composite media and asymptotic dirichlet forms we introduce a metric topology on the space of dirichlet forms and study compactness and closure properties of families of local and non local forms

hilbert s projective metric on cones of dirichlet forms abstract the construction of diffusions on finitely ramified fractals is straightforward if a certain nonlinear eigenvalue problem can be solved usually this problem is attacked probabilistically using brouwer s fixed point theorem we will translate this problem into the theory of dirichlet forms and apply a different fixed point approach hilbert s projective metric on cones this allows one to prove new results about the eigenvalue problem especially about the uniqueness and the approximation of solutions and about the structure of fixed point sets

estimates of transition densities for brownian motion on nested fractals we obtain upper and lower bounds for the transition densities of brownian motion on nested fractals compared with the estimate on the sierpinski gasket the results require the introduction of a new exponent dj related to the shortest path metric and chemical exponent on nested fractals further holder order of the resolvent densities sample paths and local times are obtained the results are obtained using the theory of multi type branching processes

the decorated teichmuller space of punctured surfaces a principal 5 bundle over the usual teichmuller space of ans times punctured surface is introduced the bundle is mapping class group equivariant and admits an invariant foliation several coordinatizations of the total space of the bundle are developed there is furthermore a natural cell decomposition of the bundle finally we compute the coordinate action of the mapping class group on the total space the total space is found to have a rich equivariant geometric structure we sketch some connections with arithmetic groups diophantine approximations and certain problems in plane euclidean geometry furthermore these investigations lead to an explicit scheme of integration over the moduli spaces and to the construction of a universal teichmuller space which we hope will provide a formalism for understanding some connections between the teichmuller theory the kp hierarchy and the virasoro algebra these latter applications are pursued elsewhere

strongly pseudoconvex domains as subvarieties of complex manifolds in this paper we obtain existence and approximation results for closed complex subvarieties that are normalized by strongly pseudoconvex stein domains our sufficient condition for the existence of such subvarieties in a complex manifold x is expressed in terms of the morse indices and the number of positive levi eigenvalues of an exhaustion function on x examples show that our conditions cannot be weakened in general we obtain optimal results for subvarieties of this type in complements of compact complex submanifolds with griffiths positive normal bundle in the projective case these generalize classical theorems of remmert bishop and narasimhan concerning proper holomorphic maps and embeddings to bbb c n bbb p n backslash bbb p n 1

closed holomorphic 1 forms without zeros on stein manifolds we prove that a stein mainfold admits a closed holomorphic 1 form without zeros in every class of the first cohomology group we also prove an approximation result for closed holomorphic 1 forms without zeros defined in a neighborhood of a compact subset

a homotopy theoretic realization of string topology let m be a closed oriented manifold of dimension d let lm be the space of smooth loops in m in 2 chas and sullivan defined a product on the homology h lm of degree d they then investigated other structure that this product induces including a batalin vilkovisky structure and a lie algebra structure on the s1 equivariant homology h s 1 lm these algebraic structures as well as others came under the general heading of the string topology of m in this paper we will describe a realization of the chas sullivan loop product in terms of a ring spectrum structure on the thom spectrum of a certain virtual bundle over the loop space we also show that an operad action on the homology of the loop space discovered by voronov has a homotopy theoretic realization on the level of thom spectra this is the cactus operad defined in 6 which is equivalent to operad of framed disks in this operad action realizes the chas sullivan bv structure on h lm we then describe a cosimplicial model of this ring spectrum and by applying the singular cochain functor to this cosimplicial spectrum we show that this ring structure can be interpreted as the cup product in the hochschild cohomology hh c m c m

improved homological stability for the mapping class group with integral or twisted coefficients in this paper we prove stability results for the homology of the mapping class group of a surface we get a stability range that is near optimal and extend the result to twisted coefficients

stability of the homology of the mapping class groups of orientable surfaces the mapping class group of f fgs r is f rgs wo a where a is the topological group of orientation preserving diffeomorphisms of f which are the identity on df and fix the s punctures when r 0 and 2g s 3 an important feature of f is that it acts properly discontinuously on the teichmiller space us the quotient is the moduli space f of isometry classes of complete hyperbolic metrics of finite area on f since us is homeomorphic to euclidean space and f is virtually torsion free we have

on the oka principle in a banach space ii let x be one of the banach spaces c0 lp 1 p x pseudoconvex open a holomorphic banach vector bundle with a banach lie group g for structure group we show that a suitable runge type approximation hypothesis on x g which we also prove for g a solvable lie group implies the vanishing of the sheaf cohomology groups hq e q 1 with coefficients in the sheaf of germs of holomorphic sections of e further letting be the sheaf of germs of holomorphic continuous sections of a banach lie group bundle with banach lie groups g g for fiber group and structure group we show that a suitable runge type approximation hypothesis on x g g which we prove again for g g solvable lie groups implies the injectivity and for x l1 also the surjectivity of the grauert oka map h1 h1 of multiplicative cohomology sets

brownian motion on the sierpinski gasket we construct a brownian motion taking values in the sierpinski gasket a fractal subset of 2 and study its properties this is a diffusion process characterized by local isotropy and homogeneity properties we show for example that the process has a continuous symmetric transition density p t x y with respect to an appropriate hausdorff measure and obtain estimates on p t x y

associahedra cyclohedra and a topological solution to the a deligne conjecture abstract we give a topological solution to the a deligne conjecture using associahedra and cyclohedra to this end we construct three cw complexes whose cells are indexed by products of polytopes giving new explicit realizations of the polytopes in terms of different types of trees we are able to show that the cw complexes are cell models for the little discs the cellular chains of one complex in particular which is built out of associahedra and cyclohedra naturally act on the hochschild cochains of an a algebra yielding an explicit topological and minimal solution to the a deligne conjecture along the way we obtain new results about the cyclohedra such as new decompositions into products of cubes and simplices which can be used to realize them via a new iterated blow up construction

harmonization and homogenization on fractals this paper suggests a direct approach to define the laplacian the spectral dimension of nested fractals and the pre sierpinski carpet conductivity we find a geometric construction of the harmonic functions on the gasket and therefore can describe effectively the dense set of functions having finite energy the paper is mostly aimed at the homogenization on the pre sierpinski gasket whose horizontal and nonhorizontal bonds have different conductivities a andb respectively we prove the convergence of the rescaled energies on the pre sierpinski gasket to a b e where e is the standard energy on the gasket with uniform conductivities we also find an explicit expression for the effective conductivity a b and deduce that its set of singularities turns out to be the julia set of a certain rational function a special section is devoted to the problem of the pre sierpinski carpet conductivity asymptotic behavior for this problem a new proof of barlow bass inequalities with sharper constants is given

extending holomorphic sections from complex subvarieties in this paper we treat the classical problem of extending holomorphic mappings and sections from closed complex subvarieties of a complexmanifold our main results theorems 1 1 and 1 4 extend those of grauert gr2 and cartan car results of this type are commonly referred to as the okagrauert principle on stein manifolds our methods are similar to those developed in the papers of gromov gro henkin and leiterer hl2 and the authors fp1 fp2 the main addition here is the interpolation of a given holomorphic section on a complex subvariety of a stein manifold our globalization scheme follows very closely the one developed in fp2 to state our first result we recall the notion of a dominating spray introduced by m gromov gro sect 0 5 given a holomorphic vector bundle p e y over a complex manifold y we denote by 0y ey p 1 y the zero element in the fiberey and we observe thatey is ac linear subspace of the tangent space t0ye definition 1 a spray on a complex manifold y is a holomorphic vector bundle p e y together with a holomorphic map s e y such that for each y y s 0y y and the derivative ds t0ye tyy maps ey surjectively onto tyy

hochschild homology of structured algebras abstract we give a general method for constructing explicit and natural operations on the hochschild complex of algebras over any prop with a multiplication we think of such algebras as a algebras with extra structure as applications we obtain an integral version of the costello kontsevich soibelman moduli space action on the hochschild complex of open tcfts the tradler zeinalian and kaufmann actions of sullivan diagrams on the hochschild complex of strict frobenius algebras and give applications to string topology in characteristic zero our main tool is a generalization of the hochschild complex

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

transition density estimates for brownian motion on affine nested fractals a class of affine nested fractals is introduced which have different scale factors for different similitudes but still have the symmetry assumptions of nested fractals for these fractals estimates on the transition density for the brownian motion are obtained using the associated dirichlet form an upper bound for the diagonal can be found using a nash type inequality then probabilistic techniques are used to obtain the off diagonal bound the approach differs from previous treatments as it uses only the dirichlet form and no estimates on the resolvent the bounds obtained are expressed in terms of an intrinsic metric on the fractal

laplacians on finitely ramified graph directed fractals a finitely ramified graph directed fractal is approximated by an adapted sequence of increasingly refined graphs the scaling problem for a corresponding sequence of discrete laplacians is rephrased via a renormalization map comparing two subsequent graphs a limit set dichotomy for this map is proved the forward orbit always accumulates at periodic points even if the corresponding models are disconnected thus these periodic points can be numerically approximated by an iteration scheme based on the schur complement this allows to turn numerical information via the short cut test into theorems on the existence of laplacians on fractals

the virtual cohomological dimension of the mapping class group of an orientable surface let f f r be the mapping class group of a surface f of genus g with s punctures and r boundary components the purpose of this paper is to establish cohomology properties of f parallel to those of the arithmetic groups if g is a linear algebraic group defined over q x is the symmetric space associated to g and a is an arithmetic subgroup of g then a is virtually torsion free and acts properly discontinuously on x the rational homology of a is the same as that of x a furthermore there is a bordification of x bs to a manifold with corners 3 and an extension of the action of a to a properly discontinuous action on jf so that the quotient x a is compact the boundary of jf is homotopy equivalent to a wedge of spheres say of dimension d and the virtual cohomological dimension of a is n d 1 where n is the dimension of x in the case of the mapping class group there is no analog for g ivanov unpublished has proven that f is not arithmetic nevertheless f acts properly discontinuously on teichmiiller space r which is homeomorphic to euclidean space of dimension 6 g 6 2 s 3 will play the role of the symmetric space the quotient of 9by f is the moduli space of curves whose rational homology is then identified with that of e harvey har has constructed a borel serre bordification j of 3 by analytic methods in the case where f has punctures we will build g by a different combinatorial method in addition we will explicitly describe inside y a cell complex y of dimension 4 g 4 s onto which j may be f equivariantly retracted thus establishing an analog of the constructions for sl n by serre soul6 sol and ash a this complex will be of the lowest possible dimension because we will use j to prove our main result

a relative oka grauert principle for holomorphic submersions over 1 convex spaces this paper presents the proof of the relative oka grauert principle for holomorphic submersions over 1 convex spaces using conic neighbourhoods of holomorphic sections over 1 convex spaces a proof of a version of cartan s theorem a for 1 convex spaces is also given

homogenization on nested fractals we study the homogenization problem on nested fractals letxt be the continuous time markov chain on the pre nested fractal given by puttingi i d random resistors on each cell it is proved that under some conditions alpha n x t e n t converges in law to a constant time change of the brownian motion on the fractal asn where is the contraction rate andte is a time scale constant as the brownian motion on fractals is not a semi martingale we need a different approach from the well developed martingale method

the unstable integral homology of the mapping class groups of a surface with boundary we construct a graph complex calculating the integral homology of the bordered mapping class groups we compute the homology of the bordered mapping class groups of the surfaces s 1 1 s 1 2 and s 2 1 using the circle action on this graph complex we build a double complex and a spectral sequence converging to the homology of the unbordered mapping class groups we compute the homology of the punctured mapping class groups associated to the surfaces s 1 1 and s 2 1 finally we use miller s operad to get the first kudo araki and browder operations on our graph complex we also consider an unstable version of the higher kudo araki dyer lashoff operations

cyclic homology and equivariant homology the purpose of this paper is to explore the relationship between the cyclic homology and cohomology theories of connes 9 11 see also loday and quillen 20 and if equivariant homology and cohomology theories here ii is the circle group the most general results involve the definitions of the cyclic homology of cyclic chain complexes and the notions of cyclic and cocyclic spaces so precise statements will be postponed until w 3 in this introduction we explain some of the formal similarities between the cyclic theory and the equivariant theory and give two examples where the general results apply let a be an associative algebra over a commutative ring k then one can form the cyclic homology hc a and cohomology hc a of a these groups have periodicity operators

oka s principle for holomorphic sections of elliptic bundles 2 the h principle for locally trivial fibrations 2 1 the h principle over small subsets 2 2 the h principle over totally real submanifolds 2 3 local extension of the h principle 2 4 localizable extensions 2 5 the h principle for cm bundles 2 6 manifolds with totally real souls 2 7 totally real extensions 2 8 nicely localizable extensions 2 9 the basic h principle for z x 2 10 homomorphisms and holomorphic maps of rank r

holomorphic flexibility properties of complex manifolds we obtain results on approximation of holomorphic maps by algebraic maps the jet transversality theorem for holomorphic and algebraic maps between certain classes of manifolds and the homotopy principle for holomorphic submersions of stein manifolds to certain algebraic manifolds

topological conformal field theories and calabi yau categories abstract this is the first of two papers which construct a purely algebraic counterpart to the theory of gromov witten invariants at all genera these gromov witten type invariants depend on a calabi yau a category which plays the role of the target in ordinary gromov witten theory when we use an appropriate a version of the derived category of coherent sheaves on a calabi yau variety this constructs the b model at all genera when the fukaya category of a compact symplectic manifold x is used it is shown under certain assumptions that the usual gromov witten invariants are recovered the assumptions are that open closed gromov witten theory can be constructed for x and that the natural map from the hochschild homology of the fukaya category of x to the ordinary homology of x is an isomorphism

dirichlet forms on fractals poincare constant and resistance we study dirichlet forms associated with random walks on fractal like finite grahs we consider related poincare constants and resistance and study their asymptotic behaviour we construct a markov semi group on fractals as a subsequence of random walks and study its properties finally we construct self similar diffusion processes on fractals which have a certain recurrence property and plenty of symmetries

oka s principle for holomorphic submersions with sprays the oka principle is a fundamental principle of complex analysis which says that on stein manifolds closed complex submanifolds of affine spaces analytic problems of cohomological or even homotopical nature have only topological obstructions the story began in 1939 when k oka proved that on a domain of holomorphy a secondcousin problem is solvable by holomorphic functions if it is solvable by continuous functions oka oka s result has the following equivalent formulation ife x is a holomorphicc bundle over adomainof holomorphy then every continuous section of e is homotopic to a holomorphic section in 1957 h grauert gra extended oka s result to sections of holomorphic fiber bundles with complex homogeneous fibers over stein manifolds see also the papers car fr hl in 1989 m gromov gr1 announced a major generalization of grauert s theorem replacing complex lie groups and homogeneous spaces by a larger class of complex manifolds with a spray a spray on a complex manifold y is a holomorphic vector bundle p e y with a holomorphic map s e y the spray map which is the identity on the zero section and whose restriction to each fibre of e is a submersion at zero such sprays exist on complex homogeneous spaces on complements of affine algebraic subvarieties of codimension at least two and more generally on manifolds whose holomorphic tangent bundle is spanned by finitely many c complete holomorphic vector fields the notion of spray extends in a natural way to submersions the following is theorem 4 5 in gr1 if h z x is a holomorphic submersion onto a stein manifold x such that every point x x has an open neighborhood u x such that z u h 1 u u admits a fiber dominating spray then the inclusion of the

the oka grauert principle without induction over the base dimension we give a new proof of grauert s theorem on oka s principle gra1 gra2 gra3 see also c in the case of smooth stein manifolds which does not use induction over the base dimension instead we use induction over the levels of a strictly plurisubharmonic exhausting function grauert s bump method the present paper is an edited version of our preprint h l2 from 1986 which is difficult to find and of bad printing quality we did not publish the paper in a journal at that time because we planned to write a book containing it but the book has not been written until now on the other hand in the meantime some interest to this proof appeared see e g the work of gromov gro therefore we think a publication of our proof could be useful even with a delay of 10 years this proof gives also some oka principle for arbitrary pseudoconvex manifolds see theorem 1 3 below for vector bundles vector bundles without further mention will refer to topological vector bundles with fibre cr this oka principle contains the following