathcal o rightarrow mathbb r for each differential form beta of middle degree satisfying beta wedge omega 0 and an exactness condition if the exactness condition does not hold mathcal c is defined on the universal cover of mathcal o a particular instance of mathcal c recovers the calabi homomorphism if beta is the imaginary part of a holomorphic volume form the critical points of mathcal c are special lagrangian submanifolds we present evidence that mathcal c is related by mirror symmetry to a functional introduced by donaldson to study einstein hermitian metrics on holomorphic vector bundles in particular we show that mathcal c is convex on an open subspace mathcal o subset mathcal o as a prerequisite we define a riemannian metric on mathcal o and analyze its geodesics finally we discuss a generalization of the flux homomorphism to the space of lagrangian submanifolds and a lagrangian analog of the flux conjecture

invariable generation with elements of coprime prime power orders abstract a finite group g is coprimely invariably generated if there exists a set of generators g 1 g d of g with the property that the orders g 1 g d are pairwise coprime and that for all x 1 x d g the set g 1 x 1 g d x d generates g in the particular case when g 1 g d can be chosen to be prime powers we say that g is prime power coprimely invariably generated we will discuss these properties proving also that the second one is stronger than the first but that in the particular case of finite soluble groups they are equivalent