spherical functions on symmetric cones in this note we obtain a recursive formula for the spherical functions associated with the symmetric cone of a formally real jordan algebra we use this formula as an inspiration for a similar recursive formula involving the jack polynomials

self adjoint block operator matrices with non separated diagonal entries and their schur complements in this paper self adjoint 2 2 block operator matrices a in a hilbert space h1 h2 are considered for an interval which does not intersect the spectrum of at least one of the diagonal entries of a we prove angular operator representations for the corresponding spectral subspace l a of a and we study the supporting subspace in this angular operator representation of l a which is the orthogonal projection of l a to the corresponding component h1 or h2 our main result is a description of a special direct complement of this supporting subspace in its component in terms of spectral subspaces of the values of the corresponding schur complement of a in the endpoints of

matrix formulation of vector operations an efficient technique is developed to simplify the computations in the area of vector analysis the evaluation of vector algebraic and differential operations becomes more simple and straightforward by simply converting the vector operations into matrix operations the matrix formulation technique is especially useful when there are mixed coordinate basis involved in the vector operations

invariant functions on lie groups and hamiltonian flows of surface group representations si est le groupe fondamental d une surface orientee fermee s et g est un groupe de lie satisfaisant des conditions tres generales alors l espace hom g g des classes de conjugaison de representation g a une structure symplectique naturelle on etudie la geometrie de cette structure symplectique a l aide d une famille naturelle de fonctions sur hom g g

symplectic structure of the moduli space of flat connection on a riemann surface we consider the canonical symplectic structure on the moduli space of flat g connections on a riemann surface of genus g with a marked points for a being a semisimple lie algebra we obtain an explicit efficient formula for this symplectic form and prove

spectral inclusion for unbounded block operator matrices abstract in this paper we establish a new analytic enclosure for the spectrum of unbounded linear operators a admitting a block operator matrix representation for diagonally dominant and off diagonally dominant block operator matrices we show that the recently introduced quadratic numerical range w 2 a contains the eigenvalues of a and that the approximate point spectrum of a is contained in the closure of w 2 a this provides a new method to enclose the spectrum of unbounded block operator matrices by means of the non convex set w 2 a several examples illustrate that this spectral inclusion may be considerably tighter than the one by the usual numerical range or by perturbation theorems both in the non self adjoint case and in the self adjoint case applications to dirac operators and to two channel hamiltonians are given

solution of maxwell s equations by matrix formulation the usual approach in solving maxwell s equations with given current distribution is by means of finding the vector potential function due to the current source this approach is generally intensive for it contains vector differential operations in addition to the integration this paper presents the detailed derivation of useful formulas for finding electromagnetic fields such that the complicated vector differential operations are replaced by the simple vector algebraic operations the resulting solutions are exact and can be explicitly written in matrix formulation such that both the current source and the resulting fields may be conveniently expressed in terms of any desired coordinate systems if the fields in the near or far region are of primary interest this formulation only requires a matrix multiplication whenever the radiation vector is determined

on the symplectic geometry of deformations of a hyperbolic surface let r be a riemann surface in this manuscript we consider a geometry on the moduli space x r for r which we regard as the space of equivalence classes of constant curvature metrics on the underlying smooth manifold of r classically the space of flat metrics for a torus is the locally symmetric space 0 2 sl 2 r sl 2 z we shall describe a symplectic geometry for the space of hyperbolic metrics on a surface of negative euler characteristic the teichmiiller space t r a covering of the moduli space is a complex kihler manifold a kuhler metric for t r defined in terms of the petersson product for automorphic forms was introduced by weil 1 the weil petersson metric is invariant under the covering transformations and so projects to the moduli space x r the metric provides a link between the function theory of r and the geometry of x r in the fenchel nielsen manuscript 8 a deformation based on an amalgamation construction for fuchsian groups is introduced the deformation is defined geometrically by cutting the surface along a simple closed geodesic rotating one side of the cut relative to the other and attaching the sides in their new position the hyperbolic metric in the complement of the cut extends to a hyperbolic metric on the new surface choose a free homotopy class a on the surface r then for each marked surface r realize a by the closed geodesic ar the fenchel nielsen deformations for the athen define a 1 parameter group of diffeomorphisms of t r whose infinitesimal generator by definition is the fenchel nielsen vector field ta in 21 the fenchel nielsen deformation was described in terms of quasiconformal mappings and an investigation of the vector fields t was begun the fenchel nielsen vector fields were found to be related to the geodesic length functions 1 introduced by fricke klein to provide coordinates for t r

the block numerical range of matrix polynomials block numerical ranges of matrix polynomials especially the quadratic numerical range are considered the main results concern spectral inclusion boundedness of the block numerical range an estimate of the resolvent in terms of the quadratic numerical range geometrical properties of the quadratic numerical range and inclusion between block numerical ranges of the matrix polynomials for refined block decompositions as an application we connect the quadratic numerical range with the localization of the spectrum of matrix polynomials

perturbation of spectra and spectral subspaces we consider the problem of variation of spectral subspaces for linear self adjoint operators with emphasis on the case of off diagonal perturbations we prove a number of new optimal results on the shift of the spectrum and obtain sharp estimates on the norm of the difference of two spectral projections associated with isolated parts of the spectrum of the perturbed and unperturbed operators respectively

an integrable discretization of the rational mathfrak su 2 gaudin model and related systems the first part of the present paper is devoted to a systematic construction of continuous time finite dimensional integrable systems arising from the rational su 2 gaudin model through certain contraction procedures in the second part we derive an explicit integrable poisson map discretizing a particular hamiltonian flow of the rational su 2 gaudin model then the contraction procedures enable us to construct explicit integrable discretizations of the continuous systems derived in the first part of the paper

deformation spaces associated to compact hyperbolic manifolds in this paper we take a first step toward understanding representations of cocompact lattices in so n 1 into arbitrary lie groups by studying the deformations of rational representations see proposition 5 1 for a rather general existence result this proposition has a number of algebraic applications for example we remark that such deformations show that the margulis super rigidity theorem see 30 cannot be extended to the rank 1 case we remark also that if so n 1 is one of the standard arithmetic examples described in section 7 then has a faithful representation in so n 1 the galois conjugate of the uniformization representation and proposition 5 1 may be used to deform the direct sum of and the trivial representation in so n 2 thereby constructing non trivial families of irreducible orthogonal representations of however most of this paper is devoted to studying certain spaces of representations which are of interest in differential geometry in a sense which we now explain

semiclassical mechanics of the wigner 6j symbol the semiclassical mechanics of the wigner 6j symbol is examined from the standpoint of wkb theory for multidimensional integrable systems to explore the geometrical issues surrounding the ponzano regge formula the relations among the methods of roberts and others for deriving the ponzano regge formula are discussed and a new approach based on the recoupling of four angular momenta is presented a generalization of the yutsis type of spin network is developed for this purpose special attention is devoted to symplectic reduction the reduced phase space of the 6j symbol the 2 sphere of kapovich and millson and the reduction of poisson bracket expressions for semiclassical amplitudes general principles for the semiclassical study of arbitrary spin networks are laid down some of these were used in our recent derivation of the asymptotic formula for the wigner 9j symbol

deformations of polyhedra and polygons by the unitary group we introduce the set of framed convex polyhedra with n faces as the symplectic quotient c2n su 2 a framed polyhedron is then parametrized by n spinors living in c2 satisfying suitable closure constraints and defines a usual convex polyhedron plus extra u 1 phases attached to each face we show that there is a natural action of the unitary group u n on this phase space which changes the shape of faces and allows to map any framed polyhedron onto any other with the same total boundary area this identifies the space of framed polyhedra to the grassmannian space u n su 2 u n 2 we show how to write averages of geometrical observables polynomials in the faces area and the angles between them over the ensemble of polyhedra distributed uniformly with respect to the haar measure on u n as polynomial integrals over the unitary group and we provide a few methods to compute these integrals systematically we also use the itzykson zuber formula from matrix models as the generating function

a kahler structure on the moduli space of isometric maps of a circle into euclidean space we study the spaces and and lip of smooth resp non degenerate lipschitz isometric maps of a circle into euclidean space modulo orientation preserving euclidean motions we prove that and lip are infinite dimensional kahler manifolds in particular they are complex frechet resp banach manifolds this is proved by an infinite dimensional version of the kirwan kempf ness theorem kir84 kn78 nes84 relating symplectic quotients to holomorphic quotients applied to the action ofpsl2 on the free loop space ofs2

geometric and algebraic reduction for singular momentum maps this paper concerns the reduction of singular constraint sets of symplectic manifolds it develops a geometric reduction procedure as well as continues the work of sniatycki and patrick on reduction a la dirac the relationships among the dirac geometric and sniatycki weinstein algebraic reduction procedures are studied primary emphasis is placed on the case where the constraints are given by the vanishing of a singular momentum map associated to the hamiltonian action of a compact lie group specifically an explicit local normal form is given for the momentum map which is used to show that these various reductions are all well defined and a necessary and sufficient condition for them to agree is derived related conditions are investigated and shown to be sufficient in the case of torus actions numerous examples are computed illustrating the results some discussion and examples are given for the noncompact case

variational principles and eigenvalue estimates for unbounded block operator matrices and applications in this paper we establish variational principles eigenvalue estimates and asymptotic formulae for eigenvalues of three different classes of unbounded block operator matrices the results allow to characterise eigenvalues that are not necessarily located at the boundary of the spectrum applications to an example from magnetohydrodynamics and to dirac operators on certain manifolds are given

the cohomology ring of weight varieties and polygon spaces we use a theorem of s tolman and j weitsman the cohomology rings of abelian symplectic quotients math dg 9807173 to find explicit formulae for the rational cohomology rings of the symplectic reduction of flag varieties in cn or generic coadjoint orbits of su n by maximal torus actions we also calculate the cohomology ring of the moduli space of n points in cpk which is isomorphic to the grassmannian of k planes in cn by realizing it as a degenerate coadjoint orbit

novel coordinate transformations for antenna applications coordinate transformations commonly encountered in antenna design and analysis are presented the introduction of a new coordinate system polarization coordinates enables one to construct a complete flowchart which can be used to express any coordinate transformation matrix more directly and explicitly without going through the usual step by step derivation consequently the computation procedure can often be greatly reduced by avoiding tedious manipulations

a new concept for block operator matrices the quadratic numerical range abstract in this paper a new concept for 2 2 block operator matrices the quadratic numerical range is studied the main results are a spectral inclusion theorem an estimate of the resolvent in terms of the quadratic numerical range factorization theorems for the schur complements and a theorem about angular operator representations of spectral invariant subspaces which implies e g the existence of solutions of the corresponding riccati equations and a block diagonalization all results are new in the operator as well as in the matrix case

existence and uniqueness of contractive solutions of some riccati equations abstract in this paper we establish criteria for the existence and uniqueness of contractive solutions k of the riccati equation kbk ka dk c 0 under the assumption that the spectra of a and d are disjoint

a variational principle for block operator matrices and its application to the angular part of the dirac operator in curved spacetime abstract the operator associated to the angular part of the dirac equation in the kerr newman background metric is a block operator matrix with bounded diagonal and unbounded off diagonal entries the aim of this paper is to establish a variational principle for block operator matrices of this type and to derive thereof upper and lower bounds for the angular operator mentioned above in the last section these analytic bounds are compared with numerical values from the literature

intersection numbers of polygon spaces we study the intersection ring of the space m 1 m of polygons in r 3 we find homology cycles dual to generators of this ring and prove a recursion relation in m the number of edges for their intersection numbers this result is an analog of the recursion relation appearing in the work of witten and kontsevich on moduli spaces of punctured curves and in the work of weitsman on moduli spaces of flat connections on two manifolds of genus g with m marked points based on this recursion formula we obtain an explicit expression for the computation of the intersection numbers of polygon spaces and use it in several examples among others we study the special case of equilateral polygon spaces where all i s are the same and compare our results with the expressions for these particular spaces that have been determined by kamiyama and tezuka finally we relate our explicit formula for the intersection numbers with the generating function for intersection pairings of the moduli space of flat connections of yoshida as well as with equivalent expressions for polygon spaces obtained by takakura and konno through different techniques

the symplectic nature of fundamental groups of surfaces soit 7b g la categorie des groupes de lie g avec une forme bilineaire symetrique non singuliere b sur l algebre de lie soit le groupe fondamental d une surface orientee close de genre 1 et soit 7b s la categorie des out espaces symplectiques alors la correspondance g b hom g g b definit un foncteur 7b g 7b s ou b est la structure symplectique sur hom g g definie par b

around polygons in ℝ3 and s3 we survey certain moduli spaces in low dimensions and some of the geometric structures that they carry and then construct identifications among all of these spaces in particular we identify the moduli spaces of polygons in 3 and s 3 the moduli space of restricted representations of the fundamental group of a punctured 2 sphere the moduli space of flat connections on a punctured sphere the moduli space of parabolic bundles on a sphere the moduli space of weighted points on 1 and the symplectic quotient of so 3 acting diagonally on s 2 n all of these spaces depend on parameters and some the above identifications require the parameters to be small one consequence of this work is that these spaces are all biholomorphic with respect to the most natural complex structures they can each be given

the fenchel nielsen deformation the uniformization theorem provides that a riemann surface s of negative euler characteristic has a metric of constant curvature 1 a hyperbolic structure can be understood in terms of its deformations unfortunately the variation of the hyperbolic metric arising from a deformation is not determined by local data if z is a conformal coordinate for s and a ii h ii 00 1 a beltrami differential then dw dz p dj defines a new conformal coordinate and structure si the tensor p may vanish on the open set 0 c s and yet the hyperbolic metrics of s and si do not coincide on this phenomenon presents a s erious difficulty in the deformation theory of hyperbolic structures the fenchel nielsen twist deformation does not share this defect the deformation is defined by cutting the surface along a simple closed geodesic rotating one side of the cut relative to the other and attaching the sides in their new position a geodesic intersecting the cut is deformed to a broken geodesic the hyperbolic structure in the complement of the cut extends naturally to a hyperbolic structure on the new surface the deformation is concentrated at the cut the basic idea for the deformation appears in the work of fricke klein dehn and fenchel nielsen 71 8 in the fenchel nielsen manuscript the deformation was considered extensively coordinates for teichmiller space are defined in terms of the deformation the proper definition is given by a fuchsian group construction calculations are made in terms of unimodular matrices this approach is often cumbersome our basic objective is to give a description of the deformation in terms of quasiconformal mappings the bers embedding of teichmiiller space is central to the analytic theory of deformations using our characterization we calculate the first variation of the bers embedding for the fenchel nielsen deformation the result is a poincare series 6 which already appears in the work of petersson 16

birkhoff s theorem and multidimensional numerical range abstract we show that under certain conditions birkhoff s theorem on doubly stochastic matrices remains valid for countable families of discrete probability spaces which have nonempty intersections using this result we study the relation between the spectrum of a self adjoint operator a and its multidimensional numerical range it turns out that the multidimensional numerical range is a convex set whose extreme points are sequences of eigenvalues of the operator a every collection of eigenvalues which can be obtained by the rayleigh ritz formula generates an extreme point of the multidimensional numerical range however it may also have other extreme points

the equations for the moduli space of n points on the line a central question in invariant theory is that of determining the relations among invariants geometric invariant theory git quotients come with a natural ample line bundle and hence often a natural projective embedding this question translates to determining the equations of the moduli space under this embedding this article deals with one of the most classical quotients the space of ordered points on the projective line we show that under any weighting of the points this quotient is cut out scheme theoretically by a particularly simple set of quadric relations with the single exception of the segre cubic threefold the space of six points with equal weight we also show that the ideal of relations is generated in degree at most four and we give an explicit description of the generators if all the weights are even e g in the case of equal weight for odd n then we show that the ideal of relations is generated by quadrics

hyperpolygon spaces and moduli spaces of parabolic higgs bundles abstract given an n tuple of positive real numbers we consider the hyperpolygon space x the hyperkahler quotient analogue to the kahler moduli space of polygons in r 3 we prove the existence of an isomorphism between hyperpolygon spaces and moduli spaces of stable rank 2 holomorphically trivial parabolic higgs bundles over c p 1 with fixed determinant and trace free higgs field this isomorphism allows us to prove that hyperpolygon spaces x undergo an elementary transformation in the sense of mukai as crosses a wall in the space of its admissible values we describe the changes in the core of x as a result of this transformation as well as the changes in the nilpotent cone of the corresponding moduli spaces of parabolic higgs bundles moreover we study the intersection rings of the core components of x in particular we find generators of these rings prove a recursion relation in n for their intersection numbers and use it to obtain explicit formulas for the computation of these numbers using our isomorphism we obtain similar formulas for each connected component of the nilpotent cone of the corresponding moduli spaces of parabolic higgs bundles thus determining their intersection rings as a final application of our isomorphism we describe the cohomology ring structure of these moduli spaces of parabolic higgs bundles and of the components of their nilpotent cone

so 2n coherent states for loop quantum gravity a su 2 intertwiner with n legs can be interpreted as the quantum state of a convex polyhedron with n faces when working in 3d we show that the intertwiner hilbert space carries a representation of the non compact group so 2n this group can be viewed as the subgroup of the symplectic group sp 4n r which preserves the su 2 invariance we construct the associated perelomov coherent states and discuss the notion of semi classical limit which is more subtle than we could expect our work completes the work by freidel and livine j math phys 51 082502 2010 and j math phys 52 052502 2011 which focused on the u n subgroup of so 2n

real polarization of the moduli space of flat connections on a riemann surface we prove that the moduli space of flatsu 2 connections on a riemann surface has a real polarization that is a foliation by lagrangian subvarieties this polarization may provide an alternative quantization of the chern simons gauge theory in higher genus in line with the results of 11 for genus one

the generation of random equilateral polygons freely jointed random equilateral polygons serve as a common model for polymer rings reflecting their statistical properties under theta conditions to generate equilateral polygons researchers employ many procedures that have been proved or at least are believed to be random with respect to the natural measure on the space of polygonal knots as a result the random selection of equilateral polygons as well as the statistical robustness of this selection is of particular interest in this research we study the key features of four popular methods the polygonal folding the crankshaft rotation the hedgehog and the triangle methods in particular we compare the implementation and efficacy of these procedures especially in regards to the population distribution of polygons in the space of polygonal knots the distribution of edge vectors the local curvature and the local torsion in addition we give a rigorous proof that the crankshaft rotation method is ergodic

gaudin models and bending flows a geometrical point of view in this paper we discuss the bi hamiltonian formulation of the rational xxx gaudin models of spin spin interaction generalized to the case of sl r valued spins we only consider the classical case using recent results concerning the quantum models as guiding principles in particular we focus on the so called homogeneous xxx models we find a pencil of poisson brackets that recursively define a complete set of integrals of the motion alternative to the set of integrals associated with the standard lax representation of the gaudin model these integrals in the case of su 2 coincide with the hamiltonians of the bending flows in the moduli space of polygons in the euclidean space introduced by kapovich and millson we finally address the problem of separability of these flows and explicitly find separation coordinates and separation relations for the sl 2 case