test fields on compact space times in this paper some basic aspects of lorentzian field theory on compact lorentz manifolds are studied all compact space times are acausal i e possess closed timelike curves this makes them a useful testbed in analyzing some new notions of causality that will be introduced for more general acausal space times in addition studying compact space times in their own right raises a wide range of fascinating mathematical problems some of which will be explored it will be shown that it is reasonable to expect lorentzian field theory on a compact space time to provide information on the topology of the underlying manifold if this is true then this information is likely to be orthogonal or complementary to the information obtained through the study of euclidean field theory

invariant and homogeneous bundles on g γ abstract let be a cocompact lattice in a connected complex lie group g given an invariant holomorphic vector bundle e on g we show that there is a trivial holomorphic subbundle f e such that any holomorphic section of e factors through holomorphic sections of f given two homomorphisms 1 and 2 from to a complex linear algebraic lie group h with relatively compact image we prove that any holomorphic isomorphism between the associated holomorphic principal h bundles e h 1 and e h 2 is automatically g equivariant

the atiyah bott patodi method in deformation quantization we give a new proof of the index theorem for deformation quantization following the atiyah bott patodi scheme the main ingredient consists of tamarkin s theorem on topological invariants of symplectic connections which is a symplectic analog of the well known theorems for riemannian connections

a differential complex for poisson manifolds construction du complexe canonique d une variete de poisson homologie canonique des varietes symplectiques application a l homologie de hochschild des algebres non commutatives

immersing almost geodesic surfaces in a closed hyperbolic three manifold let m 3 be a closed hyperbolic three manifold we construct closed surfaces that map by immersions into m 3 so that for each one the corresponding mapping on the universal covering spaces is an embedding or in other words the corresponding induced mapping on fundamental groups is an injection

batalin vilkovisky quantization and the algebraic index abstract into a geometric setting we import the physical interpretation of index theorems via semi classical analysis in topological quantum field theory we develop a direct relationship between fedosov s deformation quantization of a symplectic manifold x and the batalin vilkovisky bv quantization of a one dimensional sigma model with target x this model is a quantum field theory of aksz type and is quantized rigorously using costello s homotopic theory of effective renormalization we show that fedosov s abelian connections on the weyl bundle produce solutions to the effective quantum master equation moreover bv integration produces a natural trace map on the deformation quantized algebra this formulation allows us to exploit a rigorous localization argument in quantum field theory to deduce the algebraic index theorem via semi classical analysis i e one loop feynman diagram computations

on numerical cubatures of singular surface integrals in boundary element methods we present and analyze methods for the accurate and efficient evaluation of weakly cauchy and hypersingular integrals over piecewise analytic curved surfaces in 3 the class of admissible integrands includes all kernels arising in the numerical solution of elliptic boundary value problems in three dimensional domains by the boundary integral equation method the possibly not absolutely integrable kernels of boundary integral operators in local coordinates are pseudohomogeneous with analytic characteristics depending on the local geometry of the surface at the source point this rules out weighted quadrature approaches with a fixed singular weight for weakly singular integrals it is shown that duffy s triangular coordinates leadalways to a removal of the kernel singularity also asymptotic estimates of the integration error are provided as the size of the boundary element patch tends to zero these are based on the rabinowitz richter estimates in connection with an asymptotic estimate of domains of analyticity in 2 it is further shown that the modified extrapolation approach due to lyness is in the weakly singular case always applicable corresponding error and asymptotic work estimates are presented for the weakly singular as well as for cauchy and hypersingular integrals which e g arise in the study of crack problems we analyze a family of product integration rules in local polar coordinates these rules are hierarchically constructed from finite part integration formulas in radial and gaussian formulas in angular direction again we show how the rabinowitz richter estimates can be applied providing asymptotic error estimates in terms of orders of the boundary element size

symplectic calabi yau manifolds minimal surfaces and the hyperbolic geometry of the conifold given an so 3 bundle with connection the associated two sphere bundle carries a natural closed 2 form asking that this be symplectic gives a curvature inequality first considered by rezn ikov 34 we study this inequality in the case when the base has dimension four with three main aims firstly we use this approach to construct symplectic six manifolds with c 1 0 which are never kahler e g we produce such manifolds with b 1 0 b 3 and also with c 2 w

electrostatic fields without singularities theory algorithms and error analysis the following problems that arise in the computation of electrostatic forces and in the boundary element method are considered given two convex interior disjoint polyhedra in 3 space endowed with a volume charge density which is a polynomial in the cartesian coordinates of r 3 compute the coulomb force acting on them given two interior disjoint polygons in 3 space endowed with a surface charge density which is polynomial in the cartesian coordinates of r 3 compute the normal component of the coulomb force acting on them for both problems adaptive gaussian approximation algorithms are given which for n gaussian points in time o n achieve absolute error o c n for a constant c 1 such a result improves upon previously known best asymptotic bounds this result is achieved by blending techniques from integral geometry computational geometry and numerical analysis in particular integral geometry is used in order to represent the forces as integrals whose kernal is free from singularities

globalization for perturbative quantization of nonlinear split aksz sigma models on manifolds with boundary we describe a covariant framework to construct a globalized version for the perturbative quantization of nonlinear split aksz sigma models on manifolds with and without boundary and show that it captures the change of the quantum state as one changes the constant map around which one perturbs this is done by using concepts of formal geometry moreover we show that the globalized quantum state can be interpreted as a closed section with respect to an operator that squares to zero this condition is a generalization of the modified quantum master equation as in the bv bfv formalism which we call the modified differential quantum master equation

the finite element method with nonuniform mesh sizes for unbounded domains the finite element method with nonuniform mesh sizes is employed to approximately solve elliptic boundary value problems in unbounded domains consider the following model problem du f in w sup c u g on partialw partialu partialr 1 r u o 1 r as r vertical barxvertical bar infinity where w sup c is the complement in r sup 3 three dimensional eucidean space of a bounded set w with smooth boundary partialw f and g are smooth functions and f has bounded support this problem is approximately solved by introducing an artifical bound g sub r near infinity e g a sphere of sufficiently large radius r the intersection of this sphere with w sup c is denoted by w sub r sup c and the given problem is replaced by du sub r f in w sup c sub r u sub r g on partialw partialu sub r partialr 1 r u sub r 0 on g sub r this problem is then solved approximately by the finite element method resulting in an approximate solution u sub r sup h for each h 0 in order to obtain a reasonably small error for u u sub r sup h u u sub r u sub r u sub r sup h it is necessary to make r large this necessitates the solution of amore large number of linear equations so that this method is often not very good when a uniform mesh size h is employed it is shown that a nonuniform mesh may be introduced in such a way that optimal error estimates hold and the number of equations is bounded by ch sup 3 with c independent of h and r less

batalin vilkovisky algebras and two dimensional topological field theories by a batalin vilkovisky algebra we mean a graded commutative algebraa together with an operator a a 1 such that 2 0 and a a is a graded derivation ofa for alla a in this article we show that there is a natural structure of a batalin vilkovisky algebra on the cohomology of a topological conformal field theory in two dimensions we make use of a technique from algebraic topology the theory of operads

separable subsets of gferf negatively curved groups a word hyperbolic group g is called gferf if every quasiconvex subgroup coincides with the intersection of finite index subgroups containing it we show that in any such group the product of finitely many quasiconvex subgroups is closed in the profinite topology on g

global rigidity of holomorphic riemannian metrics on compact complex 3 manifolds we study compact complex 3 manifolds m admitting a locally homogeneous holomorphic riemannian metric g we prove the following i if the killing lie algebra of g has a non trivial semi simple part then it preserves some holomorphic riemannian metric on m with constant sectional curvature ii if the killing lie algebra of g is solvable then up to a finite unramified cover m is a quotient g where is a lattice in g and g is either the complex heisenberg group or the complex sol group

on closed anti de sitter spacetimes this article deals with the structure of the fundamental group of compact anti de sitter spacetimes i e lorentz manifolds with constant negative curvature algebraically such a manifold is the quotient of the universal cover of the homogeneous space so 2 n so 1 n by a discrete group gamma acting properly and co compactly on it this exists if and only if n is even indeed as this was observed by kulkarni u 1 d is contained in so 2 2d and acts properly transitively on so 2 2d so 1 2d it then suffices to take gamma as a co compact lattice in u 1 d the results of the present article give evidence to the question in dimension 3 are all compact anti de sitter spacetimes constructed in this way

kodaira spencer theory of gravity and exact results for quantum string amplitudes we develop techniques to compute higher loop string amplitudes for twistedn 2 theories with 3 i e the critical case an important ingredient is the discovery of an anomaly at every genus in decoupling of brst trivial states captured to all orders by a master anomaly equation in a particular realization of then 2 theories the resulting string field theory is equivalent to a topological theory in six dimensions the kodaira spencer theory which may be viewed as the closed string analog of the chern simons theory using the mirror map this leads to computation of the number of holomorphic curves of higher genus curves in calabi yau manifolds it is shown that topological amplitudes can also be reinterpreted as computing corrections to superpotential terms appearing in the effective 4d theory resulting from compactification of standard 10d superstrings on the correspondingn 2 theory relations withc 1 strings are also pointed out

supersymmetry and morse theory it is shown that the morse inequalities can be obtained by consideration of a certain supersymmetric quantum mechanics hamiltonian some of the implications of modern ideas in mathematics for supersymmetric theories are discussed

on the b twisted topological sigma model and calabi yau geometry we provide a rigorous perturbative quantization of the b twisted topological sigma model via a first order quantum field theory on derived mapping space in the formal neighborhood of constant maps we prove that the first chern class of the target manifold is the obstruction to the quantization via batalin vilkovisky formalism when the first chern class vanishes i e on calabiyau manifolds the factorization algebra of observables gives rise to the expected topological correlation functions in the b model we explain a twisting procedure to generalize to the landau ginzburg case and show that the resulting topological correlations coincide with vafa s residue formula

autour de la conjecture de l markus sur les varietes affines for any subgroupg of n we introduce some integer discg n called thediscompacity ofg this number measures to what extent the closure ofg is not compact the markus conjecture says that a compact affinely flat unimodular manifold is complete our main result called the discompact theorem is that this conjecture is true under the assumption that the linear holonomy i e the parallel transport has discompacity 1 because discso n 1 1 1 this ensures that a compact flat lorentz manifoldm is geodesically complete hence by a previous result of w goldman and y kamishima gk such am is up to finite covering a solvmanifold this achieves the proof of a bieberbach s theorem for compact lorentz flat manifolds

margulis spacetimes via the arc complex we study strip deformations of convex cocompact hyperbolic surfaces defined by inserting hyperbolic strips along a collection of disjoint geodesic arcs properly embedded in the surface we prove that any deformation of the surface that uniformly lengthens all closed geodesics can be realized as a strip deformation in an essentially unique way the infinitesimal version of this result gives a parameterization by the arc complex of the moduli space of margulis spacetimes with fixed convex cocompact linear holonomy as an application we provide a new proof of the tameness of such margulis spacetimes m by establishing the crooked plane conjecture which states that m admits a fundamental domain bounded by piecewise linear surfaces called crooked planes the noninfinitesimal version gives an analogous theory for noncompact complete anti de sitter 3 manifolds

geometry of batalin vilkovisky quantization the geometry ofp manifolds odd symplectic manifolds andsp manifolds p manifolds provided with a volume element is studied a complete classification of these manifolds is given this classification is used to prove some results about batalin vilkovisky procedure of quantization in particular to obtain a very general result about gauge independence of this procedure

a simple proof of geodesical completeness for compact space times of zero curvature any riemannian metric on a compact manifold is geodesically complete by contrast it is widely known that incomplete compact lorentzian manifolds exist in this article it will be proven that a compact manifold with a smooth lorentz metric must be complete if the metric is flat everywhere this question of completeness which has significant implications for the classification of compact lorentzian space forms has remained unresolved until quite recently when it was finally established by carriere as a special case of an even more general result here a simple alternative proof of this special case of carriere s theorem is given the proof requires minimal mathematical machinery but still involves the use of some intriguing connections between the topology and global geometry of a compact flat space time

fedosov deformation quantization as a brst theory the relationship is established between the fedosov deformation quantization of a general symplectic manifold and the bfv brst quantization of constrained dynamical systems the original symplectic manifold is presented as a second class constrained surface in the fibre bundle which is a certain modification of a usual cotangent bundle equipped with a natural symplectic structure the second class system is converted into the first class one by continuation of the constraints into the extended manifold being a direct sum of and the tangent bundle t this extended manifold is equipped with a nontrivial poisson bracket which naturally involves two basic ingredients of fedosov geometry the symplectic structure and the symplectic connection the constructed first class constrained theory being equivalent to the original symplectic manifold is quantized through the bfv brst procedure the existence theorem is proven for the quantum brst charge and the quantum brst invariant observables the adjoint action of the quantum brst charge is identified with the abelian fedosov connection while any observable being proven to be a unique brst invariant continuation for the values defined in the original symplectic manifold is identified with the fedosov flat section of the weyl bundle the fedosov fibrewise star multiplication is thus recognized as a conventional product of the quantum brst invariant observables

morse theory and finiteness properties of groups we examine the finiteness properties of certain subgroups of right angled artin groups in particular we find an example of a group that is of type fp z but is not finitely presented

flat compact hermite lorentz manifolds in dimension 4 we give a classification up to finite cover of flat compact complete hermite lorentz manifolds up to complex dimension 4

a proof of the tsygan formality conjecture for chains abstract we extend the kontsevich formality l morphism u t poly r d d poly r d to an l morphism of l modules over t poly r d u c a a r d a c r d the construction of the map u is given in kontsevich type integrals the conjecture that such an l morphism exists is due to boris tsygan formality conjecture for chains math qa 9904132 as an application we obtain an explicit formula for isomorphism a a a a a a a is the kontsevich deformation quantization of the algebra a by a poisson bivector field and is the poisson bracket we also formulate a conjecture extending the kontsevich theorem on cup products to this context the conjecture implies a generalization of the duflo formula and many other things

deformation of proper actions on reductive homogeneous spaces let g be a real reductive lie group and h a closed reductive subgroup of g we investigate the deformation of standard compact quotients of g h that is of quotients of g h by discrete groups that are uniform lattices in some closed reductive subgroup l of g acting properly and cocompactly on g h for l of real rank 1 we prove that after a small deformation in g such a group keeps acting properly discontinuously and cocompactly on g h more generally we prove that the properness of the action of any convex cocompact subgroup of l on g h is preserved under small deformations and we extend this result to reductive homogeneous spaces g h over any local field as an application we obtain compact quotients of so 2n 2 u n 1 by zariski dense discrete subgroups of so 2n 2 acting properly discontinuously

an efficient higher order fast multipole boundary element solution for poisson boltzmann based molecular electrostatics in order to compute polarization energy of biomolecules we describe a boundary element approach to solving the linearized poisson boltzmann equation our approach combines several important features including the derivative boundary formulation of the problem and a smooth approximation of the molecular surface based on the algebraic spline molecular surface state of the art software for numerical linear algebra and the kernel independent fast multipole method is used for both simplicity and efficiency of our implementation we perform a variety of computational experiments testing our method on a number of actual proteins involved in molecular docking and demonstrating the effectiveness of our solver for computing molecular polarization energy

three dimensional affine crystallographic groups those groups r which act properly discontinuously and aillnely on ii with compact fundamental domain are classified first it is shown that such a group f contains a solvable subgroup of finite index thus establishing a conjecture of auslander in dimension three then unimodular simply transitive altine actions on ir are classified this leads to a classification of attine crystallographic groups acting on ir3 a characterization of which abstract groups admit such an action is given moreover it is proved that every isomorphism between virtually solvable atline crystallographic groups respectively simply transitive afline groups is induced by conjugation by a polynomial automorphism of the affrne space a characterization is given of which closed 3 manifolds can be represented as quotients of ir by groups of afftne transformations a closed 3 manifold m admits a complete atline structure if and only if m has a finite covering homeomorphic or homotopy equivaient to a 2 torus bundle over the circle

fuchsian affine actions of surface groups let q be the irreducible representation of sl 2 r i nsl q r define a fuchsian subgroup of sl q r to be a subgroup conjugate to a discrete subgroup of q sl 2 r we prove in this paper that the fundamental group of a compact surface does not act properly on the affine space by affine tranformations if its linear part is fuchsian

note on solutions of the strominger system from unitary representations of cocompact lattices of sl 2 mathbb c this note is motivated by a recently published paper biswas and mukherjee in commun math phys 322 2 373 384 2013 we prove a no go result for the existence of suitable solutions of the strominger system in a compact complex parallelizable manifold for this we assume g to be non abelian the hermitian metric to be induced from a right invariant metric on g the bianchi identity to be satisfied using the chern connection and furthermore the gauge field to be flat in biswas and mukherjee commun math phys 322 2 373 384 2013 it is claimed that one such solution exists on our result contradicts the main result in biswas and mukherjee commun math phys 322 2 373 384 2013

discretization of the induced charge boundary integral equation boundary element methods bems for solving integral equations numerically have been used in many fields to compute the induced charges at dielectric boundaries in this paper we consider a more accurate implementation of bem in the context of ions in aqueous solution near proteins but our results are applicable more generally the ions that modulate protein function are often within a few angstroms of the protein which leads to the significant accumulation of polarization charge at the protein solvent interface computing the induced charge accurately and quickly poses a numerical challenge in solving a popular integral equation using bem in particular the accuracy of simulations can depend strongly on seemingly minor details of how the entries of the bem matrix are calculated we demonstrate that when the dielectric interface is discretized into flat tiles the qualocation method of tausch et al ieee trans comput comput aided des 20 1398 2001 to compute the bem matrix elements is always more accurate than the traditional centroid collocation method qualocation is not more expensive to implement than collocation and can save significant computational time by reducing the number of boundary elements needed to discretize the dielectric interfaces

on the asymptotic convergence of collocation methods we prove quasioptimal and optimal order estimates in various sobolev norms for the approximation of linear strongly elliptic pseudodifferential equations in one independent variable by the method of nodal collocation by odd degree polynomial splines the analysis pertains in particular to many of the boundary element methods used for numerical computation in engineering applications equations to which the analysis is applied include fredholm integral equations of the second kind certain first kind fredholm equations singular integral equations involving cauchy kernels a variety of integro differential equations and two point boundary value problems for ordinary differential equations the error analysis is based on an equivalence which we establish between the collocation methods and certain nonstandard galerkin methods we compare the collocation method with a standard galerkin method using splines of the same degree showing that the galerkin method is quasioptimal in a sobolev space of lower index and furnishes optimal order approximation for a range of sobolev indices containing and extending below that for the collocation method and so the standard galerkin method achieves higher rates of convergence

on the zariski closure of the linear part of a properly discontinuous group of affine transformations let be a subgroup of the group of affine transformations of the affine space r 2n 1 suppose acts properly discontinuously on r 2n 1 the paper deals with the question which subgroups of gl 2n 1 r occur as zariski closure of the linear part of such a group the two main results of the paper say that so n 1 n does occur as of such a group if n is odd but does not if n is even

proper affine actions and geodesic flows of hyperbolic surfaces let to c 0 2 1 be a schottky group and let h2 to be the corresponding hyperbolic surface let s denote the space of unit length geodesic currents on s the cohomology group if1 to v parametrizes equivalence classes of affine deformations tu of to acting on an irreducible representation v of 0 2 1 we define a continuous biaffine map i s x hl to v u which is linear on the vector space h1 fo v an affine deformation tu acts properly if and only if x u 7 0 for all jl e s consequently the set of proper affine actions whose linear part is a schottky group identifies with a bundle of open convex cones in hl to v over the fricke teichmiiller space of e

numerical quadratures for layer potentials over curved domains in r 3 numerical quadratures for calculating simple and double layer potentials over curved domains in mathbb r 3 are given more generally these numerical quadratures are shown to be useful for evaluating surface integral involving kernels which are pseudohomogeneous of degree 1 in addition to potential flow problems such kernels are encountered when dealing with helmholtz problem stokes flow problems or linear elasticity problems

a formality theorem for hochschild chains abstract we prove tsygan s formality conjecture for hochschild chains of the algebra of functions on an arbitrary smooth manifold m using the fedosov resolutions proposed in math qa 0307212 and the formality quasi isomorphism for hochschild chains of r y 1 y d proposed in paper math qa 0010321 by shoikhet this result allows us to describe traces on the quantum algebra of functions on an arbitrary poisson manifold

on the asymptotic convergence of collocation methods with spline functions of even degree we investigate the collocation of linear one dimensional strongly elliptic integro dif ferential or more generally pseudo differential equations on closed curves by even degree polynomial splines the equations are collocated at the respective midpoints subject to uniform nodal grids of the even degree s splines we prove quasioptimal and optimal order asymptotic error estimates in a scale of sobolev spaces the results apply in particular to boundary element methods used for numerical computations in engineering applications the equations considered include fredholm integral equations of the second and the first kind singular integral equations involving cauchy kernels and integro differential equations having convolutional or constant coefficient principal parts respectively the error analysis is based on an equivalence between the collocation and certain varia tional methods with different degree splines as trial and as test functions we further need to restrict our operators essentially to pseudo differential operators having convolutional prin cipal part this allows an explicit fourier analysis of our operators as well as of the spline spaces in terms of trigonometric polynomials providing babuska s stability condition based on strong ellipticity our asymptotic error estimates extend partly those obtained by d n arnold and w l wendland from the case of odd degree splines to the case of even degree splines 1 introduction in this paper we investigate the asymptotic convergence of the collocation method using even degree polynomial splines applied to strongly elliptic systems of pseudo differential equations on closed curves with convolutional prin cipal part the collocation here employs the gauss points of one point integration as collocation points which corresponds to the usual boundary element collocation in applications 5 12 17 19 26 31 34 this is in contrast to the method investigated by g schmidt in 29 where one collocates at the break points of even order splines the asymptotic convergence properties for the standard galerkin method with splines of arbitrary orders are well known 22 for the collocation method however asymptotic convergence up to now has been shown for strongly elliptic systems only in the case of odd order splines by d n arnold and w l wendland 7

on the globalization of the poisson sigma model in the bv bfv formalism we construct a formal global quantization of the poisson sigma model in the bv bfv formalism using the perturbative quantization of aksz theories on manifolds with boundary and analyze the properties of the boundary bfv operator moreover we consider mixed boundary conditions and show that they lead to quantum anomalies i e to a failure of the modified differential quantum master equation we show that it can be restored by adding boundary terms to the action at the price of introducing corner terms in the boundary operator we also show that the quantum grothendieck bfv operator on the total space of states is a differential i e squares to zero which is necessary for a well defined bv cohomology

quadrature methods for 2d and 3d problems in this paper we give an overview on well known stability and convergence results for simple quadrature methods based on low order composite quadrature rules and applied to the numerical solution of integral equations over smooth manifolds first we explain the methods for the case of second kind equations then we discuss what is known for the analysis of pseudodifferential equations we explain why these simple methods are not recommended for integral equations over domains with dimension higher than one finally for the solution of a two dimensional singular integral equation we prove a new result on a quadrature method based on product rules

deformation of compact clifford klein forms of indefinite riemannian homogeneous manifolds 1 1 let g be a lie group and h a closed subgroup of g if a discrete subgroup of g acts properly discontinuously and freely on g h then the double coset space g h carries naturally a manifold structure such that the quotient map g h g h is locally diffeomorphic the manifold g h is said to be a clifford klein form of g h if it is compact then is said to be a uniform lattice for g h a typical example is a compact riemann surface mg with genus g 2 which is biholomorphic to a compact clifford klein form of the poincare disk g h sl 2 r so 2 by the uniformization theorem it is important from geometric view point that a clifford klein form g h inherits any g invariant local geometric structure on g h such as indefinite riemannian metric complex structure symplectic structure causal structure and so on

high order numerical quadratures for layer potentials over curved domains in ℝ3 abstract high order numerical quadratures for approximating layer potentials over curved domains in 3 are given more generally these numerical quadratures are shown to be useful for evaluating surface integral involving kernels which are pseudo homogeneous of degree 1 a numerical application is presented

imaginary cones and limit roots of infinite coxeter groups let w s be an infinite coxeter system to each geometric representation of w is associated a root system while a root system lives in the positive side of the isotropic cone of its associated bilinear form an imaginary cone lives in the negative side of the isotropic cone precisely on the isotropic cone between root systems and imaginary cones lives the set e of limit points of the directions of roots in this article we study the close relations of the imaginary cone with the set e which leads to new fundamental results about the structure of geometric representations of infinite coxeter groups in particular we show that the w action on e is minimal and faithful and that e and the imaginary cone can be approximated arbitrarily well by sets of limit roots and imaginary cones of universal root subsystems of w i e root systems for coxeter groups without braid relations the free object for coxeter groups finally we discuss open questions as well as the possible relevance of our framework in other areas such as geometric group theory

adaptive integration techniques for almost singular functions in the boundary element method abstract three numerical adaptive integration schemes for almost singular functions occurring in the boundary element method which use subdivision of a domain or moving of nodes are discussed some numerical examples show efficiency of the methods

fast potential theory ii layer potentials and discrete sums we present three new families of fast algorithms for classical potential theory based on ewald summation and fast transforms of gaussians and fourier series ewald summation separates the green function for a cube into a high frequency localized part and a rapidly converging fourier series each part can then be evaluated efficiently with appropriate fast transform algorithms our algorithms are naturally suited to the use of green functions with boundary conditions imposed on the boundary of a cube rather than free space green functions our first algorithm evaluates classical layer potentials on the boundary of a d dimensional domain with d equal to two or three the quadrature error is o hm e where h is the mesh size on the boundary and m is the order of quadrature used the algorithm evaluates the discretized potential using n elements at o n points in o n log n arithmetic operations the constant in o n log n depends logarithmically on the desired error tolerance our second scheme evaluates a layer potential on the domain itself with the same accuracy it produces md values using n boundary elements in o n md log m arithmetic operations our third method evaluates a discrete sum of values of the green function of the type which occur in particle methods it attains error at a cost o na log n where a 2 1 dd and d is the hausdorff dimension of the set where the sources concentrate in the limit n thus it is o n log n when the sources do not cluster too much and close to o n log n in the important practical case when the points are uniformly distributed over a hypersurface we also sketch an o n log n algorithm based on special functions two dimensional numerical results are presented for all three algorithms layer potentials are evaluated to second order accuracy in times which exhibit considerable speedups even over a reasonably sophisticated direct calculation discrete sum calculations are speeded up astronomically our algorithm reduces the cpu time required for a calculation with 40 000 points from six months to one hour

numerical method for a nonlinear boundary integral equation abstract we study a boundary integral equation method for solving laplace s equation u 0 with nonlinear boundary conditions this nonlinear boundary value problem is reformulated as a nonlinear boundary integral equation with u on the boundary as the solution being sought the integral equation is solved numerically by using the wavelet galerkin method with legendre wavelet functions used as approximations to u numerical example are given for problem

classification of semisimple symmetric spaces with proper actions we give a complete classification of irreducible symmetric spaces for which there exist proper sl 2 mathbb r actions as isometries using the criterion for proper actions by t kobayashi and combinatorial techniques of nilpotent orbits in particular we classify irreducible symmetric spaces that admit surface groups as discontinuous groups combining this with benoist s theorem

a path integral approach to the kontsevich quantization formula we give a quantum field theory interpretation of kontsevich s deformation quantization formula for poisson manifolds we show that it is given by the perturbative expansion of the path integral of a simple topological bosonic open string theory its batalin vilkovisky quantization yields a superconformal field theory the associativity of the star product and more generally the formality conjecture can then be understood by field theory methods as an application we compute the center of the deformed algebra in terms of the center of the poisson algebra

mixed 3 manifolds are virtually special letm be a compact oriented irreducible 3 manifold which is neither a graph manifold nor a hyperbolic manifold we prove that 1m is virtually special

an analysis of quadrature errors in second kind boundary integral methods this paper considers the effect of quadrature errors on galerkin methods for approximating the solution of second kind fredholm integral equations with weakly singular kernels defined on a smooth closed surface in three dimensions a quadrature strategy for galerkin methods that use piecewise poly nomials both continuous and discontinuous on a quadrilateral based mesh is proposed the error resulting from this further discretization is proved to be no larger than the original discretization error and moreover the amount of work needed to compute the resulting matrix for the corresponding linear system is shown to be proportional to the number of matrix entries

algebraic index theorem we prove the atiyah singer index theorem where the algebra of pseudodifferential operators is replaced by an arbitrary deformation quantization of the algebra of functions on a symplectic manifold

chern simons perturbation theory ii we study the perturbation theory for three dimensional chern simons quantum field theory on a general compact three manifold without boundary we show that after a simple change of variables the action obtained by brs gauge fixing in the lorentz gauge has a superspace formulation the basic properties of the propagator and the feynman rules are written in a precise manner in the language of differential forms using the explicit description of the propagator singularities we prove that the theory is finite finally the anomalous metric dependence of the 2 loop partition function on the riemannian metric which was introduced to define the gauge fixing can be cancelled by a local counterterm as in the 1 loop case in fact the counterterm is equal to the chern simons action of the metric connection normalized precisely as one would expect based on the framing dependence of witten s exact solution invited talk at the xxth conference on differential geometric methods in physics new york june 1991

one dimensional chern simons theory we study a one dimensional toy version of the chern simons theory we construct its simplicial version which comprises features of a low energy effective gauge theory and of a topological quantum field theory in the sense of atiyah

numerical simulation of tridimensional electromagnetic shaping of liquid metals we describe a numerical method to compute free surfaces in electromagnetic shaping and levitation of liquid metals we use an energetic variational formulation and optimization techniques to compute a critical point the surfaces are represented by piecewise linear finite elements each step of the algorithm requires solving an elliptic boundary value problem in the exterior of the intermediate surfaces this is done by using an integral representation on these surfaces

coxeter groups are virtually special abstract in this paper we prove that every finitely generated coxeter group has a finite index subgroup that is the fundamental group of a special cube complex some consequences include every f g coxeter group is virtually a subgroup of a right angled coxeter group every word hyperbolic coxeter group has separable quasiconvex subgroups

conformality for a robust class of non conformal attractors in this paper we investigate the hausdorff dimension of limit sets of anosov representations in this context we revisit and extend the framework of hyperconvex representations and establish a convergence property for them analogue to a differentiability property as an application of this convergence we prove that the hausdorff dimension of the limit set of a hyperconvex representation is equal to a suitably chosen critical exponent in the appendix in collaboration with m bridgeman we extend a classical result on the hessian of the hausdorff dimension on purely imaginary directions

the fundamental group of a compact flat lorentz space form is virtually polycyclic a flat lorentz space form is a geodesically complete lorentzian manifold of zero curvature it is well known see auslander markus 3 that such a space m may be represented as a quotient r where r is an dimensional minkowski space n equals the dimension of m and is a group of lorentz isometries acting properly discontinuously and freely on r in particular the universal covering of m is isometric to r and the fundamental group m is isomorphic to theorem let m be a compact flat lorentz space form then m is virtually poly cyclic recall that a group is virtually polycyclic if it can be built by iterated extensions from finitely many finite groups and cyclic groups this result affirms a conjecture of milnor 13 in a special case for discussion of this conjecture as well as another special case we refer to fried goldman 8 the importance of this result is that it reduces the classification of compact flat lorentz space forms to fairly elementary problems concerning lie algebras and lattices in solvable lie groups we hope to pursue this classification in a future publication for a description of this reduction and the classification in dimension 3 see fried goldman 8 in dimension 4 the classification is worked out in fried 7 one immediate consequence of the structure theory developed in fried goldman 8 1 and kamishima 12 is the following corollary let m be the compact flat lorentz space form then m has a finite covering which is diffeomorphic to a solvmanifold the outline of this paper is as follows in the first section we collect some basic facts about the group e n 1 1 of isometries of dimensional minkowski space in the second section we prove the theorem in the special

error bounds for gaussian quadrature and weighted l 1 polynomial approximation error bounds for gaussian quadrature are given in terms of the number of quadrature points and smoothness properties of the function whose integral is being approximated an intermediate step invol

on the existence of closed timelike geodesics in compact spacetimes in 19 tipler has shown that a compact spacetime having a regular globally hyperbolic covering space with compact cauchy surfaces necessarily contains a closed timelike geodesic the restriction to compact spacetimes with just regular globally hyperbolic coverings i e the cauchy surfaces are not required to be compact is still an open question here we shall answer this question negatively by providing examples of compact flat lorentz space forms without closed timelike geodesics and shall give some criterion for the existence of such geodesics more generally we will show that in a compact spacetime having a regular globally hyperbolic covering each free timelike homotopy class determined by a central deck transformation must contain a closed timelike geodesic whether or not a compact flat spacetime contains closed nonspacelike geodesics is as far as we know an open question we shall answer this question affirmatively we shall also introduce the notion of timelike injectivity radius for a spacetime relative to a free timelike homotopy class and shall show that it is finite whenever the corresponding deck transformation is central

a boundary criterion for cubulation we give a criterion in terms of the boundary for the existence of a proper cocompact action of a word hyperbolic group on a rm cat 0 cube complex we describe applications towards lattices and hyperbolic 3 manifold groups in particular by combining the theory of special cube complexes the surface subgroup result of kahn markovic and agol s criterion we find that every subgroup separable closed hyperbolic 3 manifold is virtually fibered