closed form solution of local shape from shading at critical points in the theory of shape from shading behaviours of the local solution around a critical point of the image play an important role this paper shows that the second derivatives of the object surface can be locally determined at these image critical points closed form expressions of the surface second derivatives in terms of the second derivatives of the image brightness and of the reflectance map are shown they are derived as follows by differentiating the image irradiance equation twice at an image critical point a set of polynomial equations is obtained that contains the second derivatives of the surface of the image brightness and of the reflectance map regarding these equations as simultaneous equations for unknown surface second derivatives they are algebraically solved and their explicit expressions are derived such a derivation is possible only at image critical points and is impossible at any other image point the applicability of the derived expressions to noisy images is tested using synthetic images

the center of thin gaussian groups abstract thin gaussian groups are a natural generalization of spherical artin groups namely groups of fractions of monoids in which the existence of least common multiples is kept as an hypothesis but the relations between the generators are not supposed to necessarily be of coxeter type here we completely describe the center of thin gaussian groups by constructing a minimal generating set for the quasi center we deduce that every thin gaussian group is an iterated crossed product of thin gaussian groups with a cyclic center

hochschild cohomology and atiyah classes abstract in this paper we prove that on a smooth algebraic variety the hkr morphism twisted by the square root of the todd genus gives an isomorphism between the sheaf of poly vector fields and the sheaf of poly differential operators both considered as derived gerstenhaber algebras in particular we obtain an isomorphism between hochschild cohomology and the cohomology of poly vector fields which is compatible with the lie bracket and the cupproduct the latter compatibility is an unpublished result by kontsevich our proof is set in the framework of lie algebroids and so applies without modification in much more general settings as well

a viscosity solutions approach to shape from shading the problem of recovering a lambertian surface from a single two dimensional image may be written as a first order nonlinear equation which presents the disadvantage of having several continuous and even smooth solutions a new approach based on hamilton jacobi bellman equations and viscosity solutions theories enables one to study non uniqueness phenomenon and thus to characterize the surface among the various solutions a consistent and monotone scheme approximating the surface is constructed thanks to the dynamic programming principle and numerical results are presented

the eikonal equation some results applicable to computer vision in this paper we investigate certain first order partial differential equations which formulate the relationship between the light reflected from a surface and its shape particular emphasis is given to eikonal equations two results are presented first we prove that a special type of eikonal equation has only one convex and positive c2 solution in some neighborhood of a singular point using this result we show that a restricted form of this equation has exactly two solutions these results have application in scanning electron microscopy

quantization of poisson manifolds from the integrability of the modular function we discuss a framework for quantizing a poisson manifold via the quantization of its symplectic groupoid combining the tools of geometric quantization with the results of renault s theory of groupoid c algebras this setting allows very singular polarizations in particular we consider the case when the modular function is multiplicatively integrable i e when the space of leaves of the polarization inherits a groupoid structure if suitable regularity conditions are satisfied then one can define the quantum algebra as the convolution algebra of the subgroupoid of leaves satisfying the bohr sommerfeld conditions

noise reduction in surface reconstruction from a given gradient field we present a gradient space technique for noise reduction in surfaces reconstructed from a noisy gradient field we first analyze the error sources in the recovered gradient field of a surface using a three image photometric stereo method based on this analysis we propose an additive noise model to describe the errors in the surface gradient estimates we then use a vector space formulation and construct a multiscale orthonormal expansion for gradient fields using the sparse representation properties of this expansion we develop techniques for reducing the gradient field noise by coefficient selection with thresholding the simulation results indicate that the proposed technique provides significant improvement on the noise levels of both the estimated gradient fields and the reconstructed surfaces under heavy noise levels furthermore the experiments using noisy photometric stereo image triplets of real range data suggest that the additive model remains viable after the nonlinear photometric stereo operation to provide accurate noise removal

right l groups geometric garside groups and solutions of the quantum yang baxter equation abstract right lattice ordered groups are introduced and studied as a general framework for garside groups and related groups without a garside element every right l group g has a two sided partially ordered subgroup n g which generalizes the quasi centre of an artin tits group the group n g splits into copies of z if g is noetherian the positive cone of a right l group is described as a structure that is known from algebraic logic a pair of left and right self similar hoops for noetherian right l groups g modularity of the lattice structure is characterized in terms of an operation on the set x g of atoms it is proved that modular garside groups are equivalent to finite projective spaces with a non degenerate labelling a concept of duality for x g is introduced and applied in the distributive case this gives a one to one correspondence between noetherian right l groups with duality and non degenerate unitary set theoretic solutions of the quantum yang baxter equation the description of garside groups via garside germs is extended to right l groups which yields a one sided enhancement and a new proof of dvurecenskij s non commutative extension of mundici s correspondence between abelian l groups and mv algebras

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

garside families and garside germs garside families have recently emerged as a relevant context for extending results involving garside monoids and groups which themselves extend the classical theory of generalized braid groups here we establish various characterizations of garside families that is equivalently various criteria for establishing the existence of normal decompositions of a certain type

the use of a legendre pseudospectral viscosity technique to solve a class of nonlinear dynamic hamilton jacobi equations in this research we study the problem of finding the approximate solution of a class of hamilton jacobi equations namely the eikonal equation we employ the legendre pseudospectral viscosity method to solve this problem this method basically consists of adding a spectral viscosity to the equation this spectral viscosity which is sufficiently small to retain the formal spectral accuracy is large enough to stabilize the numerical scheme several test problems are considered and the numerical results are given to show the efficiency of the proposed method

interpretation of af c algebras in łukasiewicz sentential calculus on donne un critere pour la non simplicite d une af c algebre 7b u en termes de proprietes de la theorie de la recursivite du groupe dimension k 0 7b u

existence and uniqueness in photometric stereo we analyze the problem of recovering the shape of a smooth lambertian surface from two images obtained by consecutive illumination of the surface by distant point light sources from two different directions we discuss the issues of existence and uniqueness for the corresponding system of two first order nonlinear partial differential equations we show that generically this system has up to a constant a unique solution we also examine exceptional cases for which there is no such uniqueness

a generic and provably convergent shape from shading method for orthographic and pinhole cameras we describe a mathematical and algorithmic study of the lambertian shape from shading problem for orthographic and pinhole cameras our approach is based upon the notion of viscosity solutions of hamilton jacobi equations this approach provides a mathematical framework in which we can show that the problem is well posed we prove the existence of a solution and we characterize all the solutions our contribution is threefold first we model the camera both as orthographic and as perspective pinhole whereas most authors assume an orthographic projection see horn and brooks 1989 for a survey of the sfs problem up to 1989 and zhang et al 1999 kozera 1998 durou et al 2004 for more recent ones thus we extend the applicability of shape from shading methods to more realistic acquisition models in particular it extends the work of prados et al 2002a and rouy and tourin 1992 we provide some novel mathematical formulations of this problem yielding new partial differential equations results about the existence and uniqueness of their solutions are also obtained second by introducing a generic hamiltonian we define a general framework allowing to deal with both models orthographic and perspective thereby simplifying the formalization of the problem thanks to this unification each algorithm we propose can compute numerical solutions corresponding to all the modeling third our work allows us to come up with two new generic algorithms for computing numerical approximations of the continuous solution of the shape from shading problem as well as a proof of their convergence toward that solution moreover our two generic algorithms are able to deal with discontinuous images as well as images containing black shadows

on the set theoretical yang baxter equation we propose a general way of constructing set theoretical solutions of the yangbaxter equation we study the properties of the construction we also show that our construction includes the earlier ones given by weinstein xu and etingof schedlersoloviev

a multiresolution approach for shape from shading coupling deterministic and stochastic optimization shape from shading is an ill posed inverse problem for which there is no completely satisfactory solution in the existing literature in this paper we address shape from shading as an energy minimization problem we first show that the deterministic approach provides efficient algorithms in terms of cpu time but reaches its limits since the energy associated with shape from shading can contain multiple deep local minima we derive an alternative stochastic approach using simulated annealing the obtained results strongly outperform the results of the deterministic approach the shortcoming is an extreme slowness of the optimization therefore we propose a hybrid approach which combines the deterministic and stochastic approaches in a multiresolution framework

fedosov differentials and catalan numbers the aim of the paper is to establish a non recursive formula for the general solution of fedosov s quadratic fixed point equation fedosov 1994 j diff geom 40 213 38 fedosov s geometrical fixed point equation for a differential is rewritten in a form similar to the functional equation for the generating function of catalan numbers this allows us to guess the solution an adapted example for kaehler manifolds of constant sectional curvature is considered in detail also for every connection on a manifold a familiar classical differential will be introduced

decomposition of garside groups and self similar l algebras abstract picantin s iterated crossed product representation of garside monoids is extended and reproved for a wide class of not necessarily noetherian partially ordered groups with a right invariant lattice structure it is shown that the tree like structure of such an iterated crossed product is equivalent to a partial cycle set closely related to a class of set theoretic solutions of the quantum yang baxter equation the decomposition of finite square free solutions is related to the crossed product representation of the corresponding structure group

formality and kontsevich duflo type theorems for lie pairs newcommand poly operatorname poly bullet newcommand td operatorname td l a nabla frac 1 2 newcommand cx 1 operatorname tot big gamma lambda bullet a vee otimes r mathcal 1 poly big newcommand cy 1 mathbb h bullet operatorname ce a mathcal 1 poly kontsevich s formality theorem states that there exists an l infty quasi isomorphism from the dgla t poly m of polyvector fields on a smooth manifold m to the dgla d poly m of polydifferential operators on m which extends the classical hochschild kostant rosenberg map in this paper we extend kontsevich s formality theorem to lie pairs a framework which includes a range of diverse geometric contexts such as complex manifolds foliations and mathfrak g manifolds the spaces cx t and cx d associated with a lie pair l a each carry an l infty algebra structure canonical up to l infty isomorphism these two spaces serve as replacements for the spaces of polyvector fields and polydifferential operators respectively their corresponding cohomology groups cy t and cy d admit canonical gerstenhaber algebra structures we establish the following formality theorem for lie pairs there exists an l infty quasi isomorphism from cx t to cx d whose first taylor coefficient is equal to operatorname hkr circ td here td acts on cx t by contraction furthermore we prove a kontsevich duflo type theorem for lie pairs the hochschild kostant rosenberg map twisted by the square root of the todd class of the lie pair l a is an isomorphism of gerstenhaber algebras from cy t to cy d as applications we establish formality theorems and kontsevich duflo type theorems for complex manifolds foliations and mathfrak g manifolds in the case of complex manifolds we recover the kontsevich duflo theorem of complex geometry

uniqueness in shape from shading for general images of smooth objects wholly contained in the field of view and for illumination symmetric around the viewing direction it is proven that shape is uniquely determined by shading thus shape from shading is a well posed problem under these illumination conditions and regularization is unnecessary for surface reconstruction and should be avoided generic properties of surfaces and images are established questions of existence are also discussed under the conditions above it is argued that most images are effectively impossible with no corresponding physically reasonable surface and that any image can be rendered effectively impossible by a small perturbation of its intensities this is explicitly illustrated for a synthetic image the proofs are based on ideas of dynamical systems theory and global analysis

impossible and ambiguous shading patterns a smooth object depicted in a monochrome image will often exhibit brightness variation or shading a problem much studied in computer vision has been that of how object shape may be recovered from image shading when the imaging conditions are such that an overhead point source illuminates a smooth lambertian surface the problem may be formulated as that of finding a solution to an eikonal equation this article will focus on the existence and uniqueness of such solutions reporting recent results obtained with regard to existence shading patterns are exhibited for which there is no corresponding object shape specifically a necessary and sufficient condition is presented for a circularly symmetric eikonal equation to admit exclusively unbounded solutions additionally a sufficient condition is given for an eikonal equation to have no solution whatsoever in connection with uniqueness we consider eikonal equations defined over a disc such that the euclidean norm of the gradient of any solution is circularly symmetric vanishes exactly at the disc center and diverges to infinity as the circumference of the disc is approached contrary to earlier influential work a class of such eikonal equations is shown to possess simultaneously circularly symmetric and noncircularly symmetric bounded smooth solutions

a parallel fast sweeping method for the eikonal equation we present an algorithm for solving in parallel the eikonal equation the efficiency of our approach is rooted in the ordering and distribution of the grid points on the available processors we utilize a cuthill mckee ordering the advantages of our approach is that 1 the efficiency does not plateau for a large number of threads we compare our approach to the current state of the art parallel implementation of zhao 2007 14 and 2 the total number of iterations needed for convergence is the same as that of a sequential implementation i e our parallel implementation does not increase the complexity of the underlying sequential algorithm numerical examples are used to illustrate the efficiency of our approach

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 covering theory for non involutive set theoretic solutions to the yang baxter equation abstract an algebraic theory of non involutive set theoretic solutions to the yang baxter equation by means of a non commutative version of braces is established the structure group and the permutation group of a solution is put into a context of discrete lie theory this leads to a natural concept of covering which shares the good properties of coverings in geometry and topology in the special case of quandle coverings in knot theory known anomalies are removed

quantization of the nonlinear sigma model revisited we revisit the subject of perturbatively quantizing the nonlinear sigma model in two dimensions from a rigorous mathematical point of view our main contribution is to make precise the cohomological problem of eliminating potential anomalies that may arise when trying to preserve symmetries under quantization the symmetries we consider are twofold i diffeomorphism covariance for a general target manifold ii a transitive group of isometries when the target manifold is a homogeneous space we show that there are no anomalies in case i and that ii is also anomaly free under additional assumptions on the target homogeneous space in agreement with the work of friedan we carry out some explicit computations for the o n model finally we show how a suitable notion of the renormalization group establishes the ricci flow as the one loop renormalization group flow of the nonlinear sigma model

a decomposition theorem for square free unitary solutions of the quantum yang baxter equation abstract it is known that every skew polynomial ring with generating set x and binomial relations in the sense of gateva ivanova trans amer math soc 343 1994 203 is an artin schelter regular domain of global dimension x moreover every such ring gives rise to a non degenerate unitary set theoretical solution r x 2 x 2 of the quantum yang baxter equation which fixes the diagonal of x 2 gateva ivanova s conjecture talk at the international algebra conference miskolc hungary 1996 states that conversely every such solution r comes from a skew polynomial ring with binomial relations an equivalent conjecture duke math j 100 1999 169 says that the underlying set x is r decomposable we prove these conjectures and construct an indecomposable solution r with x which shows that an extension to infinite x is false

exploiting the generic viewpoint assumption the generic viewpoint assumption states that an observer is not in a special position relative to the scene it is commonly used to disqualify scene interpretations that assume special viewpoints following a binary decision that the viewpoint was either generic or accidental in this paper we apply bayesian statistics to quantify the probability of a view and so derive a useful tool to estimate scene parameters generic variables can include viewpoint object orientation and lighting position by considering the image as a differentiable function of these variables we derive the probability that a set of scene parameters created a given image this scene probability equation has three terms the fidelity of the scene interpretation to the image data the prior probability of the scene interpretation and a new genericity term which favors scenes likely to produce the observed image the genericity term favors image interpretations for which the image is stable with respect to changes in the generic variables it results from integration over the generic variables using a low noise approximation common in bayesian statistics this approach may increase the scope and accuracy of scene estimates it applies to a range of vision problems we show shape from shading examples where we rank shapes or reflectance functions in cases where these are otherwise unknown the rankings agree with the perceived values

a right invariant lattice order on groups of paraunitary matrices abstract in 14 rump defined and characterized noncommutative universal groups g x for generalized orthomodular lattices x we give an explicit description of g x in terms of paraunitary matrix groups whenever x is the orthomodular lattice of subspaces of a finite dimensional k vector space v that is equipped with an anisotropic symmetric k bilinear form

on complete integrals and uniqueness in shape from shading in this paper we analyze the problem of representing solutions of a first order partial differential equation in terms of complete integrals we show that some of the existing results referring to the representability problem are incomplete additionally we analyze in the context of complete integrals the uniqueness problem for the shape recovery of a smooth lambertian surface from an image obtained by illuminating this surface by an overhead distant point light source specifically we revisit the uniqueness results already existing in the shape from shading literature that concern eikonal equations corresponding to the images of a lambertian hemisphere and a lambertian plane we show that the latter results are incomplete and indicate how to fill the gaps in the corresponding proofs

a combinatorial approach to the set theoretic solutions of the yang baxter equation a bijective map r x2 x2 where x x1 xn is a finite set is called a set theoretic solution of the yang baxter equation ybe if the braid relation r12r23r12 r23r12r23 holds in x3 a nondegenerate involutive solution x r satisfying r xx xx for all x x is called square free solution there exist close relations between the square free set theoretic solutions of ybe the semigroups of i type the semigroups of skew polynomial type and the bieberbach groups as it was first shown in a joint paper with michel van den bergh in this paper we continue the study of square free solutions x r and the associated yang baxter algebraic structures the semigroup s x r the group g x r and the k algebra a k x r over a field k generated by x and with quadratic defining relations naturally arising and uniquely determined by r we study the properties of the associated yang baxter structures and prove a conjecture of the present author that the three notions a square free solution of set theoretic ybe

some remarks on the shape from shading problem in computer vision the shape from shading problem in computer vision see horn 11 deals with the reconstruction of the shape of an object from a two dimensional image we consider an object illuminated by a point source the reflected light being recorded on a photographic plate at a fixed position the intensity of the reflected light is a function of the position of the source the position of the plate and the normal to the surface of the object if the surface is given as u u x jj this leads see horn i to a first order partial differential equation of the form

on l spaces and left orderable fundamental groups examples suggest that there is a correspondence between l spaces and three manifolds whose fundamental groups cannot be left ordered in this paper we establish the equivalence of these conditions for several large classes of manifolds in particular we prove that they are equivalent for any closed connected orientable geometric three manifold that is non hyperbolic a family which includes all closed connected orientable seifert fibred spaces we also show that they are equivalent for the twofold branched covers of non split alternating links to do this we prove that the fundamental group of the twofold branched cover of an alternating link is left orderable if and only if it is a trivial link with two or more components we also show that this places strong restrictions on the representations of the fundamental group of an alternating knot complement with values in text homeo s 1

brst quantization of quasisymplectic manifolds and beyond we consider a class of factorizable poisson brackets which includes almost all reasonable poisson structures a particular case of the factorizable brackets are those associated with symplectic lie algebroids the brst theory is applied to describe the geometry underlying these brackets as well as to develop a deformation quantization procedure in this particular case this can be viewed as an extension of the fedosov deformation quantization to a wide class of irregular poisson structures in a more general case the factorizable poisson brackets are shown to be closely connected with the notion of n algebroid a simple description is suggested for the geometry underlying the factorizable poisson brackets based on construction of an odd poisson algebra bundle equipped with an abelian connection it is shown that the zero curvature condition for this connection generates all the structure relations for the n algebroid as well as a generalization of the yang baxter equation for the symplectic structure

symmetric shape from shading using self ratio image in this paper we present a symmetric shape from shading sfs approach to recover both shape and albedo for symmetric objects lambertian surfaces with unknown varying albedo and orthographic projections are assumed in our formulation of symmetric sfs we have two image irradiance equations one is the standard equation used in sfs and the other is a self ratio image irradiance equation this new image irradiance equation relates the self ratio image which is defined as the ratio of two halves of the input image to light source and surface shape the introduction of the self ratio image facilitates the direct use of symmetry cue based on the self ratio image a new model based symmetric source from shading algorithm is also presented we then propose symmetric sfs algorithms to recover both shape and albedo from a single image and present experimental results the new symmetric sfs scheme has one important property the existence of a unique global solution which consists of unique local solutions at each point simultaneously obtained using the intensity information at that point and the surrounding local region and the assumption of a cal c 2 surface proofs for the existence of a unique solution in the cases of unknown constant and non constant albedos are provided

numerical shape from shading for discontinuous photographic images the height u x y of a continuous lambertian surface of known albedo i e grayness is related to u x y information recoverable from a black and white flash photograph of the surface by the partial differential equation sqrt u 2 x u 2 y n 0 we review the notion of a unique viscosity solution for this equation when n is continuous and a recent unique extension of the viscosity solution when n is discontinuous we prove convergence to this extension for a wide class of the numerical algorithms that converge when n is continuous after discussing the properties of the extension and the order of error in the algorithms simulating the extension we point out warning signs which when observed in the numerical solution usually indicate that the surface is not continuous or that the viscosity solution or its extension does not correspond to the actual surface finally we discuss a method that in some of these cases allows us to correct the simulation and recover the actual surface again

retractability of set theoretic solutions of the yang baxter equation abstract it is shown that square free set theoretic involutive non degenerate solutions of the yang baxter equation whose associated permutation group referred to as an involutive yang baxter group is abelian are retractable in the sense of etingof schedler and soloviev this solves a problem of gateva ivanova in the case of abelian iyb groups it also implies that the corresponding finitely presented abelian by finite groups called the structure groups are poly z groups secondly an example of a solution with an abelian involutive yang baxter group which is not a generalized twisted union is constructed this answers in the negative another problem of gateva ivanova the constructed solution is of multipermutation level 3 retractability of solutions is also proved in the case where the natural generators of the iyb group are cyclic permutations moreover it is shown that such solutions are generalized twisted unions

the group of fractions of a torsion free lcm monoid is torsion free abstract we improve and shorten the argument given in j algebra 210 1998 291 297 in particular the fact that artin braid groups are torsion free now follows from garside s results almost immediately

noetherian properties of skew polynomial rings with binomial relations in this work we study standard finitely presented associative algebras over a fixed field k a restricted class of skew polynomial rings with quadratic relations considered in an earlier work of m artin and w schelter will be studied we call them binomial skew polynomial algebras we establish necessary and sufficient conditions for such an algebra to be a noetherian domain

gaussian groups are torsion free abstract assume thatgis a group of fractions of a cancellative monoid where lower common multiples exist and divisibility has no infinite descending chain thengis torsion free the result applies in particular to all finite coxeter type artin groups

quadratic algebras yang baxter equation and artin schelter regularity abstract we study two classes of quadratic algebras over a field k the class c n of all n generated pbw algebras with polynomial growth and finite global dimension and the class of quantum binomial algebras we show that a pbw algebra a is in c n iff its hilbert series is h a z 1 1 z n furthermore the class c n contains a unique up to isomorphism monomial algebra a k x 1 x n x j x i 1 i j n a surprising amount can be said when a is a quantum binomial algebra that is its defining relations are nondegenerate square free binomials x y c x y z t c x y k our main result shows that for an n generated quantum binomial algebra a the following conditions are equivalent i a is an artin schelter regular pbw algebra ii a is a yang baxter algebra that is the set of relations defines canonically a solution of the yang baxter equation iii a is a binomial skew polynomial ring with respect to an enumeration of x iv the koszul dual a is a quantum grassmann algebra

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

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

height and gradient from shading the method described here for recovering the shape of a surface from a shaded image can deal with complex wrinkled surfaces integrability can be enforced easily because both surface height and gradient are represented the robustness of the method stems in part from linearization of the reflectance map about the current estimate of the surface orientation at each picture cell the new scheme can find an exact solution of a given shape from shading problem even though a regularizing term is included this is a reflection of the fact that shape from shading problems are it not ill posed when boundary conditions are available or when the image contains singular points

yang baxter maps and symmetries of integrable equations on quad graphs a connection between the yang baxter relation for maps and the multidimensional consistency property of integrable equations on quad graphs is investigated the approach is based on the symmetry analysis of the corresponding equations it is shown that the yang baxter variables can be chosen as invariants of the multiparameter symmetry groups of the equations we use the classification results by adler bobenko and suris to demonstrate this method some new examples of yang baxter maps are derived in this way from multifield integrable equations

covariant and equivariant formality theorems abstract we give a proof of kontsevich s formality theorem for a general manifold using fedosov resolutions of algebras of polydifferential operators and polyvector fields the main advantage of our construction of the formality quasi isomorphism is that it is based on the use of covariant tensors unlike kontsevich s original proof which is based on jets of polydifferential operators and polyvector fields using our construction we prove that if a group g acts smoothly on a manifold m and m admits a g invariant affine connection then there exists a g equivariant quasi isomorphism of formality this result implies that if a manifold m is equipped with a smooth action of a finite or compact group g or equipped with a free action of a lie group g then m admits a g equivariant formality quasi isomorphism in particular this gives a solution of the deformation quantization problem for an arbitrary poisson orbifold

from local to global deformation quantization of poisson manifolds we give an explicit construction of a deformation quantization of the algebra of functions on a poisson manifold based on m kontsevich s local formula the deformed algebra of functions is realized as the algebra of horizontal sections of a vector bundle with flat connection

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

formality theorems for hochschild chains in the lie algebroid setting in this paper we prove lie algebroid versions of tsygan s formality conjecture for hochschild chains both in the smooth and holomorphic settings in the holomorphic setting our result implies a version of tsygan s formality conjecture for hochschild chains of the structure sheaf of any complex manifold and in the smooth setting this result allows us to describe quantum traces for an arbitrary poisson lie algebroid the proofs are based on the use of kontsevich s quasi isomorphism for hochschild cochains of r y 1 y d shoikhet s quasi isomorphism for hochschild chains of r y 1 y d and fedosov s resolutions of the natural analogues of hochschild co chain complexes associated with a lie algebroid

l algebras self similarity and l groups abstract every set x with a binary operation satisfying x y x z y x y z corresponds to a solution of the quantum yang baxter equation if the left multiplication is bijective w rump a decomposition theorem for square free unitary solutions of the quantum yang baxter equation adv math 193 2005 40 55 the same equation becomes a true statement of propositional logic if the binary operation is interpreted as implication in the present paper l algebras essentially defined by the mentioned equation are introduced and studied for example hilbert algebras locales left hoops pseudo mv algebras and l group cones are l algebras the main result states that every l algebra admits a self similar closure in a further step a structure group g x is associated to any l algebra x one more equation implies that the structure group g x is lattice ordered as an application we characterize the l algebras with a natural embedding into the negative cone of an l group in particular this implies mundici s equivalence between mv algebras and unital abelian l groups and dvurecenskij s non commutative generalization

involutive yang baxter groups in 1992 drinfeld posed the question of finding the set theoretic solutions of the yang baxter equation recently gateva ivanova and van den bergh and etingof schedler and soloviev have shown a group theoretical interpretation of involutive non degenerate solutions namely there is a one to one correspondence between involutive non degenerate solutions on finite sets and groups of i type a group g of i type is a group isomorphic to a subgroup of fa n sym n so that the projection onto the first component is a bijective map where fa n is the free abelian group of rank n and sym n is the symmetric group of degree n the projection of g onto the second component sym n we call an involutive yang baxter group iyb group this suggests the following strategy to attack drinfeld s problem for involutive non degenerate set theoretic solutions first classify the iyb groups and second for a given iyb group g classify the groups of i type with g as associated iyb group it is known that every iyb group is solvable in this paper some results supporting the converse of this property are obtained more precisely we show that some classes of groups are iyb groups we also give a non obvious method to construct infinitely many groups of i type and hence infinitely many involutive non degenerate set theoretic solutions of the yang baxter equation with a prescribed associated iyb group

deformation quantization in algebraic geometry abstract we study deformation quantizations of the structure sheaf o x of a smooth algebraic variety x in characteristic 0 our main result is that when x is d affine any formal poisson structure on x determines a deformation quantization of o x canonically up to gauge equivalence this is an algebro geometric analogue of kontsevich s celebrated result

twisted deformation quantization of algebraic varieties abstract let x be a smooth algebraic variety over a field k containing the real numbers we introduce the notion of twisted associative resp poisson deformation of the structure sheaf o x these are stack like versions of usual deformations we prove that there is a twisted quantization operation from twisted poisson deformations to twisted associative deformations which is canonical and bijective on gauge equivalence classes this result extends work of kontsevich and our own earlier work on deformation quantization of algebraic varieties

set theoretic solutions of the yang baxter equation rc calculus and garside germs abstract building on a result by w rump we show how to exploit the right cyclic law x y x z y x y z in order to investigate the structure groups and monoids attached with involutive nondegenerate set theoretic solutions of the yang baxter equation we develop a sort of right cyclic calculus and use it to obtain short proofs for the existence both of the garside structure and of the i structure of such groups we describe finite quotients that play for the considered groups the role that coxeter groups play for artin tits groups

braid groups and left distributive operations the decidability of the word problem for the free left distributive law is proved by introducing a structure group which describes the underlying identities this group is closely connected with artin s braid group b braid colourings associated with free left distributive structures are used to show the existence of a unique ordering on the braids which is compatible with left translation and such that every generator ai is preponderant over all ak with k i this ordering is a linear ordering the first goal of the present paper is to give a proof of the following result which had been conjectured for several years theorem there is an effective algorithm for deciding whether a given identity is or is not a consequence of the left distributivity identity x yz xy xz until recently this question has had a rather unusual status conditional solutions were given independently in 6 and 24 where the decision problem was reduced to a specific algebraic hypothesis one which had been shown by richard laver to be a consequence of a very strong set theoretical axiom of a type which certainly cannot be derived from the usual axioms of set theory namely a large cardinal axiom the question as to whether a strong axiom of this type was actually needed remained open opinions were in fact divided a connection between large cardinals and a purely finitistic problem of this caliber would seem paradoxical but it is well known that problems of a combinatorial type can embody surprisingly strong proof principles see for instance 28 and some work on free distributive structures has shown that they do give rise to intrinsically complex objects typically nonprimitive recursive ones cf 12 13 we will show that in the present case a solution which is purely algebraic in terms of the methods employed and the spirit of the argument can in fact be given in particular no unusual set theoretical axioms are required for this argument the decision method described in 6 was fully effective but the proof of the correctness of the algorithm amounted to a direct invocation of this specific algebraic hypothesis which followed from a large cardinal axiom with no hint of a direct proof we shall refine this decision method below introducing uniqueness at each step of the process and the correctness will then be seen to follow very naturally received by the editors february 5 1993 1991 mathematics subject classification primary 20f36 1 7a30 1 7a50

gaussian groups and garside groups two generalisations of artin groups it is known that a number of algebraic properties of the braid groups extend to arbitrary finite coxeter type artin groups here we show how to extend the results to more general groups that we call garside groups define a gaussian monoid to be a finitely generated cancellative monoid where the expressions of a given element have bounded lengths and where left and right lowest common multiples exist a garside monoid is a gaussian monoid in which the left and right lowest common multiples satisfy an additional symmetry condition a gaussian group is the group of fractions of a gaussian monoid and a garside group is the group of fractions of a garside monoid braid groups and more generally finite coxeter type artin groups are garside groups we determine algorithmic criteria in terms of presentations for recognizing gaussian and garside monoids and groups and exhibit infinite families of such groups we describe simple algorithms that solve the word problem in a gaussian group show that these algorithms have a quadratic complexity if the group is a garside group and prove that garside groups have quadratic isoperimetric inequalities we construct normal forms for gaussian groups and prove that in the case of a garside group the language of normal forms is regular symmetric and geodesic has the 5 fellow traveller property and has the uniqueness property this shows in particular that garside groups are geodesically fully biautomatic finally we consider an automorphism of a finite coxeter type artin group derived from an automorphism of its defining coxeter graph and prove that the subgroup of elements fixed by this automorphism is also a finite coxeter type artin group that can be explicitly determined 1991 mathematics subject classification primary 20f05 20f36 secondary 20b40 20m05

semigroups ofi type abstract assume thatsis a semigroup generated by x1 xn and let u be the multiplicative free commutative semigroup generated by u1 un we say thatsis ofi typeif there is a bijectivev u ssuch that for alla u v u1a v una x1v a xnv a this condition appeared naturally in the work on sklyanin algebras by john tate and the second author in this paper we show that the condition for a semigroup to be ofi type is related to various other mathematical notions found in the literature in particular we show that semigroups ofi type appear in the study of the set theoretic solutions of the yang baxter equation in the theory of bieberbach groups and in the study of certain skew binomial polynomial rings which were introduced by the first author

one sided orthogonality orthomodular spaces quantum sets and a class of garside groups abstract the classification of orthomodular lattices by means of their structure group is extended to non symmetric orthogonality relations spaces with a non hermitian sesquilinear form quantum sets and a new class of garside groups are given as examples

braces radical rings and the quantum yang baxter equation abstract non degenerate cycle sets are equivalent to non degenerate unitary set theoretical solutions of the quantum yang baxter equation we embed such cycle sets into generalized radical rings braces and study their interaction in this context we establish a galois theory between ideals of braces and quotient cycle sets our main result determines the relationship between two square free cycle sets operating transitively on each other