lie theory for nilpotent l algebras the deligne groupoid is a functor from nilpotent differential graded lie algebras concentrated in positive degrees to groupoids in the special case of lie algebras over a field of characteristic zero it gives the associated simply connected lie group we generalize the deligne groupoid to a functor from l algebras concentrated in degree n to n groupoids we actually construct the nerve of the n groupoid which is an enriched kan complex the construction of gamma is quite explicit it is based on dupont s proof of the de rham theorem and yields higher dimensional analogues of holonomy and of the campbell hausdorff formula in the case of abelian l algebras i e chain complexes the functor is the dold kan simplicial set

factorization homology i higher categories abstract we construct a pairing which we call factorization homology between framed manifolds and higher categories the essential geometric notion is that of a vari framing of a stratified manifold which is a framing on each stratum together with a coherent system of compatibilities of framings along links between strata our main result constructs labeling systems on disk stratified vari framed n manifolds from n categories these n categories in contrast with the literature to date are not required to have adjoints this allows the following conceptual definition the factorization homology m c of a framed n manifold m with coefficients in an n category c is the classifying space of c labeled disk stratifications over m the core calculation underlying our main result is the following for any disk stratified manifold the space of conically smooth diffeomorphisms which preserve a vari framing is discrete

on the reconstruction problem in mirror symmetry abstract let m b be a lagrangian torus fibration with singularities such that the fibers are of maslov index zero and unobstructed the paper constructs a rigid analytic space m 0 over the novikov field which is a deformation of the semi flat complex structure of the dual torus fibration over the smooth locus b 0 b of transition functions of m 0 are obtained via a homomorphisms which capture the wall crossing phenomenon of moduli spaces of holomorphic disks

deformations of coisotropic submanifolds and strong homotopy lie algebroids in this paper we study deformations of coisotropic submanifolds in a symplectic manifold first we derive the equation that governs c deformations of coisotropic submanifolds and define the corresponding c moduli space of coisotropic submanifolds modulo the hamiltonian isotopies this is a non commutative and non linear generalization of the well known description of the local deformation space of lagrangian submanifolds as the set of graphs of closed one forms in the darboux weinstein chart of a given lagrangian submanifold we then introduce the notion of strong homotopy lie algebroid or l algebroid and associate a canonical isomorphism class of strong homotopy lie algebroids to each pre symplectic manifold y and identify the formal deformation space of coisotropic embeddings into a symplectic manifold in terms of this strong homotopy lie algebroid the formal moduli space then is provided by the gauge equivalence classes of solutions of a version of the maurer cartan equation or the master equation of the strong homotopy lie algebroid and plays the role of the classical part of the moduli space of quantum deformation space of coisotropic a branes we provide a criterion for the unobstructedness of the deformation problem and analyze a family of examples that illustrates that this deformation problem is obstructed in general and heavily depends on the geometry and dynamics of the null foliation

formality theorem for gerbes abstract the main result of the present paper is an analogue of kontsevich formality theorem in the context of the deformation theory of gerbes we construct an l deformation of the schouten algebra of multi vectors which controls the deformation theory of a gerbe

on hamiltonian perturbations of hyperbolic systems of conservation laws ii universality of critical behaviour hamiltonian perturbations of the simplest hyperbolic equation u t a u u x 0 are studied we argue that the behaviour of solutions to the perturbed equation near the point of gradient catastrophe of the unperturbed one should be essentially independent on the choice of generic perturbation neither on the choice of generic solution moreover this behaviour is described by a special solution to an integrable fourth order ode

on reduced convex qp formulations of monotone lcps techniques for transforming convex quadratic programs qps into monotone linear complementarity problems lcps and vice versa are well known we describe a class of lcps for which a reduced qp formulation one that has fewer constraints than the standard qp formulation is available we mention several instances of this class including the known case in which the coefficient matrix in the lcp is symmetric

fixed points for fuzzy mappings in this paper the problem of the existence and computation of fixed points for fuzzy mappings is approached a fuzzy mapping r over a set x is defined to be a function attaching to each x in x a fuzzy subset r of x an element x of x is called fixed point of r iff its membership degree to r is at least equal to the membership degree to r of any y x i e r r y y x two existence theorems for fixed points of a fuzzy mapping are proved and an algorithm for computing approximations of such a fixed point is described the convergence theorem of our algorithm is proved under the restrictive assumption that for any x in x the membership function of r has a complementary function examples of fuzzy mappings having this property are given but the problem of proving general criteria for a function to have a complementary remain open

bihamiltonian cohomologies and integrable hierarchies i a special case we present some general results on properties of the bihamiltonian cohomologies associated to bihamiltonian structures of hydrodynamic type and compute the third cohomology for the bihamiltonian structure of the dispersionless kdv hierarchy the result of the computation enables us to prove the existence of bihamiltonian deformations of the dispersionless kdv hierarchy starting from any of its infinitesimal deformations

derivations the lawrence sullivan interval and the fiorenza manetti mapping cone we describe the rational homotopy type of any component of the based mapping space map x y as an explicit l algebra defined on the desuspended and positive derivations between quillen models of x and y when considering the lawrence sullivan model of the interval we obtain an l model of the contractible path space of y we then relate this in a geometrical and natural manner to the l structure on the fiorenza manetti mapping cone of any differential graded lie algebra morphism two in principal different algebraic objects in which bernoulli numbers appear

descent for n bundles abstract given a lie group g one constructs a principal g bundle on a manifold x by taking a cover u x specifying a transition cocycle on the cover and descending the trivialized bundle u g along the cover we demonstrate the existence of an analogous construction for local n bundles for general n we establish analogues for simplicial lie groups of moore s results on simplicial groups these imply that bundles for strict lie n groups arise from local n bundles our construction leads to simple finite dimensional models of lie 2 groups such as string n from cocycle data

boundary values as hamiltonian variables i new poisson brackets the ordinary poisson brackets in field theory do not fulfill the jacobi identity if boundary values are not reasonably fixed by special boundary conditions it is shown that these brackets can be modified by adding some surface terms to lift this restriction the new brackets generalize a canonical bracket considered by lewis marsden montgomery and ratiu for the free boundary problem in hydrodynamics the definition of poisson brackets used herein permits the treating of to treat boundary values of a field on equal footing with its internal values and the direct estimation of estimate the brackets between both surface and volume integrals this construction is applied to any local form of poisson brackets

the wess zumino witten term of the m5 brane and differential cohomotopy we combine rational homotopy theory and higher lie theory to describe the wess zumino witten wzw term in the m5 brane sigma model we observe that this term admits a natural interpretation as a twisted 7 cocycle on super minkowski spacetime with coefficients in the rational 4 sphere this exhibits the wzw term as an element in twisted cohomology with the twist given by the cocycle of the m2 brane we consider integration of this rational situation to differential cohomology and differential cohomotopy

poisson bracket on 1 forms and evolutionary partial differential equations we introduce a bracket on 1 forms defined on i e the infinite jet extension of the space of loops and prove that it satisfies the standard properties of a poisson bracket using this bracket we show that certain hierarchies appearing in the framework of f manifolds with a compatible flat connection m are hamiltonian in a generalized sense moreover we show that if a metric g compatible with is also invariant with respect to then this generalized hamiltonian setup reduces to the standard one

a version of the goldman millson theorem for filtered l algebras abstract in this paper we consider l algebras equipped with complete descending filtrations we prove that under some mild conditions an l quasi isomorphism u l l induces a weak equivalence between the deligne getzler hinich dgh groupoids corresponding to l and l respectively this paper may be considered as a modest addition to foundational paper 10 by ezra getzler

l infinity pairs and applications to singularities abstract over a field of characteristic zero every deformation problem with cohomology constraints is controlled by a pair consisting of a differential graded lie algebra together with a module unfortunately these pairs are usually infinite dimensional we show that every deformation problem with cohomology constraints is controlled by a typically finite dimensional l pair as a first application we show that for complex algebraic varieties with no weight zero 1 cohomology classes the components of the cohomology jump loci of rank one local systems containing the constant sheaf are tori this imposes restrictions on the fundamental groups the same holds for links and milnor fibers

lagrangian approach to sheaves of vertex algebras we explain how sheaves of vertex algebras are related to mathematical structures encoded by a class of lagrangians the exposition is focused on two examples the wzw model and the 1 1 supersymmetric model we conclude by showing how to construct a family of vertex algebras with base the barannikov kontsevich moduli space thus furnishing the b model moduli for witten s half twisted model

semiregularity and obstructions of complete intersections abstract we prove that on a smooth projective variety over an algebraically closed field of characteristic 0 the semiregularity map annihilates every obstruction to embedded deformations of a local complete intersection subvariety with extendable normal bundle the proof is based on the theory of l algebras and tamarkin tsygan calculus on the de rham complex of dg schemes

presenting higher stacks as simplicial schemes abstract we show that an n geometric stack may be regarded as a special kind of simplicial scheme namely a duskin n hypergroupoid in affine schemes where surjectivity is defined in terms of covering maps yielding artin n stacks deligne mumford n stacks and n schemes as the notion of covering varies this formulation adapts to all hag contexts so in particular works for derived n stacks replacing rings with simplicial rings we exploit this to describe quasi coherent sheaves and complexes on these stacks and to draw comparisons with kontsevich s dg schemes as an application we show how the cotangent complex controls infinitesimal deformations of higher and derived stacks

on bi hamiltonian deformations of exact pencils of hydrodynamic type in this paper we are interested in nontrivial bi hamiltonian deformations of the poisson pencil deformations are generated by a sequence of vector fields x2 x3 x4 where each xk is homogeneous of degree k with respect to a grading induced by rescaling constructing recursively the vector fields xk one obtains two types of relations involving their unknown coefficients one set of linear relations and an other one which involves quadratic relations we prove that the set of linear relations has a geometric meaning using miura quasitriviality the set of linear relations expresses the tangency of the vector fields xk to the symplectic leaves of 1 and this tangency condition is equivalent to the exactness of the pencil moreover extending the results of lorenzoni p 2002 j geom phys 44 331 75 we construct the nontrivial deformations of the poisson pencil up to the eighth order in the deformation parameter showing therefore that deformations are unobstructed and that both poisson structures are polynomial in the derivatives of u up to that order

implementation of a continuation method for normal maps this paper presents an implementation of a nonsmooth continuation method of which the idea was originally put forward by alexander et al we show how the method can be computationally implemented and present numerical results for variational inequality problems in up to 96 variables

a darboux getzler theorem for scalar difference hamiltonian operators in this paper we extend the notion of poisson lichnerowicz cohomology an object encapsulating the building blocks for the theory of deformations of hamiltonian operators to the difference case a local scalar difference hamiltonian operator is a polynomial in the shift operator and its inverse with coefficients in the algebra of difference functions endowing the space of local functionals with the structure of a lie algebra its poisson lichnerowicz cohomology carries information about the center the symmetries and the admissible deformations of such an algebra the analogue notion for the differential case has been widely investigated the first and most important result is the triviality of all but the lowest cohomology for first order hamiltonian differential operators due to getzler we study the poisson lichnerowicz cohomology for the operator k 0 mathcal s mathcal s 1 which is the normal form for 1 1 order scalar difference hamiltonian operators we obtain the same result as getzler did namely h p k 0 0 forall p 1 and explicitly compute h 0 k 0 and h 1 k 0 we then apply our main result to the classification of lower order scalar hamiltonian operators recently obtained by de sole kac valeri and wakimoto

configurations braids and homotopy groups the main results of this article are certain connections between braid groups and the homotopy groups of the 2 sphere the connections are given in terms of brunnian braids over the disk and over the 2 sphere the techniques arise from the natural structure of simplicial and structures on fundamental groups of configuration spaces

deformations of affine varieties and the deligne crossed groupoid abstract let x be a smooth affine algebraic variety over a field k of characteristic 0 and let r be a complete parameter k algebra e g r k we consider associative resp poisson r deformations of the structure sheaf o x the set of r deformations has a crossed groupoid i e strict 2 groupoid structure our main result is that there is a canonical equivalence of crossed groupoids from the deligne crossed groupoid of normalized polydifferential operators resp polyderivations of x to the crossed groupoid of associative resp poisson r deformations of o x the proof relies on a careful study of adically complete sheaves in the associative case we also have to use ring theory ore localizations and the properties of the hochschild cochain complex the results of this paper extend previous work by various authors they are needed for our work on twisted deformation quantization of algebraic varieties

poincare koszul duality we prove a duality for factorization homology which generalizes both usual poincare duality for manifolds and koszul duality for mathcal e n algebras the duality has application to the hochschild homology of associative algebras and enveloping algebras of lie algebras we interpret our result at the level of topological quantum field theory

deformation theory of objects in homotopy and derived categories i general theory abstract this is the first paper in a series we develop a general deformation theory of objects in homotopy and derived categories of dg categories namely for a dg module e over a dg category we define four deformation functors def h e codef h e def e codef e the first two functors describe the deformations and co deformations of e in the homotopy category and the last two in the derived category we study their properties and relations these functors are defined on the category of artinian not necessarily commutative dg algebras

frobenius manifolds and central invariants for the drinfeld sokolov bihamiltonian structures the drinfeld sokolov construction associates a hierarchy of bihamiltonian integrable systems with every untwisted affine lie algebra we compute the complete sets of invariants of the related bihamiltonian structures with respect to the group of miura type transformations

hamiltonian pdes deformations integrability solutions we review recent classification results on the theory of systems of nonlinear hamiltonian partial differential equations with one spatial dimension including a perturbative approach to the integrability theory of such systems and discuss universality conjectures describing critical behaviour of solutions to such systems

deformations and homotopy theory of relative rota baxter lie algebras we determine the l infty algebra that controls deformations of a relative rota baxter lie algebra and show that it is an extension of the dg lie algebra controlling deformations of the underlying mathsf lie mathsf rep pair by the dg lie algebra controlling deformations of the relative rota baxter operator consequently we define the cohomology of relative rota baxter lie algebras and relate it to their infinitesimal deformations a large class of relative rota baxter lie algebras is obtained from triangular lie bialgebras and we construct a map between the corresponding deformation complexes next the notion of a homotopy relative rota baxter lie algebra is introduced we show that a class of homotopy relative rota baxter lie algebras is intimately related to pre lie infty algebras

jacobi structures of evolutionary partial differential equations abstract in this paper we introduce the notion of infinite dimensional jacobi structure to describe the geometrical structure of a class of nonlocal hamiltonian systems which appear naturally when applying reciprocal transformations to hamiltonian evolutionary pdes we prove that our class of infinite dimensional jacobi structures is invariant under the action of reciprocal transformations that only change the spatial variable the main technical tool is in a suitable generalization of the classical schouten nijenhuis bracket to the space of the so called quasi local multi vectors and a simple realization of this structure in the framework of supermanifolds these constructions are used to compute the lichnerowicz jacobi cohomologies and to prove a darboux theorem for jacobi structures with hydrodynamic leading terms we also introduce the notion of bi jacobi structures and consider the integrability of a system of evolutionary pdes that possesses a bi jacobi structure

continuity of fuzzy multivalued mappings continuity of multivalued mappings is characterized by two types of semi continuity lower and upper semi continuity in this paper different equivalent ways in which these concepts can be expressed are investigated and then used to define lower and upper semi continuous fuzzy multivalued mappings some shortcomings in existing definitions of upper semi continuous multivalued mappings are addressed several relationships between the lower semi continuity of fuzzy multivalued mappings and the classical lower semi continuity of their cuts or the lower semi continuity w r t the level topologies are established in the setting of stratified fuzzy topological spaces two different types of upper semi continuity of fuzzy multivalued mappings are introduced inspired by the two equivalent ones in the crisp case the relationships between these two types are studied completely many interesting properties of lower and upper semi continuous fuzzy multivalued mappings are presented including the composition and union of lower and upper semi continuous fuzzy multivalued mappings and the compactness preserving property of upper semi continuous and compact valued fuzzy multivalued mappings

dispersive deformations of hamiltonian systems of hydrodynamic type in 2 1 dimensions abstract we develop a theory of integrable dispersive deformations of 2 1 dimensional hamiltonian systems of hydrodynamic type following the scheme proposed by dubrovin and his collaborators in 1 1 dimensions our results show that the multi dimensional situation is far more rigid and generic hamiltonians are not deformable as an illustration we discuss a particular class of two component hamiltonian systems establishing the triviality of first order deformations and classifying hamiltonians possessing nontrivial deformations of the second order

algebra of the infrared and secondary polytopes abstract we study algebraic structures l and a algebras introduced by gaiotto moore and witten in their recent work devoted to certain supersymmetric 2 dimensional massive field theories we show that such structures can be systematically produced in any number of dimensions by using the geometry of secondary polytopes esp their factorization properties in particular in 2 dimensions we produce out of a polyhedral coefficient system a dg category r with a semi orthogonal decomposition and an l algebra g we show that g is quasi isomorphic to the ordered hochschild complex of r governing deformations preserving the semi orthogonal decomposition this allows us to give a more precise mathematical formulation of the conjectural alternative description of the fukaya seidel category of a kahler manifold endowed with a holomorphic morse function

deformation quantization of gerbes this is the first in a series of articles devoted to deformation quantization of gerbes we introduce basic definitions interpret deformations of a given stack as maurer cartan elements of a differential graded lie algebra dgla and classify deformations of a given gerbe in terms of maurer cartan elements of the dgla of hochschild cochains twisted by the cohomology class of the gerbe we also classify all deformations of a given gerbe on a symplectic manifold as well as provide a deformation theoretic interpretation of the first rozansky witten class

algebraic models of non connected spaces and homotopy theory of l algebras we develop a homotopy theory of l algebras based on the lawrence sullivan construction a complete differential graded lie algebra which as we show satisfies the necessary properties to become the right cylinder in this category as a result we obtain a general procedure to algebraically model the rational homotopy type of non connected spaces 2012 elsevier ltd

on the structure of the classes of stable and pure fuzzy sets in this paper two interesting classes of sets with respect to a multivalued mapping are considered the class of stable sets and the class of pure sets in addition to the results existing in the literature several new properties of stable and pure sets are established a generalization of the concepts of stable and pure sets to fuzzy multivalued mappings is developed the properties of stable and pure fuzzy sets are studied extensively it is obtained that the stable resp pure fuzzy sets with respect to a normalized fuzzy multivalued mapping constitute a complete lattice and also a stratified fuzzy topology the notions of level stable and level pure fuzzy sets are introduced interactions between stability resp purity and level stability resp level purity are established and expressed in terms of relationships between particular stratified fuzzy topologies some additional relationships between the stability resp purity of a fuzzy set with respect to a multivalued mapping and the stability resp purity of its cuts with respect to this mapping are established by imposing certain conditions on the triangular norm and the implication operator involved a one to one correspondence between the class of stable fuzzy sets and the class of pure fuzzy sets with respect to a normalized fuzzy multivalued mapping that has a normalized inverse is obtained in the last section it is shown that a normalized fuzzy multivalued mapping is continuous with respect to the stratified fuzzy topologies constituted by the stable fuzzy sets and the pure fuzzy sets with respect to this fuzzy multivalued mapping

renormalization for holomorphic field theories we introduce the concept of a holomorphic field theory on any complex manifold in the language of the batalin vilkovisky formalism when the complex dimension is one this setting agrees with that of chiral conformal field theory our main result concerns the behavior of holomorphic theories under renormalization group flow namely we show that holomorphic theories are one loop finite we use this to completely characterize holomorphic anomalies in any dimension throughout we compare our approach to holomorphic field theories to more familiar approaches including that of supersymmetric field theories

exact poisson pencils τ structures and topological hierarchies abstract we discuss in the framework of the dubrovin zhang perturbative approach to integrable evolutionary pdes in 1 1 dimensions the role of a special class of poisson pencils called exact poisson pencils in particular we show that in the semisimple case exactness of the pencil is equivalent to the constancy of the so called central invariants of the theory that were introduced by dubrovin liu and zhang

fuzzy topology and fuzzy multifunctions the fundamental concept of a fuzzy set was first introduced by zadeh in his classical paper 141 since then work has been done by many authors in several directions which has resulted in the formation of a new mathematical field called fuzzy mathematics the theory of fuzzy topological spaces is a branch of such mathematics chang in 2 was the first to introduce the notion of fuzzy topology several others continued the work in this area for example see 12 131 we remark that general point set topology can be regarded as a special case of fuzzy topology where all membership functions are just characteristic functions in this paper we introduce several fuzzy topological concepts and examine their properties our basic tool in that process is the notion of the fuzzy neighborhood of a point introduced by warren in ill we also introduce the notion of fuzzy multifunctions which we strongly believe will be useful in the study of fuzzy games and general fuzzy optimization problems we study the continuity and measurability properties of these multifunctions also we introduce linear fuzzy multifunctions and study their properties in the light of the recent work of katsaras 5 katsaras and liu s and lowen 9 on fuzzy vector spaces

homological mirror symmetry for curves of higher genus abstract this paper is devoted to homological mirror symmetry conjecture for curves of higher genus it was proposed by katzarkov as a generalization of original kontsevich s conjecture a version of this conjecture in the case of the genus two curve was proved by seidel 25 based on the paper of seidel we prove the conjecture in the same version for curves of genus g 3 namely we relate the fukaya category of a genus g curve to the category of singularities of zero fiber in the mirror dual landau ginzburg model we also prove a kind of reconstruction theorem for hypersurface singularities namely formal type of hypersurface singularity i e a formal power series up to a formal change of variables can be reconstructed with some technical assumptions from its d z 2 g category of landau ginzburg branes the precise statement is theorem 1 2

bihamiltonian cohomology of kdv brackets using spectral sequences techniques we compute the bihamiltonian cohomology groups of the pencil of poisson brackets of dispersionless kdv hierarchy in particular this proves a conjecture of liu and zhang about the vanishing of such cohomology groups

universality of the break up profile for the kdv equation in the small dispersion limit using the riemann hilbert approach we obtain an asymptotic expansion for the solution of the cauchy problem for the korteweg de vries kdv equation u t 6uu x epsilon 2 u xxx 0 u x t 0 epsilon u 0 x for epsilon small near the point of gradient catastrophe x c t c for the solution of the dispersionless equation u t 6uu x 0 the sub leading term in this expansion is described by the smooth solution of a fourth order ode which is a higher order analogue to the painleve i equation this is in accordance with a conjecture of dubrovin suggesting that this is a universal phenomenon for any hamiltonian perturbation of a hyperbolic equation using the deift zhou steepest descent method applied on the riemann hilbert problem for the kdv equation we are able to prove the asymptotic expansion rigorously in a double scaling limit

minimal models of quantum homotopy lie algebras via the bv formalism using the batalin vilkovisky formalism of mathematical physics an explicit construction for the minimal model of a quantum l algebra is given as a formal super integral the approach taken herein to these formal integrals is axiomatic and they can be approached using perturbation theory to obtain combinatorial formulae as shown in the appendix additionally there exists a canonical differential graded lie algebra morphism mapping formal functions on homology to formal functions on the whole space an inverse l algebra morphism to this differential graded lie algebra morphism is constructed as a formal super integral using the batalin vilkovisky formalism of mathematical physics an explicit construction for the minimal model of a quantum l algebra is given as a formal super integral the approach taken herein to these formal integrals is axiomatic and they can be approached using perturbation theory to obtain combinatorial formulae as shown in the appendix additionally there exists a canonical differential graded lie algebra morphism mapping formal functions on homology to formal functions on the whole space an inverse l algebra morphism to this differential graded lie algebra morphism is constructed as a formal super integral

on hamiltonian perturbations of hyperbolic systems of conservation laws i quasi triviality of bi hamiltonian perturbations we study the general structure of formal perturbative solutions to the hamiltonian perturbations of spatially one dimensional systems of hyperbolic pdes vt v x 0 under certain genericity assumptions it is proved that any bi hamiltonian perturbation can be eliminated in all orders of the perturbative expansion by a change of coordinates on the infinite jet space depending rationally on the derivatives the main tool is in constructing the so called quasimiura transformation of jet coordinates eliminating an arbitrary deformation of a semisimple bi hamiltonian structure of hydrodynamic type the quasi triviality theorem we also describe following 35 the invariants ofsuchbi hamiltonian structures with respect to the group of miura type transformations depending polynomially on the derivatives c

a reduction method for variational inequalities this paper explains a method by which the number of variables in a variational inequality having a certain form can be substantially reduced by changing the set over which the variational inequality is posed the method applies in particular to certain economic equilibrium problems occurring in applications we explain and justify the method and give examples of its application including a numerical example in which the solution time for the reduced problem was approximately 2 of that for the problem in its original form 1998 the mathematical programming society inc published by elsevier science b v

the thom sebastiani theorem for the euler characteristic of cyclic l algebras abstract let l be a cyclic l algebra of dimension 3 with finite dimensional cohomology only in dimension one and two by transfer theorem there exists a cyclic l algebra structure on the cohomology h l the inner product plus the higher products of the cyclic l algebra defines a superpotential function f on h 1 l we associate with an analytic milnor fiber for the formal function f and define the euler characteristic of l is to be the euler characteristic of the etale cohomology of the analytic milnor fiber in this paper we prove a thom sebastiani type formula for the euler characteristic of cyclic l algebras as applications we prove the joyce song formulas about the behrend function identities for semi schur objects in the derived category of coherent sheaves over calabi yau threefolds a motivic thom sebastiani type formula and a conjectural motivic joyce song formula for the motivic milnor fiber of cyclic l algebras are also discussed

what do homotopy algebras form abstract in paper 4 we constructed a symmetric monoidal category s lie mc whose objects are shifted and filtered l algebras here we fix a cooperad c and show that algebras over the operad cobar c naturally form a category enriched over s lie mc following 4 we integrate this s lie mc enriched category to a simplicial category hoalg c whose mapping spaces are kan complexes the simplicial category hoalg c gives us a particularly nice model of an 1 category of cobar c algebras we show that the homotopy category of hoalg c is the localization of the category of cobar c algebras and morphisms with respect to quasi isomorphisms finally we show that the homotopy transfer theorem is a simple consequence of the goldman millson theorem

on almost continuous and weakly continuous fuzzy multifunctions abstract the paper deals with the study of almost continuous and weakly continuous multifunctions in fuzzy setting several characterizations and certain properties of these fuzzy multifunctions along with their mutal relationships are established attempts are also made to correlate them with fuzzy lower and upper semi continuous multifunctions

on the moduli space of deformations of bihamiltonian hierarchies of hydrodynamic type we investigate the deformation theory of the simplest bihamiltonian structure of hydrodynamic type that of the dispersionless kdv hierarchy we prove that all of its deformations are quasi trivial in the sense of b dubrovin and y zhang that is trivial after allowing transformations where the first partial derivative u of the field is inverted we reformulate the question about deformations as a question about the cohomology of a certain double complex and calculate the appropriate cohomology group

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

a cellular topological field theory we present a construction of cellular bf theory in both abelian and non abelian variants on cobordisms equipped with cellular decompositions partition functions of this theory are invariant under subdivisions satisfy a version of the quantum master equation and satisfy atiyah segal type gluing formula with respect to composition of cobordisms

a physical origin for singular support conditions in geometric langlands theory we explain how the nilpotent singular support condition introduced into the geometric langlands conjecture by arinkin and gaitsgory arises naturally from the point of view of 4 dimensional mathcal n 4 supersymmetric gauge theory we define what it means in topological quantum field theory to restrict a category of boundary conditions to the full subcategory of objects compatible with a fixed choice of vacuum both in functorial field theory and in the language of factorization algebras for b twisted mathcal n 4 gauge theory with gauge group g the moduli space of vacua is equivalent to mathfrak h w and the nilpotent singular support condition arises by restricting to the vacuum 0 in mathfrak h w we then investigate the categories obtained by restricting to points in larger strata and conjecture that these categories are equivalent to the geometric langlands categories with gauge symmetry broken to a levi subgroup and furthermore that by assembling such for the groups mathrm gl n with n ge 1 one finds a hidden factorization algebra structure for the geometric langlands theory

disconnected rational homotopy theory we construct two algebraic versions of homotopy theory of rational disconnected topological spaces one based on differential graded commutative associative algebras and the other one on complete differential graded lie algebras as an application of the developed technology we obtain results on the structure of maurer cartan spaces of complete differential graded lie algebras

dg resolutions of nc smooth thickenings and nc fourier mukai transforms we give a construction of nc smooth thickenings a notion defined by kapranov j reine angew math 505 73 118 1998 of a smooth variety equipped with a torsion free connection we show that a twisted version of this construction realizes all nc smooth thickenings as 0 th cohomology of a differential graded sheaf of algebras similarly to fedosov s construction in j differ geom 40 213 238 1994 we use this dg resolution to construct and study sheaves on nc smooth thickenings in particular we construct an nc version of the fourier mukai transform from coherent sheaves on a commutative curve to perfect complexes on the canonical nc smooth thickening of its jacobian we also define and study analytic nc manifolds we prove nc versions of some of gaga theorems and give a c infty construction of analytic nc thickenings that can be used in particular for kahler manifolds with constant holomorphic sectional curvature finally we describe an analytic nc thickening of the poincare line bundle for the jacobian of a curve and the corresponding fourier mukai functor in terms of a infty structures

deformations of algebroid stacks abstract in this paper we consider deformations of an algebroid stack on an etale groupoid we construct a differential graded lie algebra dgla which controls this deformation theory in the case when the algebroid is a twisted form of functions we show that this dgla is quasiisomorphic to the twist of the dgla of hochschild cochains on the algebra of functions on the groupoid by the characteristic class of the corresponding gerbe