mckay correspondence for symplectic quotient singularities let v be a finite dimensional complex vector space and g sl v a finite subgroup the quotient variety x v g is usually singular if dim v 2 then there exists a canonical smooth projective resolution y x of the singular variety x this resolution is crepant which in this case means that the manifold y has trivial canonical bundle we recall the precise definitions in definition 1 2 the fiber 1 0 y over the singular point 0 x v g is a rational curve whose components are numbered by the nontrivial conjugacy classes in g the homology classes of these components freely generate h2 y q all the other homology groups of y are trivial except for h0 y q q this situation was described by j mckay in mck it is known as the mckay correspondence in higher dimensions the picture splits into two parts one can consider two separate questions 1 when does the quotient x v g admit a smooth crepant resolution y x 2 assuming there exists a smooth crepant resolution y x what can one say about the homology h y q

normality of marsden weinstein reductions for representations of quivers we prove that the marsden weinstein reductions for the moment map associated to representations of a quiver are normal varieties we give an application to conjugacy classes of matrices

reflection functors and symplectic reflection algebras for wreath products we construct reflection functors on categories of modules over deformed wreath products of the preprojective algebra of a quiver these functors give equivalences of categories associated to generic parameters which are in the same orbit under the weyl group action we give applications to the representation theory of symplectic reflection algebras of wreath product groups

sheaves on fibered threefolds and quiver sheaves this paper classifies a class of holomorphic d branes closely related to framed torsion free sheaves on threefolds fibered in resolved ade surfaces over a general curve c in terms of representations with relations of a twisted kronheimer nakajima type quiver in the category coh c of coherent sheaves on c for the local calabi yau case c cong bf a 1 and special choice of framing one recovers the n 1 ade quiver studied by cachazo katz vafa

noncommutative resolutions of ade fibered calabi yau threefolds in this paper we construct noncommutative resolutions of a certain class of calabi yau threefolds studied by cachazo et al geometric transitions and n 1 quiver theories http arxiv org abs hep th 0108120v2 2001 the threefolds under consideration are fibered over a complex plane with the fibers being deformed kleinian singularities the construction is in terms of a noncommutative algebra introduced by ginzburg calabi yau algebras http arxiv org abs math 0612139v2 2006 which we call the n 1 ade quiver algebra

poisson deformations of symplectic quotient singularities abstract we establish a connection between smooth symplectic resolutions and symplectic deformations of a possibly singular affine poisson variety in particular let v be a finite dimensional complex symplectic vector space and g sp v a finite subgroup our main result says that the so called calogero moser deformation of the orbifold v g is in an appropriate sense a versal poisson deformation that enables us to determine the algebra structure on the cohomology h x c of any smooth symplectic resolution x v g multiplicative mckay correspondence we prove further that if g gl h is an irreducible weyl group and v h h then no smooth symplectic resolution of v g exists unless g is of types a b c

symplectic resolutions for nilpotent orbits in this paper firstly we calculate picard groups of a nilpotent orbit in a classical complex simple lie algebra and discuss the properties of being factorial and factorial for the normalization tilde of the closure of then we consider the problem of symplectic resolutions for tilde our main theorem says that for any nilpotent orbit in a semi simple complex lie algebra equipped with the kostant kirillov symplectic form if for a resolution z tilde the 2 form defined on 1 extends to a symplectic 2 form on z then z is isomorphic to the cotangent bundle t g p of a projective homogeneous space and is the collapsing of the zero section it proves a conjecture of cho miyaoka shepherd barron in this special case using this theorem we determine all varieties tilde which admit such a resolution

a multiscale measure for mixing abstract we present a multiscale measure for mixing that is based on the concept of weak convergence and averages the mixedness of an advected scalar field at various scales this new measure referred to as the mix norm resolves the inability of the l 2 variance of the scalar density field to capture small scale variations when advected by chaotic maps or flows in addition the mix norm succeeds in capturing the efficiency of a mixing protocol in the context of a particular initial scalar field wherein lyapunov exponent based measures fail to do so we relate the mix norm to the classical ergodic theoretic notion of mixing and present its formulation in terms of the power spectrum of the scalar field we demonstrate the utility of the mix norm by showing how it measures the efficiency of mixing due to various discrete dynamical systems and to diffusion in particular we show that the mix norm can capture known exponential and algebraic mixing properties of certain maps we also analyze numerically the behaviour of scalar fields evolved by the standard map using the mix norm

representations of symplectic reflection algebras and resolutions of deformations of symplectic quotient singularities we give an equivalence of triangulated categories between the derived category of finitely generated representations of symplectic reflection algebras associated with wreath products with parameter t 0 and the derived category of coherent sheaves on a crepant resolution of the spectrum of the centre of these algebras

calogero moser space and kostka polynomials abstract we consider the canonical map from the calogero moser space to symmetric powers of the affine line sending conjugacy classes of pairs of n n matrices to their eigenvalues we show that the character of a natural c action on the scheme theoretic zero fiber of this map is given by kostka polynomials a similar result is proved for a cyclic version of the calogero moser space

the mckay correspondence as an equivalence of derived categories the classical mckay correspondence relates representations of a finite subgroup g sl 2 c to the cohomology of the well known minimal resolution of the kleinian singularity c2 g gonzalez sprinberg and verdier 10 interpreted the mckay correspondence as an isomorphism on k theory observing that the representation ring of g is equal to the g equivariant k theory of c2 more precisely they identify a basis of the k theory of the resolution consisting of the classes of certain tautological sheaves associated to the irreducible representations of g

accuracy and stability of integration algorithms for elastoplastic constitutive relations an analysis of accuracy and stability of algorithms for the integration of elastoplastic constitutive relations is carried out in this paper reference is made to a very general internal variable formulation of plasticity and to two families of algorithms that generalize the well known trapezoidal and midpoint rules to fit the present context other integration schemes such as the radial return mean normal and closest point procedures are particular cases of this general formulation the meaning of first and second order accuracy in the presence of the plastic consistency condition is examined in detail and the criteria derived are used to identify two second order accurate members of the proposed algorithms a general methodology is also derived whereby the numerical stability properties of integration schemes can be systematically assessed with the aid of this methodology the generalized midpoint rule is seen to have far better stability properties than the generalized trapezoidal rule finally numerical examples are presented that illustrate the performance of the algorithms

germs of arcs on singular algebraic varieties and motivic integration we study the scheme of formal arcs on a singular algebraic variety and its images under truncations we prove a rationality result for the poincare series of these images which is an analogue of the rationality of the poincare series associated to p adic points on a p adic variety the main tools which are used are semi algebraic geometry in spaces of power series and motivic integration a notion introduced by m kontsevich in particular we develop the theory of motivic integration for semi algebraic sets of formal arcs on singular algebraic varieties we prove a change of variable formula for birational morphisms and we prove a geometric analogue of a result of oesterle

application of a domain decomposition method to elastoplastic problems abstract in this paper parallel computing is applied to the solution of elastoplastic problems with strain hardening after a brief discussion of the classical variational and finite element formulations of the plasticity problem an algorithm is proposed it is based on a variant of the johnson weak formulation and on the schwarz domain decomposition method the domain is triangulated and the subdomains are defined as the union of the elements having a node in common this leads to many small subproblems which can be solved in a simple and efficient manner the algorithm has been implemented on a parallel computer the numerical results obtained for two examples a sheared plate and a perforated plate in tension compare quite favorably with simulations using a classical finite element code the computing times obtained show that the proposed method utilizes the parallel structure of the computer very efficiently

rational cherednik algebras and hilbert schemes let h c be the rational cherednik algebra of type a n 1 with spherical subalgebra u c e h c e then u c is filtered by order of differential operators with associated graded ring mbox gr u c mathbb c mathfrak h oplus mathfrak h where w is the n th symmetric group we construct a filtered mathbb z algebra b such that under mild conditions on c the category b qgr of graded noetherian b modules modulo torsion is equivalent to u c mod the associated graded mathbb z algebra mbox gr b has mbox gr b lqgr simeq coh hilb n this can be regarded as saying that u c simultaneously gives a non commutative deformation of mathfrak h oplus mathfrak h w and of its resolution of singularities hilb n rightarrow mathfrak h oplus mathfrak h as we show elsewhere this result is a powerful tool for studying the representation theory of h c and its relationship to hilb n

a generalization of the schwarz alternating method to an arbitrary number of subdomains we give a generalization of the schwarz alternating method to an arbitrary number of subdomains as applications we prove that the method converges for the main second order boundary value problems as well as for the system of linear elasticity for the dirichlet problems it is proved that the method converges geometrically and some numerical examples are given

homological aspects of noetherian pi hopf algebras and irreducible modules of maximal dimension abstract we prove that a noetherian hopf algebra of finite global dimension possesses further attractive homological properties at least when it satisfies a polynomial identity this applies in particular to quantized enveloping algebras and to quantized function algebras at a root of unity as well as to classical enveloping algebras in positive characteristic in all three cases we show that these algebras are auslander regular and macaulay we derive representation theoretic consequences concerning the coincidence of the non azumaya and singular loci for each of the above three classes of algebras

acceleration of one dimensional mixing by discontinuous mappings abstract the paper considers a simple class of models for mixing of a passive tracer into a bulk material that is essentially one dimensional we examine the relative rates of mixing due to diffusion stretch and fold operations and permutation of sections of the sample in particular we show how a combination of diffusion with permutation of sections of the sample chopping and shuffling can achieve a faster rate of mixing than pure diffusion this is done by numerical approximation of eigenvalues of certain linear operators

finite groups acting linearly hochschild cohomology and the cup product abstract when a finite group acts linearly on a complex vector space the natural semi direct product of the group and the polynomial ring over the space forms a skew group algebra this algebra plays the role of the coordinate ring of the resulting orbifold and serves as a substitute for the ring of invariant polynomials from the viewpoint of geometry and physics its hochschild cohomology predicts various hecke algebras and deformations of the orbifold in this article we investigate the ring structure of the hochschild cohomology of the skew group algebra we show that the cup product coincides with a natural smash product transferring the cohomology of a group action into a group action on cohomology we express the algebraic structure of hochschild cohomology in terms of a partial order on the group modulo the kernel of the action this partial order arises after assigning to each group element the codimension of its fixed point space we describe the algebraic structure for coxeter groups where this partial order is given by the reflection length function a similar combinatorial description holds for an infinite family of complex reflection groups

symplectic reflection algebras calogero moser space and deformed harish chandra homomorphism to any finite group sp v of automorphisms of a symplectic vector space v we associate a new multi parameter deformation h of the algebra v smash product of with the polynomial algebra on v the parameter runs over points of r where r number of conjugacy classes of symplectic reflections in the algebra h called a symplectic reflection algebra is related to the coordinate ring of a poisson deformation of the quotient singularity v this leads to a symplectic analogue of mckay correspondence which is most complete in case of wreath products if is the weyl group of a root system in a vector space and v then the algebras h are certain rational degenerations of the double affine hecke algebra introduced earlier by cherednik

deformations of algebras defined by tilting bundles abstract in this paper we produce noncommutative algebras derived equivalent to deformations of schemes with tilting bundles we do this in two settings first proving that a tilting bundle on a scheme lifts to a tilting bundle on an infinitesimal deformation of that scheme and then expanding this result to c equivariant deformations over schemes with a good c action in both these situations the endomorphism algebra of the lifted tilting bundle produces a deformation of the original endomorphism algebra and this is a graded deformation in the c equivariant case we apply our results to rational surface singularities generalising the deformed preprojective algebras of crawley boevey and holland and also to symplectic situations where the deformations produced are related to symplectic reflection algebras

non commutative resolutions of quotient singularities for reductive groups in this paper we generalize standard results about non commutative resolutions of quotient singularities for finite groups to arbitrary reductive groups we show in particular that quotient singularities for reductive groups always have non commutative resolutions in an appropriate sense moreover we exhibit a large class of such singularities which have twisted non commutative crepant resolutions we discuss a number of examples both new and old that can be treated using our methods notably we prove that twisted non commutative crepant resolutions exist in previously unknown cases for determinantal varieties of symmetric and skew symmetric matrices in contrast to almost all prior results in this area our techniques are algebraic and do not depend on knowing a commutative resolution of the singularity

a tfeti domain decomposition solver for elastoplastic problems we propose an algorithm for the efficient parallel implementation of elastoplastic problems with hardening based on the so called tfeti total finite element tearing and interconnecting domain decomposition method we consider an associated elastoplastic model with the von mises plastic criterion and the linear isotropic hardening law such a model is discretized by the implicit euler method in time and the consequent one time step elastoplastic problem by the finite element method in space the latter results in a system of nonlinear equations with a strongly semismooth and strongly monotone operator the semismooth newton method is applied to solve this nonlinear system corresponding linearized problems arising in the newton iterations are solved in parallel by the above mentioned tfeti domain decomposition method the proposed tfeti based algorithm was implemented in matlab parallel environment and its performance was illustrated on a 3d elastoplastic benchmark numerical results for different time discretizations and mesh levels are presented and discussed and a local quadratic convergence of the semismooth newton method is observed

on the schwarz alternating method with more than two subdomains for nonlinear monotone problems in a hilbert space an iterative process is defined which generalizes the schwarz alternating method for nonlinear monotone problems to more than two subdomains and the convergence of this process is proved in the case of the problems with the solution in a convex of h 0 1 omega it is proved that the method converges geometrically numerical results are given concerning the problem of elastic plastic torsion of prismatic bars and the problem of seepage of fluids through porous media

twisted graded hecke algebras abstract we generalize graded hecke algebras to include a twisting two cocycle for the associated finite group we give examples where the parameter spaces of the resulting twisted graded hecke algebras are larger than that of the graded hecke algebras we prove that twisted graded hecke algebras are particular types of deformations of a crossed product

deformed preprojective algebras and symplectic reflection algebras for wreath products abstract we determine the pbw deformations of the wreath product of a symmetric group with a deformed preprojective algebra of an affine dynkin quiver in particular we show that there is precisely one parameter which does not come from deformation of the preprojective algebra we prove that the pbw deformation is morita equivalent to a corresponding symplectic reflection algebra for wreath product

control of mixing in fluid flow a maximum entropy approach in many technological processes a fundamental stage involves the mixing of two or more fluids as a result the design of optimal mixing protocols is a problem of both fundamental and practical importance in this paper the authors formulate a prototypical mixing problem in a control framework where the objective is to determine the sequence of fluid flows that will maximize entropy by developing the appropriate ergodic theoretic tools for the determination of entropy of periodic sequences they derive the form of the protocol which maximizes entropy among all of the possible periodic sequences composed of two shear flows orthogonal to each other the authors discuss the relevance of their results in the interpretation of previous studies of mixing protocols