a unilaterally constrained quadratic minimization with adaptive finite elements we consider obstacle problems where a quadratic functional associated with the laplacian is minimized in the set of functions above a possibly discontinuous and thin but piecewise affine obstacle in order to approximate minimum point and value we propose an adaptive algorithm that relies on minima with respect to admissible linear finite element functions and on an a posteriori estimator for the error in the minimum value it is proven that the generated sequence of approximate minima converges to the exact one furthermore our numerical results in two and three dimensions indicate that the convergence rate with respect to the number of degrees of freedom is optimal in that it coincides with the one of nonlinear or adaptive approximation

convergent adaptive finite elements for the nonlinear laplacian the numerical solution of the homogeneous dirichlet problem for the p laplacian p in 1 infty is considered we propose an adaptive algorithm with continuous piecewise affine finite elements and prove that the approximate solutions converge to the exact one while the algorithm is a rather straight forward generalization of those for the linear case p 2 the proof of its convergence is different in particular it does not rely on a strict error reduction

the lambrechts stanley model of configuration spaces we prove the validity over mathbb r of a commutative differential graded algebra model of configuration spaces for simply connected closed smooth manifolds answering a conjecture of lambrechts stanley we get as a result that the real homotopy type of such configuration spaces only depends on the real homotopy type of the manifold we moreover prove if the dimension of the manifold is at least 4 that our model is compatible with the action of the fulton macpherson operad weakly equivalent to the little disks operad when the manifold is framed we use this more precise result to get a complex computing factorization homology of framed manifolds our proofs use the same ideas as kontsevich s proof of the formality of the little disks operads

kontsevich deformation quantization and flat connections in torossian j lie theory 12 2 597 616 2002 the second author used the kontsevich deformation quantization technique to define a natural connection n on the compactified configuration spaces overline c n 0 of n points on the upper half plane connections n take values in the lie algebra of derivations of the free lie algebra with n generators in this paper we show that n is flat

pentagon and hexagon equations the author will prove that drinfel d s pentagon equation implies his two hexagon equations in the lie algebra pro unipotent pro l and pro nilpotent contexts

an explicit two step quantization of poisson structures and lie bialgebras we develop a new approach to deformation quantizations of lie bialgebras and poisson structures which goes in two steps in the first step one associates to any poisson resp lie bialgebra structure a so called quantizable poisson resp lie bialgebra structure we give explicit transcendental formulae for this correspondence in the second step one deformation quantizes a quantizable poisson resp lie bialgebra structure we show again explicit transcendental formulae for this second step correspondence as a byproduct we obtain configuration space models for biassociahedron and bipermutohedron in the poisson case the first step is the most non trivial one and requires a choice of an associator while the second step quantization is essentially unique it is independent of a choice of an associator and can be done by a trivial induction we conjecture that similar statements hold true in the case of lie bialgebras the main new result is a surprisingly simple explicit universal formula which uses only smooth differential forms for universal quantizations of finite dimensional lie bialgebras

the kashiwara vergne conjecture and drinfeld s associators the kashiwara vergne kv conjecture is a property of the campbell hausdorff series put forward in 1978 it has been settled in the positive by e meinrenken and the first author in 2006 in this paper we study the uniqueness issue for the kv problem to this end we introduce a family of infinite dimensional groups krv 0 n and a group krv 2 which contains krv 0 2 as a normal subgroup we show that krv 2 also contains the grothendieck teichmuller group grt 1 as a subgroup and that it acts freely and transitively on the set of solutions of the kv problem solkv furthermore we prove that solkv is isomorphic to a direct product of affine line a 1 and the set of solutions of the pentagon equation with values in the group krv 0 3 the latter contains the set of drinfeld s associators as a subset as a by product of our construction we obtain a new proof of the kashiwara vergne conjecture based on the drinfeld s theorem on existence of associators

a posteriori error estimators for regularized total variation of characteristic functions we consider a nonuniformly elliptic double obstacle problem arising from convexifying and regularizing a minimization of a functional with total variation in the set of characteristic functions we derive a posteriori estimators for the discretization error with linear finite elements which are uniform in the regularization incorporate computable and local information on the conditioning vanish in the intersection of discrete and exact contact sets and are not affected by possible nonuniqueness moreover we integrate these estimators in an adaptive algorithm and illustrate their properties by various numerical experiments

relative non formality of the little cubes operads and the algebraic cerf lemma it is shown that the operad maps e n to e n k are formal over the reals for k geq 2 and non formal for k 1 furthermore we compute the homology of the deformation complex of the operad maps e n to e n 1 proving an algebraic version of the cerf lemma

averaging techniques yield reliable a posteriori finite element error control for obstacle problems the reliability of frequently applied averaging techniques for a posteriori error control has recently been established for a series of finite element methods in the context of second order partial differential equations this paper establishes related reliable and efficient a posteriori error estimates for the energy norm error of an obstacle problem on unstructured grids as a model example for variational inequalities the surprising main result asserts that the distance of the piecewise constant discrete gradient to any continuous piecewise affine approximation is a reliable upper error bound up to known higher order terms consistency terms and a multiplicative constant

double shuffle relation for associators it is proved that drinfel 0 d s pentagon equation implies the generalized double shue relation as a corollary an embedding from the grothen

equivalence of formalities of the little discs operad we show that kontsevich s formality of the little disk operad obtained using graphs is homotopic to tamarkin s formality for a special choice of a drinfeld associator the associator is given by parallel transport of the alekseev torossian connection

a posteriori error estimates for elliptic variational inequalities we derive hierarchical a posteriori error estimates for elliptic variational inequalities the evaluation amounts to the solution of corresponding scalar local subproblems we derive some upper bounds for the effectivity rates and the numerical properties are illustrated by typical examples

differentials on graph complexes abstract we study the cohomology of complexes of ordinary non decorated graphs introduced by m kontsevich we construct spectral sequences converging to zero whose first page contains the graph cohomology in particular these spectral sequences may be used to show the existence of an infinite series of previously unknown and provably non trivial cohomology classes and put constraints on the structure of the graph cohomology as a whole

small data oscillation implies the saturation assumption the saturation assumption asserts that the best approximation error in h 1 0 with piecewise quadratic finite elements is strictly smaller than that of piecewise linear finite elements we establish a link between this assumption and the oscillation of f delta u and prove that small oscillation relative to the best error with piecewise linears implies the saturation assumption we also show that this condition is necessary and asymptotically valid provided f in l 2

pointwise a posteriori error control for elliptic obstacle problems we consider a finite element method for the elliptic obstacle problem over polyhedral domains in d which enforces the unilateral constraint solely at the nodes we derive novel optimal upper and lower a posteriori error bounds in the maximum norm irrespective of mesh fineness and the regularity of the obstacle which is just assumed to be holder continuous they exhibit optimal order and localization to the non contact set we illustrate these results with simulations in 2d and 3d showing the impact of localization in mesh grading within the contact set along with quasi optimal meshes

multiple edges in m kontsevich s graph complexes and computations of the dimensions and euler characteristics we provide a generating function for the graded dimensions of m kontsevich s graph complexes of ordinary graphs this generating function can be used to compute the euler characteristic in each loop order furthermore we show that graphs with multiple edges can be omitted from these graph complexes

double shuffle and kashiwara vergne lie algebras abstract we prove that the double shuffle lie algebra ds dual to the space of new formal multiple zeta values injects into the kashiwara vergne lie algebra krv 2 defined and studied by alekseev and torossian the proof is based on a reformulation of the definition of krv 2 and uses a theorem of ecalle on a property of elements of ds in the final section we lift this result to an injection ds krv 2 of the corresponding pro unipotent groups

kontsevich s graph complex grt and the deformation complex of the sheaf of polyvector fields we generalize kontsevich s construction of l1 derivations of polyvector elds from the ane space to an arbitrary smooth algebraic variety more precisely we construct a map in the homotopy category from kontsevich s graph complex to the deformation complex of the sheaf of polyvector elds on a smooth algebraic variety we show that the action of deligne drinfeld

m kontsevich s graph complex and the grothendieck teichmuller lie algebra we show that the zeroth cohomology of m kontsevich s graph complex is isomorphic to the grothendieck teichmuller lie algebra mathfrak grt 1 the map is explicitly described this result has applications to deformation quantization and duflo theory we also compute the homotopy derivations of the gerstenhaber operad they are parameterized by mathfrak grt 1 up to one class or two depending on the definitions more generally the homotopy derivations of the non unital e n operads may be expressed through the cohomology of a suitable graph complex our methods also give a second proof of a result of h furusho stating that the pentagon equation for mathfrak grt 1 elements implies the hexagon equation

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

profinite completion of operads and the grothendieck teichmuller group abstract in this paper we prove that the group of homotopy automorphisms of the profinite completion of the operad of little 2 disks is isomorphic to the profinite grothendieck teichmuller group in particular the absolute galois group of q acts faithfully on the profinite completion of e 2 in the homotopy category of profinite weak operads

adaptive multilevel methods for obstacle problems the authors consider the discretization of obstacle problems for second order elliptic differential operators by piecewise linear finite elements assuming that the discrete problems are reduced to a sequence of linear problems by suitable active set strategies the linear problems are solved iteratively by preconditioned conjugate gradient iterations the proposed preconditioners are treated theoretically as abstract additive schwarz methods and are implemented as truncated hierarchical basis preconditioners to allow for local mesh refinement semilocal and local a posteriors error estimates are derived providing lower and upper estimates for the discretization error the theoretical results are illustrated by numerical computations

deformation theory of representations of prop erad s ii in this paper and its follow up merkulov and vallette j reine angew math we study the deformation theory of morphisms of properads and props thereby extending quillen s deformation theory for commutative rings to a non linear framework the associated chain complex is endowed with an l algebra structure its maurer cartan elements correspond to deformed structures which allows us to give a geometric interpretation of these results to do so we endow the category of prop erad s with a model category structure we provide a complete study of models for prop erad s a new effective method to make minimal models explicit that extends the koszul duality theory is introduced and the associated notion is called homotopy koszul as a corollary we obtain the co homology theories of al gebras over a prop erad and of homotopy al gebras as well their underlying chain complex is endowed with an l algebra structure in general and a lie algebra structure only in the koszul case in particular we make the deformation complex of morphisms from the properad of associative bialgebras explicit for any minimal model of this properad the boundary map of this chain complex is shown to be the one defined by gerstenhaber and schack as a corollary this paper provides a complete proof of the existence of an l algebra structure on the gerstenhaber schack bicomplex associated to the deformations of associative bialgebras

a posteriori error estimates for elliptic problems in two and three space dimensions let u in h be the exact solution of a given selfadjoint elliptic boundary value problem which is approximated by some tilde u in mathcal s mathcal s being a suitable finite element space efficient and reliable a posteriors estimates of the error u tilde u measuring the local quality of tilde u play a crucial role in termination criteria and in the adaptive refinement of the underlying mesh a well known class of error estimates can be derived systematically by localizing the discretized defect problem by using domain decomposition techniques in this paper we provide a guideline for the theoretical analysis of such error estimates we further clarify the relation to other concepts our analysis leads to new error estimates which are specially suited to three space dimensions the theoretical results are illustrated by numerical computations

the frobenius properad is koszul we show koszulness of the prop governing involutive lie bialgebras and also of the props governing non unital and unital counital frobenius algebras solving a long standing problem this gives us minimal models for their deformation complexes and for deformation complexes of their algebras which are discussed in detail using an operad of graph complexes we prove with the help of an earlier result of one of the authors that there is a highly non trivial action of the grothendieck teichm uller group grt 1 on completed versions of the minimal models of the properads governing lie bialgebras and involutive lie bialgebras by automorphisms as a corollary one obtains a large class of universal deformations of any involutive lie bialgebra and any frobenius algebra parameterized by elements of the grothendieck teichm uller lie algebra we also prove that for any given homotopy involutive lie bialgebra structure in a vector space there is an associated homotopy batalin vilkovisky algebra structure on the associated chevalley eilenberg complex

chaos in a class of reacting systems induced by robust asymptotic feedback chaotic behavior in a controlled reacting system is discussed the open loop dynamics of the reactor i e without control actions does not display chaos the feedback control has a unique adjust parameter l for all l l 0 the reactor under control actions converges to a prescribed point l is a parameter then the above condition implies that if l is larger than any constant l the reactor is stabilized at the prescribed point however homoclinic chaos can be induced by the continuous robust asymptotic feedback if 0 l l indeed the closed loop system i e the reactor under feedback ations displays four regions in the first one a limit cycle with large amplitude is found in the second one chaotic behavior is displayed small amplitude limit cycle is found in the third region finally in the fourth region the feedback yields asymptotic convergence of the trajectories to the prescribed point

structure of symplectic invariant lie subalgebras of symplectic derivation lie algebras abstract we study the structure of the symplectic invariant part h g 1 sp of the lie algebra h g 1 consisting of symplectic derivations of the free lie algebra generated by the rational homology group of a closed oriented surface g of genus g first we describe the orthogonal direct sum decomposition of this space which is induced by the canonical metric on it and compute it explicitly up to degree 20 in this framework we give a general constraint which is imposed on the sp invariant component of the bracket of two elements in h g 1 second we clarify the relations among h g 1 and the other two related lie algebras h g and h g which correspond to the cases of a closed surface g with and without base point g in particular based on a theorem of labute we formulate a method of determining these differences and describe them explicitly up to degree 20 third by giving a general method of constructing elements of h g 1 sp we reveal a considerable difference between two particular submodules of it one is the sp invariant part of a certain ideal j g 1 and the other is that of the johnson image finally we combine these results to determine the structure of h g 1 completely up to degree 6 including the unstable cases where the genus 1 case has an independent meaning in particular we see a glimpse of the galois obstructions explicitly from our point of view

hierarchical error estimates for the energy functional in obstacle problems we present a hierarchical a posteriori error analysis for the minimum value of the energy functional in symmetric obstacle problems the main result is that the error in the energy minimum is up to oscillation terms equivalent to an appropriate hierarchical estimator the proof does not invoke any saturation assumption we even show that small oscillation implies a related saturation assumption in addition we prove efficiency and reliability of an a posteriori estimate of the discretization error and thereby cast some light on the theoretical understanding of previous hierarchical estimators finally we illustrate our theoretical results by numerical computations

a posteriori error estimates based on hierarchical bases the authors present an analysis of an a posteriors error estimator based on the use of hierarchical basis functions the authors analyze nonlinear nonselfadjoint and indefinite problems as well as the selfadjoint positive definite case because both the analysis and the estimator itself are quite simple it is easy to see how various approximations affect the quality of the estimator as examples the authors apply the theory to some scalar elliptic equations and the stokes system of equations

a posteriori error estimators for obstacle problems another look we show that a posteriori estimators for the obstacle problem are easily obtained from the theory for linear equations the theory would be even simpler if the lagrange multiplier does not have a nonconforming contribution as it has in actual finite element computations

efficient and reliable a posteriori error estimators for elliptic obstacle problems a posteriori error estimators are derived for linear finite element approximations to elliptic obstacle problems these estimators yield global upper and local lower bounds for the discretization error here discretization error means the sum of two contributions the distance between continuous and discrete solution in the energy norm and some quantity that is related to the distance of continuous and discrete contact set moreover the local error indicators in the interior of the discrete contact set reduce to quantities that measure only data resolution

efficient and reliable hierarchical error estimates for the discretization error of elliptic obstacle problems we present and analyze novel hierarchical a posteriori error estimates for self adjoint elliptic obstacle problems our approach differs from straightforward but nonreliable estimators by an additional extra term accounting for the deviation of the discrete free boundary in the localization step we prove efficiency and reliability on a saturation assumption and a regularity condition on the underlying grid heuristic arguments suggest that the extra term is of higher order and preserves full locality numerical computations confirm our theoretical findings

an efficient and reliable residual type a posteriori error estimator for the signorini problem we derive a new a posteriori error estimator for the signorini problem it generalizes the standard residual type estimators for unconstrained problems in linear elasticity by additional terms at the contact boundary addressing the non linearity remarkably these additional contact related terms vanish in the case of so called full contact we prove reliability and efficiency for two and three dimensional simplicial meshes moreover we address the case of non discrete gap functions numerical tests for different obstacles and starting grids illustrate the good performance of the a posteriori error estimator in the two and three dimensional case for simplicial as well as for unstructured mixed meshes

homotopy graph complex for configuration and knot spaces we prove that the primitive part of the sinha homology spectral sequence e2 term for the space of long knots is rationally isomorphic to the homotopy epsilon 2 term we also define natural graph complexes computing the rational homotopy of configuration and of knot spaces

mixed tate motives over z we prove that the category of mixed tate motives over z is spanned by the motivic fundamental group of p 1 minus three points we prove a conjecture by m hoffman which states that every multiple zeta value is a q linear combination of n 1 n r where n i 2 3