invariants of 3 manifolds via link polynomials and quantum groups the aim of this paper is to construct new topological invariants of compact oriented 3 manifolds and of framed links in such manifolds our invariant of a link in a closed oriented 3 manifold is a sequence of complex numbers parametrized by complex roots of 1 for a framed link in s 3 the terms of the sequence are equale to the values of the suitably parametrized jones polynomial of the link in the corresponding roots of 1 in the case of manifolds with boundary our invariant is a sequence of finite dimensional complex linear operators this produces from each root of unity q a 3 dimensional topological quantum field theory

multi soliton and double wronskian solutions of a 2 1 dimensional modified heisenberg ferromagnetic system a 2 1 dimensional modified heisenberg ferromagnetic system which arises in the motion of magnetization vector of the isotropic ferromagnet and biological pattern formation is investigated via the hirota bilinear method multi soliton solutions of such a system are derived it is proved that the system possesses the n soliton solutions expressed in terms of the double wronskian determinant head on and overtaking elastic interactions are exhibited elastic interaction behavior between the two solitons has been interpreted through the asymptotic analysis namely amplitude and velocity of each soliton remain unchanged except for the phase shift after the interaction inelastic interactions including the soliton fusion and fission between two solitons are shown during the soliton propagation for the product of two fields the soliton with the smaller amplitude can travel faster than with the larger while for the third field the soliton with the larger amplitude can travel faster than with the smaller on the other hand the soliton for the third field may exhibit the solitoff like property with respect to the three solitons head on elastic interaction can be found

representations of semisimple lie algebras in prime characteristic and the noncommutative springer resolution we prove most of lusztig s conjectures on the canonical basis in homology of a springer ber the conjectures predict that this basis controls numerics of representations of the lie algebra of a semisimple algebraic group over an algebraically closed eld of positive characteristic we check this for almost all characteristics to this end we construct a noncommutative resolution of the nilpotent cone which is derived equivalent to the springer resolution on the one hand this noncommutative resolution is closely related to the positive characteristic derived localization equivalences obtained earlier by the present authors and rumynin on the other hand it is compatible with the t structure arising from an equivalence with the derived category of perverse sheaves on the ane ag variety of the langlands dual group this equivalence established by arkhipov and the rst author ts the framework of local geometric langlands duality the latter compatibility allows one to apply frobenius purity theorem to deduce the desired properties of the basis we expect the noncommutative counterpart of the springer resolution to be of independent interest from the perspectives of algebraic geometry and geometric langlands duality

fourier transforms and frobenius eigenvalues for finite coxeter groups abstract lusztig s classification of the unipotent characters of a finite chevalley or steinberg group involves a certain non abelian fourier transformation we construct analogous transformations for the suzuki and ree groups based on a set of axioms derived from lusztig s theory of character sheaves we also determine fourier matrices for the spetses in the sense of broue michel and the second author associated with twisted dihedral groups this completes the determination of fourier matrices for all spetses associated with finite coxeter groups we end by collecting common properties of these fourier matrices and the eigenvalues of frobenius of character sheaves and unipotent characters

character sheaves v this paper is part of a series s 131 devoted to the study of a class g of irreducible perverse sheaves called character sheaves on a connected reductive algebraic group g the numbering of chapters sections and references will continue that of s 131 this paper is a step towards the classification of character sheaves on g one of the main results is the following one under certain assumptions there is a natural surjective map with finite fibers from g to the set of all pairs 9 c up to conjugacy by the weyl group where 9 is a tame local system on the maximal torus and c is a two sided cell in the stabilizer w of 9 in the weyl group the assumptions made on g are

categorification of induced cell modules and the rough structure of generalised verma modules this paper presents categorifications of right cell modules and induced cell modules for hecke algebras of finite weyl groups fit type a we show that these categorifications depend only on the isomorphism class of the cell module not on the cell itself our main application is multiplicity formulas for parabolically induced modules over a reductive lie algebra of type a which finally determines the so called rough structure of generalised verma modules on the way we present several categorification results and give a positive answer to kostant s problem from a joseph kostant s problem goldie rank and the gelfand kirillov conjecture invent math 56 3 1980 191 213 in many cases we also present a general setup of decategorification precategorification and categorification

on solving linear programs with the ordered weighted averaging objective abstract the problem of aggregating multiple criteria to form overall objective functions is of considerable importance in many disciplines the most commonly used aggregation is based on the weighted sum the ordered weighted averaging owa aggregation introduced by yager uses the weights assigned to the ordered values i e to the worst value the second worst and so on rather than to the specific criteria this allows to model various aggregation preferences preserving simultaneously the impartiality neutrality with respect to the individual criteria in this paper we analyze solution procedures for linear programs with the owa objective functions two alternative linear programming formulations are introduced and their computational efficiency is analyzed

self similar solutions for the 1 d schrodinger map on the hyperbolic plane in this paper we will study self similar solutions for x t x x ss the equivalent in the minkowski 3 space mathbb r 2 1 to the localized induction approximation flow trying to adapt some results given by gutierrez rivas and vega we will show the existence of a one parameter family of smooth solutions developing a corner in finite time the main difference with respect to the euclidean case studied by those authors will be the proof of the boundedness of t e 1 and e 2 the equivalents of t b and n in mathbb r 2 1

traces on module categories over fusion categories abstract we consider traces on module categories over pivotal fusion categories which are compatible with the module structure it is shown that such module traces characterise the morita classes of special haploid frobenius algebras moreover they are unique up to a scale factor and they equip the dual category with a pivotal structure this implies that for each pivotal structure on a fusion category over c there exists a conjugate pivotal structure defined by the canonical module trace

the gauge equivalence of the nls and the schrodinger flow of maps in 2 1 dimensions the gauge equivalence of the analogues of the nonlinear schrodinger equation nls and those of the schrodinger flow of maps into h2 the minkowski hf model in 2 1 dimensions are proved respectively combining these with the already known results we obtain a complete understanding of the gauge equivalence of the analogous models of the nonlinear schrodinger equation for 1 or 1 and those of the heisenberg ferromagnet model for euclidean or minkowski in 2 1 dimensions

the gauge equivalence of the davey stewartson equation and 2 1 dimensional continuous heisenberg ferromagnetic model the gauge equivalence between the davey stewartson ds equation and the 2 1 dimensional continuous heisenberg ferromagnetic model is shown explicitly through the gauge transformation solutions of the dsii are obtained from the vortex type solutions of the former equation

the exotic robinson schensted correspondence abstract we study the action of the symplectic group on pairs of a vector and a flag considering the irreducible components of the conormal variety we obtain an exotic analogue of the robinson schensted correspondence conjecturally the resulting cells are related to exotic character sheaves

some integrable hierarchies in 2 1 dimensions and their twistor description the twistor space construction due to mason and sparling for solutions of the nonlinear schrodinger and korteweg devries hierarchies is generalized resulting in families of integrable models in 2 1 dimensions soliton solutions to one such hierarchy associated with the gauge group su 2 are constructed by adapting the methods used to construct instanton and monopole solutions these integrable models have the feature that their solutions depend on a number of arbitrary functions as well as arbitrary constants in contrast to the closely related davey stewartson hierachy

cells in affine weyl groups and tensor categories abstract in this paper we construct a tensor or monoidal category for any two sided cell in a finite or affine weyl group the isomorphism classes of simple objects of this category correspond to the elements of the two sided cell the tensor product arises by a certain truncation procedure from the standard convolution of perverse sheaves on a flag manifold which imitates geometrically the definition of the ring j given in g lusztig j algebra 109 1987 536 548 similarly we construct for any left cell a tensor category in which the isomorphism classes of simple objects correspond to the elements in the left cell intersected with its inverse a right cell this contains as a special case the category of equivariant perverse sheaves on an affine grassmannian which as a consequence of g lusztig asterisque 101 102 1983 208 209 is a tensor category for convolution without truncation closely related to the tensor category of representations of the group dual in the sense of langlands to the group which gives rise to the affine grassmannian but our construction can be specialized in other ways and it gives a conjectural realization of the tensor category of representations of the reductive quotient of the centralizer of any unipotent element of the langlands dual group

characterizations of character sheaves for complex reductive algebraic groups character sheaves are geometric objects associated to a reductive algebraic group they are closely related to irreducible characters of finite chevalley groups the existence of character sheaves was conjectured by g lusztig in the book characters of reductive groups over finite fields 1984 there he described what one should expect character sheaves to be for complex reductive groups a definition of character sheaves for an arbitrary characteristic quite different from this description was given by him later 1985 it is shown in this thesis in particular that the original 1984 lusztig s suggestion for describing character sheaves is equivalent to the later definition we use alternative characterizations of character sheaves for a connected complex reductive algebraic group g that are obtained in this thesis and also the one conjectured by g lusztig and g laumon that was obtained in 1988 by v ginsburg and independently by i mirkovic and k vilonen these characterizations involve microsupport and microlocalization of a sheaf a microsupport ss cal f of a complex cal f of sheaves on g is a closed conic subset of t sp g g times bf g it tells us about singularities of cal f a microlocalization mu sb m cal f of cal f along a submanifold m of g is a complex of sheaves on the conormal bundle t sbsp m g which allows us to analyze cal f on the infinitesimally small neighbourhood of m theorem let g be a complex connected reductive algebraic group cal f be a g equivariant for conjugation irreducible perverse sheaf on g let cal n be the nilpotent cone of the lie algebra g of g s may denote either g or the set of all semisimple elements of g the following are equivalent i cal f is a character sheaf ii ss cal f vert sb s subset s times cal n iii supp mu sb cal o cal f subset cal o times cal n where cal o subset s is a conjugacy class iv supp mu sb x cal f subset cal n x in s characterization ii when s g is the one given by ginsburg mirkovic and vilonen characterization iii with s being the set of semisimple elements was conjectured by j bernstein for arbitrary characteristic this kind of characterizations essentially simplifies the development and our understanding of the theory currently there is still no such description for positive characteristic an important intermediate result is the description of the microsupport of a doubc constructible sheaf cal f on a manifold x in terms of microlocalizations to the points of this manifold ss cal f overline bigcup limits sb x in x rm supp mu sb x cal f

fuzzy scheduling modelling flexible constraints vs coping with incomplete knowledge abstract an overview of some fuzzy set based approaches to scheduling is proposed emphasizing two distinct uses of fuzzy sets representing preference profiles and modelling uncertainty distributions the first setting leads to a valued non compensatory generalization of constraint directed scheduling the other setting yields a possibility theoretic counterpart of pert where probability distributions of activity durations are changed into possibility distributions for the purpose of modelling incomplete information it is pointed out that a special case of the latter interval valued pert is a difficult ill known problem regarding the determination of critical activities latest starting times and floats lastly when flexible constraints and uncertain processing times are to be jointly considered the use of possibilistic decision theory leads to the computation of robust schedules

the character sheaves on the group compactification abstract we give a definition of character sheaves on the group compactification which is equivalent to lusztig s definition in g lusztig parabolic character sheaves ii mosc math j 4 4 2004 869 896 we also prove some properties of the character sheaves on the group compactification

gauge equivalence between dimensional continuous heisenberg ferromagnetic models and nonlinear schrodinger type equations the gauge equivalence between the 2c 1 dimensional zakharov equation and the 2c 1 dimensional integrable continuous heisenberg ferromagnetic model is established their integrable reductions are also shown explicitly

the application of fuzzy integrals in multicriteria decision making abstract this paper presents a synthesis on the application of fuzzy integrals as an innovative tool for criteria aggregation in decision problems the main point is that fuzzy integrals are able to model interaction between criteria in a flexible way the methodology has been elaborated mainly in japan and has been applied there successfully in various fields such as design reliability evaluation of goods etc it seems however that this technique is still very little known in europe it is one of the aims of this review to disseminate this emerging technology in many industrial fields

crossed s matrices and character sheaves on unipotent groups abstract let k be the algebraic closure of a finite field f q of characteristic p let g be a connected unipotent group over k equipped with an f q structure given by a frobenius map f g g we will denote the corresponding algebraic group defined over f q by g 0 character sheaves on g are certain special objects in the triangulated braided monoidal category d g g of bounded conjugation equivariant q l complexes where l p is a prime number on g boyarchenko has proved that the trace of frobenius functions associated with f stable character sheaves on g form an orthonormal basis of the space of class functions on g 0 f q and that the matrix relating this basis to the basis formed by the irreducible characters of g 0 f q is block diagonal with small blocks in particular there is a partition of the set of character sheaves as well as a partition of the set of irreducible characters of g 0 f q into small families known as l packets in this paper we describe these block matrices relating character sheaves and irreducible characters corresponding to each l packet we prove that these matrices can be described as certain crossed s matrices associated with each l packet we will also derive a formula for the dimensions of the irreducible representations of g 0 f q in terms of certain modular categorical data associated with the corresponding l packet in fact we will formulate and prove more general results which hold for possibly disconnected groups g such that g is unipotent to prove our results we will establish a formula which holds for any algebraic group g which expresses the inner product of the trace of frobenius function of any f stable object of d g g with any character of g 0 f q or of any of its pure inner forms in terms of certain categorical operations

remarks on computing irreducible characters 0 1 let g be a connected reductive algebraic group defined over a finite field fq and let g fq be the finite group of all fq rational points of g we would like to present here a strategy for computing the character table of g fq under the assumption that p the characteristic of fq is sufficiently large we can assume that g has a simply connected derived group indeed in the general case we can find a connected reductive group g with simply connected derived group and a surjective homomorphism f g g whose kernel is a central torus such that g and f are defined over fq then the character table

derivative nonlinear schrodinger equations and hermitian symmetric spaces it is shown that to each hermitian symmetric space there corresponds an integrable system of generalised derivative nonlinear schrodinger equations nls the nonlinear terms are related to the curvature tensor of the associated symmetric space the hamiltonian form of the equations is presented the results are an extension of those presented in an earlier paper 1983 on generalised nls equations associated with symmetric and reductive homogeneous spaces

on the simplest 2 1 dimensional integrable spin systems and their equivalent nonlinear schrodinger equations using a moving space curve formalism geometrical as well as gauge equivalence between a 2 1 dimensional spin equation m i equation and the 2 1 dimensional nonlinear schrodinger equation nlse originally discovered by calogero discussed then by zakharov and recently rederived by strachan have been estabilished a compatible set of three linear equations are obtained and integrals of motion are discussed through stereographic projection the m i equation has been bilinearized and different types of solutions such as line and curved solitons breaking solitons induced dromions and domain wall type solutions are presented breaking soliton solutions of 2 1 dimensional nlse have also been reported generalizations of the above spin equation are discussed

computing improved optimal solutions to max min flexible constraint satisfaction problems abstract the formal framework for decision making in a fuzzy environment is based on a general max min bottleneck like optimization problem proposed by zadeh it is also the basis for extending the constraint satisfaction paradigm of artificial intelligence to accommodating flexible or prioritized constraints this paper surveys refinements of the ordering of solutions supplied by the max min formulation namely the discrimin partial ordering and the leximin complete preordering a general algorithm is given which computes all maximal solutions in the sense of these relations it also sheds light on the structure of the set of best solutions moreover classes of problems for which there is a unique best discrimin and leximin solution are exhibited namely continuous problems with convex domains and so called isotonic problems noticeable examples of such problems are fuzzy linear programming problems and fuzzy pert like scheduling problems

workers placement in an industrial environment a problem of workers placement in an industrial environment is studied in this paper an effect of workers relationship to their placement is newly included in this paper evaluating workers suitability is important for decision makers to select the better candidates under various evaluation criteria among available human resources and jobs for problems of this type an analysis using the fuzzy number approach promises to be potentially effective in order to make a more convincing and accurate decision the relationship among workers is included in the workers assignment in an industrial environment by means of aggregating the decision makers fuzzy assessment the fuzzy suitability evaluation is performed typical application examples are also presented the results demonstrate that the workers relationship is an important factor and our method is effective for the decision making process

darboux transformation and soliton solutions for the 2 1 dimensional nonlinear schrodinger hierarchy with symbolic computation abstract in this paper the n fold darboux transformation is applied to the lax pair associated with the 2 1 dimensional nonlinear schrodinger hierarchy the general explicit formula for n soliton solution is derived in terms of vandermonde like determinant expression on the zero background two different families of soliton solutions multi line soliton and multi parabola soliton solutions are constructed by applying the darboux transformation successively based on the figures for several sample solutions the soliton interactions are discussed by virtue of the physical parameters amplitude velocity and direction of the interacting solitons

local cohomology of modules of covariants abstract let g be a connected reductive algebraic group over an algebraically closed field of characteristic zero and let w u be two finite dimensional representations of g in this paper we compute the local cohomology of u sw g provided a certain relatively weak technical condition is true

on fusion categories using a variety of methods developed in the literature in particular the theory of weak hopf algebras we prove a number of general results about fusion categories in characteristic zero we show that the global dimension of a fusion category is always positive and that the s matrix of any not necessarily hermitian modular category is unitary we also show that the category of module functors between two module categories over a fusion category is semisimple and that fusion categories and tensor functors between them are undeformable generalized ocneanu rigidity in particular the number of such categories functors realizing a given fusion datum is finite finally we develop the theory of frobenius perron dimensions in an arbitrary fusion category at the end of the paper we generalize some of these results to positive characteristic

topological generation of exceptional algebraic groups abstract let g be a simple algebraic group over an algebraically closed field k and let c 1 c t be non central conjugacy classes in g in this paper we consider the problem of determining whether there exist g i c i such that g 1 g t is zariski dense in g first we establish a general result which shows that if is an irreducible subvariety of g t then the set of tuples in generating a dense subgroup of g is either empty or dense in in the special case c 1 c t by considering the dimensions of fixed point spaces we prove that this set is dense when g is an exceptional algebraic group and t 5 assuming k is not algebraic over a finite field in fact for g g 2 we only need t 4 and both of these bounds are best possible as an application we show that many faithful representations of exceptional algebraic groups are generically free we also establish new results on the topological generation of exceptional groups in the special case t 2 which have applications to random generation of finite exceptional groups of lie type in particular we prove a conjecture of liebeck and shalev on the random r s generation of exceptional groups

a discretization of the matrix nonlinear schrodinger equation in this paper we shall show that a class of solutions to the discrete coupled matrix nonlinear schrodinger equation dcmnlse is gauge equivalent to the discrete equation of the schrodinger flow of maps into the grassmannian and the realizing gauge transformation is only the discretization of a classical gauge transformation between the matrix nonlinear schrodinger equation mnlse and the schrodinger flow of maps into the grassmannian in other words from the viewpoint of gauge equivalence a class of solutions of the dcmnlse is a correct discretization of the mnlse

characteristic cycles for the loop grassmannian and nilpotent orbits x which is a linear combination of closures of conormal bundles to submanifolds of x intuitively the microlocal multiplicities c me asurethesingularity of at in settings related to representation theory a group g acts on x is g equivariant and the microlocal multiplicities play a significant but only partially understood role in representation theory see ro sv abv and kasa e g in this paper we compute microlocal multiplicities for certain cases of interest in representation theory let g be a connected reductive group with loop group lg and let p bethesubgroup of lg consisting of loops with positive fourier coefficients then p orbits on theloop grassmannian lg p correspond to the irreducible representations l of thedual re ductivegroup g for dominant weights and of a torus of

prolongation structure of the 2 1 dimensional integrable heisenberg ferromagnet model we investigate the integrable 2 1 dimensional modified heisenberg ferromagnet hf model using the prolongation structure theory for the integrable 2 1 dimensional modified hf models the corresponding geometrical equivalent counterparts such as the 2 1 dimensional nonlinear schrodinger equation and the coupled 2 1 dimensional integrable equations are also presented through the motion of minkowski space curves endowed with an additional spatial variable

character d modules via drinfeld center of harish chandra bimodules the category of character d modules is realized as drinfeld center of the abelian monoidal category of harish chandra bimodules tensor product of harish chandra bimodules is related to convolution of d modules via the long intertwining functor radon transform by a result of beilinson and ginzburg represent theory 3 1 31 1999 exactness property of the long intertwining functor on a cell subquotient of the harish chandra bimodules category shows that the truncated convolution category of lusztig adv math 129 85 98 1997 can be realized as a subquotient of the category of harish chandra bimodules together with the description of the truncated convolution category bezrukavnikov et al in isr j math 170 207 234 2009 this allows us to derive under a mild technical assumption a classification of irreducible character sheaves over obtained by lusztig by a different method

monoidal 2 structure of bimodule categories abstract we define a notion of tensor product of bimodule categories and prove that with this product the 2 category of c bimodule categories for fixed tensor c is a monoidal 2 category in the sense of kapranov and voevodsky 1991 1 we then provide a monoidal structure preserving 2 equivalence between the 2 category of c bimodule categories and z c module categories module categories over the center of c for a finite group g we show that de equivariantization is equivalent to the tensor product over rep g we derive rep g module fusion rules and show that the group of invertible rep g module categories is isomorphic to h 2 g k extending results in etingof et al 2

robust competence assessment for job assignment allocating the right person to a task or job is a key issue for improving quality and performance of achievements usually addressed using the concept of competences nevertheless providing an accurate assessment of the competences of an individual may be in practice a difficult task we suggest in this paper to model the uncertainty on the competences possessed by a person using a possibility distribution and the imprecision on the competences required for a task using a fuzzy constraint taking into account the possible interactions between competences using a choquet integral as a difference with comparable approaches we then suggest to perform the allocation of persons to jobs using a robust optimisation approach allowing to minimize the risk taken by the decision maker we first apply this framework to the problem of selecting a candidate within n for a job then extend the method to the problem of selecting c candidates for j jobs c j using the leximin criterion

bicomplex formulation and moyal deformation of 2 1 dimensional fordy kulish systems using bicomplex formalism we construct generalizations of fordy kulish systems of matrix nonlinear schrodinger equations on two dimensional space time in two respects firstly we obtain corresponding equations in three space time dimensions secondly a moyal deformation is applied to the space time coordinates and the ordinary product of functions replaced by the moyal product in a suitable way both generalizations preserve the existence of an infinite set of conservation laws

modular generalized springer correspondence iii exceptional groups we complete the construction of the modular generalized springer correspondence for an arbitrary connected reductive group with a uniform proof of the disjointness of induction series that avoids the case by case arguments for classical groups used in previous papers in the series we show that the induction series containing the trivial local system on the regular nilpotent orbit is determined by the sylow subgroups of the weyl group under some assumptions we give an algorithm for determining the induction series associated to the minimal cuspidal datum with a given central character we also provide tables and other information on the modular generalized springer correspondence for quasi simple groups of exceptional type including a complete classification of cuspidal pairs in the case of good characteristic and a full determination of the correspondence in type g 2

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

correspondences of ribbon categories much of algebra and representation theory can be formulated in the general framework of tensor categories the aim of this paper is to further develop this theory for braided tensor categories several results are established that do not have a substantial counterpart for symmetric tensor categories in particular we exhibit various equivalences involving categories of modules over algebras in ribbon categories finally we establish a correspondence of ribbon categories that can be applied to and is in fact motivated by the coset construction in conformal quantum field theory

exterior differential expression of the 1 1 dimensional nonlinear evolution equation with lax integrability abstract a 1 1 dimensional nonlinear evolution equation with lax integrability is investigated in this paper with the aid of differential forms and exterior differentials equivalent definition of the lax integrability is given relation between the lax integrability and complete integrability is discussed geometric interpretation of the lax equation is presented gauge transformation and darboux transformation are restated in terms of the differential forms and exterior differentials if i s is a darboux transformation between two lax equations the system that the matrix s should satisfy is given where is the spectral parameter and i is the identity matrix the completely integrable condition of that system is obtained according to the frobenius theorem and it is shown that s h h 1 satisfies that completely integrable condition with h and as two matrices

weighted fuzzy pattern matching the fuzzy pattern matching technique has been developed in the framework of fuzzy set and possibility theory in order to take into account the imprecision and the uncertainty pervading values which have to be compared in a matching process this technique has proved very useful for implementing patterns of approximate reasoning in expert system inference engines and for designing retrieval systems capable of managing incomplete and fuzzy information data bases and vague queries in this paper the fuzzy pattern matching procedure is improved by introducing weights assessing the relative importance of atoms in the pattern matching indices are obtained using weighted versions of the minimum and maximum operations of fuzzy set theory this approach is extended to the case of variable weights and is related to aggregation schemes proposed by yager when modeling soft quantified statements such as most of the criteria must be met a notion of soft partial matching is thus modelled

nonlinear schrodinger equations and simple lie algebras we associate a system of integrable generalised nonlinear schrodinger nls equations with each hermitian symmetric space these nls equations are considered as reductions of more general systems this time associated with a reductive homogeneous space the nonlinear terms are related to the curvature and torsion tensors of the appropriate geometrical space the hamiltonian structure is investigated using r matrix techniques and shown to be canonical for all these equations throughout the reduction procedure this hamiltonian structure does not degenerate each of the above systems of equations is gauge equivalent to a generalised ferromagnet reductions of the latter are discussed in terms of the corresponding nls type equations

legendrian knots and constructible sheaves we study the unwrapped fukaya category of lagrangian branes ending on a legendrian knot our knots live at contact infinity in the cotangent bundle of a surface the fukaya category of which is equivalent to the category of constructible sheaves on the surface itself consequently our category can be described as constructible sheaves with singular support controlled by the front projection of the knot we use a theorem of guillermou kashiwara schapira to show that the resulting category is invariant under legendrian isotopies a subsequent article establishes its equivalence to a category of representations of the chekanov eliashberg differential graded algebra we also find two connections to topological knot theory first drawing a positive braid closure on the annulus the moduli space of rank n objects maps to the space of local systems on a circle the second page of the spectral sequence associated to the weight filtration on the pushforward of the constant sheaf is the colored by n triply graded khovanov rozansky homology second drawing a positive braid closure in the plane the number of points of our moduli spaces over a finite field with q elements recovers the lowest coefficient in a of the homfly polynomial of the braid closure

generalized springer correspondence and green functions for type b c graded hecke algebras this paper presents a combinatorial approach to the tempered representation theory of a graded hecke algebra h of classical type b or c with arbitrary parameters we present various combinatorial results which together give a uniform combinatorial description of what becomes the springer correspondence in the classical situation of equal parameters more precisely by using a general version of lusztig s symbols which describe the classical springer correspondence we associate to a discrete series representation of h with central character w0c a set w0c of w0 characters where w0 is the weyl group this set w0c is shown to parametrize the central characters of the generic algebra which specialize into w0c using the parabolic classification of the central characters of h t r on the one hand and a truncated induction of weyl group characters on the other hand we define a set w0c for any central character w0c of h t r and show that this property is preserved we show that in the equal parameter situation we retrieve the classical springer correspondence by considering a set u of partitions which replaces the unipotent classes of so2n 1 c and sp2n c and a bijection between u and the central characters of h t r we end with a conjecture which basically states that our generalized springer correspondence determines h t r exactly as the classical springer correspondence does in the equal label case in particular we conjecture that w0c indexes the modules in h t r with central character w0c in the following way a module m in h t r has a natural grading for the action of w0 and the w0 representation m in its top degree is irreducible when m runs through the modules in h t r with central character w0c m runs through w0c moreover still in analogy with the equal parameter case we conjecture that the w0 structure of the modules in h t r can be computed using generalized green functions