hitchin s self duality equations on complete riemannian manifolds let m be a complete riemannian manifold let g be a real semisimple noncompact algebraic group and let k be its maximal compact subgroup let p be a principal g bundle over m fix a reduction of the structure group g of p to k and consider the space d of smooth g connections on p d is an affine space modeled on the vector space 21 ad p of 1 forms on m with values in the adjoint bundle associated to p for any connection d e d we have a decomposition d d 0 where d is a connection preserving the k structure and 0 is a 1 form on m with values in the orthogonal complement in ad p of ad pk here px is the k bundle defined by the reduction of structure group i f d is a flat connection we have

calculation of energy spectrum and eigenstates of 1d time independent short range potentials abstract we show that it is possible to approximate 1d time independent short range potentials by a sum of function potentials by the use of transfer matrix techniques it is possible to calculate the total transfer matrix as well as the s matrix which connects the incoming waves to the outgoing waves the transmission coefficient and the resonance states can be evaluated by the function approximation using the same approach in potential wells the energy spectrum as well as the eigenfunctions of the well can be constructed we examine the approximation successfully on two well known potentials the square well and the harmonic oscillator

exact convergent periodic orbit expansions of individual energy eigenvalues of regular quantum graphs we present exact explicit convergent periodic orbit expansions for individual energy levels of regular quantum graphs in the paper one simple application is the energy levels of a particle in a piecewise constant potential since the classical ray trajectories including ray splitting in such systems are strongly chaotic this result provides an explicit quantization of a classically chaotic system

a fourier method solution for the time dependent schrodinger equation as a tool in molecular dynamics a new method is presented for the solution of the time dependent schrbdinger equation in its application to physical and chemical molecular phenomena the method is based on discretizing space and time on a grid and using the fourier method to produce both spatial derivatives and second order differencing for time derivatives the method conserves norm and energy and preserves quantum mechanical commutation relations one and twodimensional examples where a comparison to analytic results is possible are investigated many phenomena of physics and chemistry have a common dynamical evolution pattern this pattern begins with an initial state which under the influence of the potentials evolves through time to produce a final asymptotic state among such phenomena are chemical reactions photodissociation unimolecular breakdown surface scattering and desorption the present study presents a numerical solution based on quantum mechanics for the common pattern or for the time dependent evolution of the state of the system the importance of exact numerical solutions is twofold first these solutions give a quantitative description of physical problems and also are of use as a bench mark to check approximation methods and to define their range of validity

combinatorial identities for binary necklaces from exact ray splitting trace formulas based on an exact trace formula for a one dimensional ray splitting system we derive novel combinatorial identities for cyclic binary sequences polya necklaces

periodic orbit theory and the statistical analysis of scaling quantum graph spectra the explicit solution to the spectral problem of quantum graphs found recently by dabaghian and blumel phys rev e 68 055201 r 2003 70 046206 2004 jetp lett 77 530 2003 is used to produce an exact periodic orbit theory description for the probability distributions of spectral statistics including the distribution for the nearest neighbor separations sn kn kn 1 and the distribution of the spectral oscillations around the average deltakn kn kn

the self duality equations on a riemann surface in this paper we shall study a special class of solutions of the self dual yang mills equations the original self duality equations which arose in mathematical physics were defined on euclidean 4 space the physically relevant solutions were the ones with finite action the so called instantons the same equations may be dimensionally reduced to euclidean 3 space by imposing invariance under translation in one direction these equations also have physical relevance the solutions which have finite action in three dimensions are the magnetic monopoles if we take the reduction process one step further and consider solutions which are invariant under two translations we obtain a set of equations in the plane here however there is no clear physical meaning and indeed attempts to find finite action solutions have failed nevertheless these are the equations we shall consider despite the lack of interesting solutions in r2 the equations have the important property conformal invariance which allows them to be defined on manifolds modelled on r2 by conformal maps namely riemann surfaces we shall consider here solutions of the self duality equations defined on a compact riemann surface there are in fact solutions as we shall show and the moduli space of all solutions turns out to be a manifold with an extremely rich geometric structure which will be the focus of our study it brings together in a harmonious way the subjects of riemannian geometry topology algebraic geometry and symplectic geometry illuminating all these facets of the same object accounts for the length of this paper the self duality equations are equations from gauge theory geometrically they are defined in terms of connections on principal bundles while the group of the principal bundle may be chosen arbitrarily for the equations to make sense we restrict attention here to the simplest case of su 2 or so 3 there are two reasons for this the first and most obvious is that it simplifies calculations and avoids the use of inductive processes which are inherent in the consideration of a general lie group of higher rank the second reason is that solutions for su 2 have an intimate relationship with the internal structure of the riemann surface as a consequence of results we shall prove about solutions to the self duality equations we learn something about the moduli space of complex structures on the surface itself namely teichmuller space

unitary local systems multiplier ideals and polynomial periodicity of hodge numbers abstract the space of unitary local systems of rank one on the complement of an arbitrary divisor in a complex projective algebraic variety can be described in terms of parabolic line bundles we show that multiplier ideals provide natural stratifications of this space we prove a structure theorem for these stratifications in terms of complex tori and convex rational polytopes generalizing to the quasi projective case results of green lazarsfeld and simpson as an application we show the polynomial periodicity of hodge numbers h q 0 of congruence covers in any dimension generalizing results of e hironaka and sakuma we extend the structure theorem and polynomial periodicity to the setting of cohomology of unitary local systems in particular we obtain a generalization of the polynomial periodicity of betti numbers of unbranched congruence covers due to sarnak adams we derive a geometric characterization of finite abelian covers which recovers the classic one and the one of pardini we use this for example to prove a conjecture of libgober about hodge numbers of abelian covers

on the monodromy of the moduli space of calabi yau threefolds coming from eight planes in mathbb p 3 it is a fundamental problem in geometry to decide which moduli spaces of polarized algebraic varieties are embedded by their period maps as zariski open subsets of locally hermitian symmetric domains in the present work we prove that the moduli space of calabi yau threefolds coming from eight planes in mathbb p 3 does not have this property we show furthermore that the monodromy group of a good family is zariski dense in the corresponding symplectic group moreover we study a natural sublocus which we call hyperelliptic locus over which the variation of hodge structures is naturally isomorphic to wedge product of a variation of hodge structures of weight one it turns out the hyperelliptic locus does not extend to a shimura subvariety of type iii siegel space within the moduli space besides general hodge theory representation theory and computational commutative algebra one of the proofs depends on a new result on the tensor product decomposition of complex polarized variations of hodge structures

green function approach for general quantum graphs we present a recursive prescription to calculate the exact green function for general quantum graphs for the closed case the expression for the poles of g which gives the individual eigenstates has the structure of a semiclassical formula where the sum over the periodic orbit is already performed as applications we discuss eigenstate localization for a three arm closed star and filter like mechanisms for transmission throughout an open trident graph

solution of the schrodinger equation by a spectral method abstract a new computational method for determining the eigenvalues and eigenfunctions of the schrodinger equation is described conventional methods for solving this problem rely on diagonalization of a hamiltonian matrix or iterative numerical solutions of a time independent wave equation the new method in contrast is based on the spectral properties of solutions to the time dependent schrodinger equation the method requires the computation of a correlation function r 0 r t from a numerical solution r t fourier analysis of this correlation function reveals a set of resonant peaks that correspond to the stationary states of the system analysis of the location of these peaks reveals the eigenvalues with high accuracy additional fourier transforms of r t with respect to time generate the eigenfunctions the effectiveness of the method is demonstrated for a one dimensional asymmetric double well potential and for the two dimensional henon heiles potential

anti self dual yang mills connections over complex algebraic surfaces and stable vector bundles on presente une correspondance entre la geometrie algebrique et la geometrie differentielle des fibres vectoriels soit une surface algebrique projective x qui a un plongement donne x cp n et soit une metrique de kahler sur x dont la classe de cohomologie associee est duale a la classe de section d hyperplan h un fibre sur x est stable par rapport au plongement projectif si et seulement si il admet une connexion irreductible d hermite einstein par rapport a la metrique cette connexion est alors unique

explicit spectral formulas for scaling quantum graphs we present an exact analytical solution of the spectral problem of quasi one dimensional scaling quantum graphs strongly stochastic in the classical limit these systems are frequently employed as models of quantum chaos we show that despite their classical stochasticity all scaling quantum graphs are explicitly solvable in the form en fsnd where n is the sequence number of the energy level of the quantum graph and f is a known function which depends only on the physical and geometrical properties of the quantum graph our method of solution motivates a new classification scheme for quantum graphs we show that each quantum graph can be uniquely assigned an integer m reflecting its level of complexity we show that a network of taut strings with piecewise constant mass density provides an experimentally realizable analogue system of scaling quantum graphs

harmonic metrics on higgs sheaves and uniformization of varieties of general type we prove a criterion for the existence of harmonic metrics on higgs bundles that are defined on smooth loci of klt varieties as one application we resolve the quasi etale uniformisation problem for minimal varieties of general type to obtain a complete numerical characterisation of singular quotients of the unit ball by discrete co compact groups of automorphisms that act freely in codimension one as a further application we establish a nonabelian hodge correspondence on smooth loci of klt varieties

an exact periodic orbit formula for the energy levels of the three pronged star graph we derive an exact periodic orbit expansion formula for the energy levels of the three pronged star graph in addition we present a proof of the one root per root cell property and derive exact periodic orbit sum rules a spin off of our periodic orbit formulae is novel arithmetic series for 2

the weil petersson geometry of the moduli space of su n 3 calabi yau manifolds i the weil petersson metric is defined on the moduli space of calabi yau manifolds the curvature of this weil petersson metrics is computed and its potential is explicitely defined it is proved that the moduli space of calabi yau manifolds is unobstructed see tian