arithmetic representations of fundamental groups i let x be a normal algebraic variety over a finitely generated field k of characteristic zero and let ell be a prime say that a continuous ell adic representation rho of pi 1 acute mathrm e text t x bar k is arithmetic if there exists a finite extension k of k and a representation tilde rho of pi 1 acute mathrm e text t x k with rho a subquotient of tilde rho pi 1 x bar k we show that there exists an integer n n x ell such that every nontrivial semisimple arithmetic representation of pi 1 acute mathrm e text t x bar k is nontrivial mod ell n as a corollary we prove that any nontrivial ell adic representation of pi 1 acute mathrm e text t x bar k which arises from geometry is nontrivial mod ell n

on the variance of fuzzy random variables abstract this paper deals with an expectation and a real valued variance of fuzzy random variables the expectation and the variance of a fuzzy random variable is characterized by frechet s principle in a metric space we study properties of the variance of a fuzzy random variable and compare it with the common variance of real valued random variables using the expectation and the variance of fuzzy random variables we consider a linear regression problem and limit theorems

a characterization of varieties whose universal cover is the polydisk or a tube domain in this article we give necessary and sufficient conditions in terms of certain tensors called semispecial tensors respectively slope zero tensors in order that the universal covering of a complex projective manifold be a symmetric domain of tube type as an application we give precisions of a result of kazhdan showing that a galois conjugate of such a manifold has the same universal covering

on effective existence of symmetric differentials of complex hyperbolic space forms for a noncompact complex hyperbolic space form of finite volume x mathbb b n vargamma we consider the problem of producing symmetric differentials vanishing at infinity on the mumford compactification overline x of x similar to the case of producing cusp forms on hyperbolic riemann surfaces we introduce a natural geometric measurement which measures the size of the infinity overline x x called canonical radius of a cusp of vargamma the main result in the article is that there is a constant r r n depending only on the dimension so that if the canonical radii of all cusps of vargamma are larger than r then there exist symmetric differentials of overline x vanishing at infinity as a corollary we show that the cotangent bundle t overline x is ample modulo the infinity if moreover the injectivity radius in the interior of overline x is larger than some constant d d n which depends only on the dimension

stable bundles and integrable systems on considere la geometrie symplectique des fibres cotangents aux espaces de modules de fibres vectoriels stables sur une surface de riemann on montre que ce sont des systemes dynamiques hamiltoniens algebriquement completement integrables

sur la convexite holomorphe des revetements lineaires reductifs d une variete projective algebrique complexe let x be a smooth connected compact projective variety over we study the shafarevich conjecture concerning holomorphic convexity along the lines of kollar s approach when the fundamental group of x admits large finite dimensionnal representations we prove that given n the topological covering space widetilde x n of a projective algebraic compact complex manifold x corresponding to the intersection of the kernels of all linear reductive representations of 1 x to gln is holomorphically convex in the surface case this is a corollary of a theorem due to katzarkov and ramachandran

defuzzification using steiner points a defuzzification function assigns to each fuzzy set a crisp value in a way that this value may intuitively be understood as the centre of the fuzzy set in the present paper this vague concept is put into a mathematically rigorous form to this end we proceed analogously to the case of sharply bordered subsets for which the steiner point is frequently used the function assigning to each convex subset its steiner point is characterised by three properties here we study functions whose domains consist of fuzzy sets and which fulfil analogous properties although uniqueness can no longer be achieved we give a complete characterisation of what we call steiner points of fuzzy sets

moments of the position of the maximum for gue characteristic polynomials and for log correlated gaussian processes we study three instances of log correlated processes on the interval the logarithm of the gaussian unitary ensemble gue characteristic polynomial the gaussian log correlated potential in presence of edge charges and the fractional brownian motion with hurst index h rightarrow 0 fbm0 in previous collaborations we obtained the probability distribution function pdf of the value of the global minimum equivalently maximum for the first two processes using the freezing duality conjecture fdc here we study the pdf of the position of the maximum x m through its moments using replica this requires calculating moments of the density of eigenvalues in the beta jacobi ensemble using jack polynomials we obtain an exact and explicit expression for both positive and negative integer moments for arbitrary beta 0 and positive integer n in terms of sums over partitions for positive moments this expression agrees with a very recent independent derivation by mezzadri and reynolds we check our results against a contour integral formula derived recently by borodin and gorin presented in the appendix 1 from these authors the duality necessary for the fdc to work is proved and on our expressions found to correspond to exchange of partitions with their dual performing the limit n rightarrow 0 and to negative dyson index beta rightarrow 2 we obtain the moments of x m and give explicit expressions for the lowest ones numerical checks for the gue polynomials performed independently by n simm indicate encouraging agreement some results are also obtained for moments in laguerre hermite gaussian as well as circular and related ensembles the correlations of the position and the value of the field at the minimum are also analyzed

random hermitian matrices and gaussian multiplicative chaos n berestycki s work is supported by epsrc grants ep l018896 1 and ep i03372x 1 m d wong is a phd student at the cambridge centre for analysis supported by epsrc grant ep l016516 1 some of this work was carried out while the first and third authors visited the university of helsinki funded in part by epsrc grant ep l018896 1 they also wish to thank the university of helsinki for its hospitality during this visit c webb wishes to thank the isaac newton institute for mathematical sciences for its hospitality during the random geometry program during which this project was initiated c webb was supported by the eemil aaltonen foundation grant stochastic dynamics on large random graphs and academy of finland grants 288318 and 308123

reduction to least squares estimates in multiple fuzzy regression analysis in this paper we deal with the problem of least squares multiple regression with fuzzy data the regression coefficients are assumed to be real crisp a formula for solving the regression coefficients in one variable models is derived if each independent variable is effective i e its corresponding regression coefficient is nonzero the multiple regression problem can be replaced with a 0 1 programming problem its optimal solution is easily computed finally we also propose effective algorithms to compute the regression coefficients in a general case

hurwitz ball quotients we consider the analogue of hurwitz curves smooth projective curves c of genus g ge 2 that realize equality in the hurwitz bound mathrm aut c le 84 g 1 to smooth compact quotients s of the unit ball in mathbb c 2 when s is arithmetic we show that mathrm aut s le 288 e s where e s is the topological euler characteristic and in the case of equality show that s is a regular cover of a particular deligne mostow orbifold we conjecture that this inequality holds independent of arithmeticity and note that work of xiao makes progress on this conjecture and implies the best known lower bound for the volume of a complex hyperbolic 2 orbifold

the nonexistence of certain level structures on abelian varieties over complex function fields let a be a principally polarized abelian variety of dimension g over a number field k the mordell weil theorem tells us that the group a k of k rational points of a is finitely generated in particular the torsion subgroup of a k is finite for the important special case of an elliptic curve e q mazur mal 2 has proved that the torsion subgroup of e q has order 16 in general one would expect the torsion subgroup of a k to have order c g k a constant depending only on the number field k and the dimension g but not

convergence theorem for fuzzy martingales abstract we study fuzzy set valued measures in a banach space and their relationships to fuzzy random variables our main result is a convergence theorem for fuzzy martingales our tool is a radon nikodym type theorem for fuzzy measures which are absolutely continuous with respect to probability measures

global jacquet langlands correspondence multiplicity one and classification of automorphic representations in this paper we generalize the local jacquet langlands correspondence to all unitary irreducible representations we prove the global jacquet langlands correspondence in characteristic zero as consequences we obtain the multiplicity one and strong multiplicity one theorems for inner forms of gl n as well as a classification of the residual spectrum and automorphic representations in analogy with results proved by m glin waldspurger and jacquet shalika for gl n

the mean square error of a random fuzzy vector based on the support function and the steiner point metrics between fuzzy values are a topic with interest for different purposes among them statistics with fuzzy data is growing in modelling and techniques largely through the use of suitable distances between such data this paper introduces a generalized actually parameterized l 2 metric between fuzzy vectors which is based on their representation in terms of their support function and steiner points consequently the metric takes into account the deviation in central location represented by the steiner points and the deviation in shape represented by a deviation defined in terms of the support function and steiner points then sufficient conditions can be given for this representation to characterize fuzzy vectors which is valuable for different aims like optimization studies properties of the metric are analyzed and its application to quantify the mean square error of a fuzzy value in estimating the value of a fuzzy vector valued random element is examined some immediate implications from this mean error are finally described

covolumes of nonuniform lattices in pu n 1 this paper studies the covolumes of nonuniform arithmetic lattices in pu n 1 we determine the smallest covolume nonuniform arithmetic lattices for each n the number of minimal covolume lattices for each n and study the growth of the minimal covolume as n varies in particular there is a unique lattice up to isomorphism in pu 9 1 of smallest euler poincare characteristic amongst all nonuniform arithmetic lattices in pu n 1 we also show that for each even n there are arbitrarily large families of nonisomorphic maximal nonuniform lattices in pu n 1 of equal covolume

conformal invariance in random cluster models i holmorphic fermions in the ising model we construct discrete holomorphic observables in the ising model at criticality and show that they have conformally covariant scaling limits as mesh of the lattice tends to zero in the sequel those observables are used to construct conformally invariant scaling limits of interfaces though ising model is often cited as a classical example of conformal invariance it seems that ours is the first paper where it is actually established

a parameterized l2 metric between fuzzy numbers and its parameter interpretation abstract when handling fuzzy number data it is a common practice to make use of a metric to quantify distances between fuzzy numbers several metrics have been suggested in the literature for this purpose when statistically analyzing fuzzy number valued data l 2 metrics become especially useful this paper introduces a new family of generalized l 2 metrics which take into account key features of the involved fuzzy numbers namely a measure of central location and two measures associated with the shape of the fuzzy numbers are used a crucial property related to these three measures is that necessary and sufficient conditions can be established for them to characterize fuzzy numbers furthermore the family of generalized l 2 metrics depends on one parameter a discussion is provided regarding the interpretation of this parameter which can guide selection of its value in practice

linear shafarevich conjecture in this paper we settle armatively shafarevich s uniformization conjecture for varieties with linear fundamental groups we prove the strongest to date uniformization result the universal covering space of a complex projective manifold with a linear fundamental group is holomorphically convex the proof is based on both known and newly developed techniques in non abelian hodge theory

defect relations for holomorphic maps between spaces of different dimensions there is now a rather rich theory of defect relations for holomorphic maps between spaces of equal dimension developed by nevanlinna stoll griffiths and many other people however for the case of maps between spaces of unequal dimension essentially there is only the theory developed by weyl weyl and ahlfors when the target space is the complex projective space and their theory is built on the projective linear structure of the projective space a partial list of references for these results is given at the end of this paper in this paper we derive a defect relation for holomorphic maps between spaces of different dimensions where the target space can be an arbitrary compact complex algebraic manifold since the special case of maps from c to a compact complex manifold of complex dimension two contains already the methods and techniques needed for the general case in order to avoid unnecessarily complicated notations and to make the arguments more easily understood we give complete details only for this special case and indicate the easy modifications needed for the general case besides the usual techniques in value distribution theory of using the poincar6 lelong formula the calculus lemma of estimating the integrand by its integral and the concavity of logarithm a new key point in our argument is to introduce a meromorphic connection so that the second fundamental form of a divisor with respect to the connection is zero more precisely for the case of a meromorphic map f from c to a compact kihler surface m to consider the defect for a nonsingular zero set of a holomorphic section s of a positive holomorphic line bundle l we introduce a meromorphic connection i v form with the following two properties i for some hermitian holomorphic line bundle f over m and some holomorphic section 0 of f

on the geometry of classifying spaces and horizontal slices we study the local properties of the moduli space of a polarized calabi yau manifold let u be a neighborhood of the moduli space then we know the universal covering space v of u is a smooth manifold suppose d is the classifying space of a polarized calabi yau manifold with the automorphism group g then we prove that the map from v to d induced by the period map is a pluriharmonic map we also give a kahler metric on v which is called the hodge metric we prove that the ricci curvature of the hodge metric is negative away from zero we also proved the nonexistence of the kahler metric on the classifying space of a calabi yau threefold which is invariant under a cocompact lattice of g 1 introduction let x be a polarized simply connected calabi yau manifold that is x is a simply connected compact kahler manifold of dimension n with zero first chern class and is akform of x such that h 2 x z in this paper we study the local properties of the moduli space of the polarized calabi yau manifold x by definition is the parameter space of the complex structures over x for the fixed polarization is a quasi projective variety by a theorem of viehweg 17 suppose x is a calabi yau manifold let n d im h 1 x tx 0

symmetric differential forms on complete intersection varieties and applications in this paper we study the cohomology of the symmetric powers of the cotangent bundle of complete intersection varieties in projective space we provide an explicit description of some of those cohomology groups in terms of the equations defining the complete intersection as a first application we give a new example illustrating the fact that the dimension of the space of holomorphic symmetric differential forms is not deformation invariant then as our main application we construct varieties with ample cotangent bundle providing new results towards a conjecture of debarre

uber die lokale zetafunktion von shimuravarietaten monodromiefiltration und verschwindende zyklen in ungleicher charakteristik

big vector bundles and complex manifolds with semi positive tangent bundles we classify compact kahler manifolds with semi positive holomorphic bisectional and big tangent bundles we also classify compact complex surfaces with semi positive tangent bundles and compact complex 3 folds of the form mathbb p t x whose tangent bundles are nef moreover we show that if x is a fano manifold such that mathbb p t x has nef tangent bundle then x cong mathbb p n

the geometry of the eisenstein picard modular group the eisenstein picard modular group rm pu 2 1 mathbb z omega is defined to be the subgroup of rm pu 2 1 whose entries lie in the ring mathbb z omega where omega is a cube root of unity this group acts isometrically and properly discontinuously on bf h 2 mathbb c that is on the unit ball in mathbb c 2 with the bergman metric we construct a fundamental domain for the action of rm pu 2 1 mathbb z omega on bf h 2 mathbb c which is a 4 simplex with one ideal vertex as a consequence we elicit a presentation of the group see theorem 5 9 this seems to be the simplest fundamental domain for a finite covolume subgroup of rm pu 2 1

scaling exponents and multifractal dimensions for independent random cascades this paper is concerned with mandelbrot s stochastic cascade measures the problems of i scaling exponents of structure functions of the measure q and ii multifractal dimensions are considered for cascades with a generator vector w1 wc of the general type these problems were previously studied for independent strongly bounded variableswi 0 a wi c consequently a broad class of models used in applications including kolmogorov s long normal model in turbulence long stable universal cascades in atmospheric dynamics has not been covered roughly speaking problems i ii are here solved under the condition that the scaling exists the function is calculated for all arguments previously this was done for positiveq and a new effect emerges the function can generally involve discontinuities in the first derivative as well as in the second

gaussian fuzzy random variables in this paper we define the concepts of a gaussian fuzzy random variable and a gaussian fuzzy random vector and discuss their properties the covariance operator the characteristic functional and the linear transformation of a gaussian fuzzy random variable are investigated

symmetric differentials and the fundamental group helene esnault asked whether a smooth complex projective variety x with infinite fundamental group must have a nonzero symmetric differential meaning that h x s x 6 0 for some i 0 this was prompted by the second author s work 23 in fact severi had wondered in 1949 about possible relations between symmetric differentials and the fundamental group 34 p 36 we know from hodge theory that the cotangent bundle x has a nonzero section if and only if the abelianization of 1x is infinite the geometric meaning of other symmetric differentials is more mysterious and it is intriguing that they may have such a direct relation to the fundamental group in this paper we prove the following result on esnault s question in the slightly broader setting of compact kahler manifolds

least squares estimation in linear regression models with vague concepts the paper is a contribution to parameter estimation in fuzzy regression models with random fuzzy sets here models with crisp parameters and fuzzy observations of the variables are investigated this type of regression models may be understood as an extension of the ordinary single equation linear regression models by integrating additionally the physical vagueness of the involved items so the significance of these regression models is to improve the empirical meaningfulness of the relationship between the items by a more sensitive attention to the fundamental adequacy problem of measurement concerning the parameter estimation the ordinary least squares method is extended the existence of estimators by the suggested method is shown and some of their stochastic properties are surveyed

linear regression with random fuzzy variables extended classical estimates best linear estimates least squares estimates this paper deals with problems which arise if for well justified statistical models like linear regression only fuzzy data are available three approaches are discussed the first one is an application of zadeh s extension principle to optimal classical estimators here the result is that they in general do not keep their optimality properties the second one is the attempt to develop a certain kind of linear theory for fuzzy random variables w r t extended addition and scalar multiplication the main problem in this connection is that fuzzy sets with these extended operations do not constitute a linear space therefore in the third approach the least squares approximation principle for fuzzy data is investigated which leads to most acceptable results

vanishing theorems for ample vector bundles the main result of this article is a general vanishing theorem for the cohomology of tensorial representations of an ample vector bundle on a smooth complex projective variety in particular we extend classical theorems of griffiths and le potier to the whole dolbeault cohomology prove a variant of an uncorrect conjecture of sommese and answer a question of demailly as an application we prove conjectures of debarre and kim for branched coverings of grassmannians and extend a well known barth lefschetz type theorem for branched covers of projective spaces due to lazarsfeld we also obtain new restriction theorems for certain degeneracy loci

the self duality equations on a riemann surface in this paper we shall study a special class of solutions of the self dual yang mills equations the original self duality equations which arose in mathematical physics were defined on euclidean 4 space the physically relevant solutions were the ones with finite action the so called instantons the same equations may be dimensionally reduced to euclidean 3 space by imposing invariance under translation in one direction these equations also have physical relevance the solutions which have finite action in three dimensions are the magnetic monopoles if we take the reduction process one step further and consider solutions which are invariant under two translations we obtain a set of equations in the plane here however there is no clear physical meaning and indeed attempts to find finite action solutions have failed nevertheless these are the equations we shall consider despite the lack of interesting solutions in r2 the equations have the important property conformal invariance which allows them to be defined on manifolds modelled on r2 by conformal maps namely riemann surfaces we shall consider here solutions of the self duality equations defined on a compact riemann surface there are in fact solutions as we shall show and the moduli space of all solutions turns out to be a manifold with an extremely rich geometric structure which will be the focus of our study it brings together in a harmonious way the subjects of riemannian geometry topology algebraic geometry and symplectic geometry illuminating all these facets of the same object accounts for the length of this paper the self duality equations are equations from gauge theory geometrically they are defined in terms of connections on principal bundles while the group of the principal bundle may be chosen arbitrarily for the equations to make sense we restrict attention here to the simplest case of su 2 or so 3 there are two reasons for this the first and most obvious is that it simplifies calculations and avoids the use of inductive processes which are inherent in the consideration of a general lie group of higher rank the second reason is that solutions for su 2 have an intimate relationship with the internal structure of the riemann surface as a consequence of results we shall prove about solutions to the self duality equations we learn something about the moduli space of complex structures on the surface itself namely teichmuller space

random conformal weldings we construct a conformally invariant random family of closed curves in the plane by welding of random homeomorphisms of the unit circle the homeomorphism is constructed using the exponential of x where x is the restriction of the 2 dimensional free field on the circle and the parameter is in the high temperature regime beta sqrt 2 the welding problem is solved by studying a non uniformly elliptic beltrami equation with a random complex dilatation for the existence a method of lehto is used this requires sharp probabilistic estimates to control conformal moduli of annuli and they are proven by decomposing the free field as a sum of independent fixed scale fields and controlling the correlations of the complex dilatation restricted to dyadic cells of various scales for the uniqueness we invoke a result by jones and smirnov on conformal removability of holder curves our curves are closely related to sle for 4

uniform boundedness of level structures on abelian varieties over complex function fields let x be a smooth quotient of a bounded symmetric domain by an arithmetic subgroup open image in new window we prove the following generalization of nadel s result for any non negative integer g there exists a finite etale cover xg g of x determined by a subgroup open image in new window depending only on g such that for any compact riemann surface r of genus g and any non constant holomorphic map f r xg from r into the satake baily borel compactification xg of xg the image f r lies in the boundary xg x g xg nadel proved it for g 0 or 1 moreover for any positive integer n and any non negative integer g 0 we show that there exists a positive number a n g depending only on n and g with the following property a principally polarized non isotrivial n dimensional abelian variety over a complex function field of genus g does not have a level n structure for n a n g this was proved by nadel for g 0 or 1 and by noguchi for arbitrary g under the additional hypothesis that the abelian variety has non empty singular fibers

fractional brownian motion with hurst index h 0 and the gaussian unitary ensemble the goal of this paper is to establish a relation between characteristic polynomials of n n gue random matrices h as n and gaussian processes with logarithmic correlations we introduce a regularized version of fractional brownian motion with zero hurst index which is a gaussian process with stationary increments and logarithmic increment structure then we prove that this process appears as a limit of d n z log det h zi on mesoscopic scales as n by employing a fourier integral representation we use this to prove a continuous analogue of a result by diaconis and shahshahani j appl probab 31a 1994 49 62 on the macroscopic scale d n x gives rise to yet another type of gaussian process with logarithmic correlations we give an explicit construction of the latter in terms of a chebyshev fourier random series

families of rationally connected varieties recall that a proper variety x is said to be rationally connected if two general points p q x are contained in the image of a map g p x this is clearly a birationally invariant property when x is smooth this turns out to be equivalent to the a priori weaker condition that two general points can be joined by a chain of rational curves and also to the a priori stronger condition that for any finite subset x there is a map g p x whose image contains and such that gtx is an ample bundle

a new family of metrics for compact convex fuzzy sets based on a generalized concept of mid and spread one of the most important aspects of the statistical analysis of imprecise data is the usage of a suitable distance on the family of all compact convex fuzzy sets which is not too hard to calculate and which reflects the intuitive meaning of fuzzy sets on the basis of expressing the metric of bertoluzza et al c bertoluzza n corral a salas on a new class of distances between fuzzy numbers mathware soft comput 2 1995 71 84 in terms of the mid points and spreads of the corresponding intervals we construct new families of metrics on the family of all d dimensional compact convex sets as well as on the family of all d dimensional compact convex fuzzy sets it is shown that these metrics not only fulfill many good properties but also that they are easy to calculate and easy to manage for statistical purposes and therefore useful from the practical point of view

asymptotic properties of least squares estimation with fuzzy observations this paper deals with the asymptotic properties of the least squares estimators for fuzzy linear regression models with fuzzy triangular input output and random error terms the asymptotic normality and strong consistency of the fuzzy least squares estimator flse are investigated a confidence region based on a class of flses is proposed the asymptotic relative efficiency of flses with respect to the crisp least squares estimators is also provided and a numerical example is given some simulation results are also presented to illustrate the behavior of flses

least squares fuzzy regression with fuzzy random variables let x y be two convex fuzzy random variables on rn using a suitable metric we prove that the conditional expectation e y x is the best approximation of y by measurable functions of x this generalizes the analogous and well known property for real random variables a further topic is the approximation of y by a linear function of x in special cases and by use of hukuhara s difference between fuzzy sets we obtain formulas which are analogous to the classical structure contrary to the classical fact however the conditional expectation of gaussian fuzzy random variables in general does not coincide with the linear regression function

on euler characteristic and fundamental groups of compact manifolds let m be a compact riemannian manifold pi widetilde m rightarrow m be the universal covering and omega be a smooth 2 form on m with pi omega cohomologous to zero suppose the fundamental group pi 1 m satisfies certain radial quadratic resp linear isoperimetric inequality we show that there exists a smooth 1 form eta on widetilde m of linear resp bounded growth such that pi omega d eta as applications we prove that on a compact kahler manifold m omega with pi omega cohomologous to zero if pi 1 m is mathrm cat 0 or automatic resp hyperbolic then m is kahler non elliptic resp kahler hyperbolic and the euler characteristic 1 frac dim mathbb r m 2 chi m ge 0 resp 0

strictly pseudoconvex domains in c n the theory which we have developed so far will be expanded in two ways 1 in this chapter we shall use the cauchy fantappie calculus to transform the bochner martinelli kernel into a kernel adapted to strictly pseudoconvex domains in n and prove a number of finiteness and vanishing theorems for these domains 2 in the next chapter we shall pass from n to arbitrary hermitian manifolds and establish the analogue of the bmk formula in that context obtaining along the way the hodge decomposition theorem on compact manifolds both lines of thought will merge in the subsequent chapters and lead to a solution of the levi problem and finiteness theorems for strictly pseudoconvex manifolds as well as to a solution of the overline partial neumann problem to be formulated later

generators of the sister of euclidean picard modular groups the sister of eisenstein picard modular group was described in parker duke math j 94 3 433 464 1998 in fact one can define the sister groups of all picard modular groups in particular we find the generators of the sister of euclidean picard modular groups by using a geometric method

symmetric differentials kahler groups and ball quotients while margulis superrigidity theorem completely describes the finite dimensional linear representations of lattices of higher rank simple real lie groups almost nothing is known concerning the representation theory of complex hyperbolic lattices the main result of this paper theorem 1 3 is a strong rigidity theorem for a certain class of cocompact arithmetic complex hyperbolic lattices it relies on the following two ingredients theorem 1 6 showing that the representations of the topological fundamental group of a compact kahler manifold x are controlled by the global symmetric differentials on x an arithmetic vanishing theorem for global symmetric differentials on certain compact ball quotients using automorphic forms in particular deep results of clozel on base change theorem 1 11

the geometry of the gauss picard modular group we give a construction of a fundamental domain for rm pu 2 1 mathbb z i that is the group of holomorphic isometries of complex hyperbolic space with coefficients in the gaussian ring of integers mathbb z i we obtain from that construction a presentation of that lattice and relate it in particular to lattices constructed by mostow

a unified approach to fuzzy random variables the concept of fuzzy random variable was introduced as an analogous notion to random variables in order to extend statistical analysis to situations when the outcomes of some random experiment are fuzzy sets but in contrary to the classical statistical methods no unique definition has been established yet in this paper a set theoretical concept of fuzzy random variable will be presented this notion provides a useful framework to compare different concepts of fuzzy random variables using methods of general topology and drawing on results from topological measure theory and the theory of analytic spaces as the main result it will be shown that the introduced concept of fuzzy random variable is a unification of the already known ones

sur la rigidite de certains groupes fondamentaux l arithmeticite des reseaux hyperboliques complexes et les faux plans projectifs the motivation of this work comes from the study of lattices in real simple lie groups the famous margulis s superrigidity theorem claims that finite dimensional reductive representations of any lattice of a real simple lie group of real rank 2 are superrigid as a corollary such a lattice is arithmetic these results extend to the real rank one case for lattices in sp n 1 and f4 20 by the work of corlette and gromov schoen on the other hand mostow and deligne mostow exhibited arithmetic lattices with non superrigid representations as well as non arithmetic lattices in the unitary group pu 2 1 a natural question is then to find simple sufficient conditions for superrigidity or arithmeticity of lattices in pu 2 1 rogawski conjectured the following let be a torsion free cocompact lattice in pu 2 1 such that the hyperbolic quotient m b2 verifies the cohomogical conditions b1 m 0 and h1 1 m h2 m then is arithmetic in this paper we consider a smooth complex projective surface m verifying the above cohomological assumptions and study zariski dense representations of the fundamental group 1 m in a simple k group h of k rank 2 where k denotes a local field our main result states that there are strong restrictions on such representations especially when k is non archimedean theorem 5 we then consider some cocompact lattices in pu 2 1 of special geometric interest recall that a fake p2 is a smooth complex surface distinct from p2 having the same betti numbers as p2 fake p2 exist by a result of mumford and are complex hyperbolic quotients h2 by yau s proof of the calabi conjecture they obviously verify the hypotheses of rogawski s conjecture in this case we prove that every zariski dense representation of in pgl 3 is superrigid in the sense of margulis theorem 3 as a corollary every fake p2 is an arithmetic quotient of the ball b2

local rigidity for complex hyperbolic lattices and hodge theory we prove general local rigidity results for cocompact complex hyperbolic lattices using hodge theory

uniqueness theorems of hermitian metrics of seminegative curvature on quotients of bounded symmetric domains in the theory of locally symmetric spaces of negative ricci curvature it has been a classical problem to study the extent to which the topology of the manifold determines the geometry and in the hermitian case the complex structure the rigidity theorem of mostow 26 asserts that for a compact locally symmetric riemannian manifold of negative ricci curvature the manifold is determined up to isometry and normalizing constants by its fundamental group among the class of such manifolds with the obvious exceptions involving compact riemann surfaces the same theorem for locally symmetric riemannian manifolds of finite volume and negative ricci curvature was proved by prasad 27 in case of rank 1 and included in the super rigidity theorem of margulis 13 in the case of rank 2 in the class of compact locally symmetric hermitian manifolds of negative ricci curvature calabi vesentini 5 and borel 3 proved vanishing theorems of certain cohomology groups which imply in particular that for x locally irreducible of complex dimension 2 there exists no non trivial deformation of x as a complex manifold in this direction siu 32 33 proved the strong rigidity of the kdhler manifold x in the sense that any compact kkhler manifold m homotopic to x is necessarily biholomorphic or conjugate biholomorphic to x if m is also locally symmetric by assumption then it follows from the uniqueness of kihler einstein metrics of negative ricci curvature yau 37 that the conjugate biholomorphism is in fact an isometry up to a normalizing constant it is thus natural to ask

counting and equidistribution in heisenberg groups we strongly develop the relationship between complex hyperbolic geometry and arithmetic counting or equidistribution applications that arises from the action of arithmetic groups on complex hyperbolic spaces especially in dimension 2 we prove a mertens formula for the integer points over a quadratic imaginary number fields k in the light cone of hermitian forms as well as an equidistribution theorem of the set of rational points over k in heisenberg groups we give a counting formula for the cubic points over k in the complex projective plane whose galois conjugates are orthogonal and isotropic for a given hermitian form over k and a counting and equidistribution result for arithmetic chains in the heisenberg group when their cygan diameter tends to 0

a decomposition theorem for fuzzy set valued random variables in this paper a decomposition theorem for a square integrable fuzzy random variable frv is proposed the paper is mainly divided into two parts in the first part for any frv x we define the hukuhara set as the family of deterministic fuzzy sets c for which the hukuhara difference email protected hc exists almost surely in particular we prove that such a family is a closed with respect to different well known metrics convex subset of the family of all fuzzy sets in the second part we prove that any square integrable frv can be decomposed up to a random translation as the sum of a frv y and an element c chosen uniquely thanks to a minimization argument in the hukuhara set this decomposition allows us to characterize all fuzzy random translations in particular a frv is a fuzzy random translation if and only if its aumann expectation equals c given by the above decomposition up to a deterministic translation examples and open problems are also presented

brownian measures on jordan virasoro curves associated to the weil petersson metric abstract in this paper existence of the brownian measure on jordan curves with respect to the weil petersson metric is established the step from brownian motion on the diffeomorphism group of the circle to brownian motion on jordan curves in c requires probabilistic arguments well beyond the classical theory of conformal welding due to the lacking quasi symmetry of canonical brownian motion on diff s 1 a new key step in our construction is the systematic use of a kahlerian diffusion on the space of jordan curves for which the welding functional gives rise to conformal martingales together with a douady earle type conformal extension of vector fields on the circle to the disk

level structures on abelian varieties kodaira dimensions and lang s conjecture abstract assuming lang s conjecture we prove that for a prime p number field k and positive integer g there is an integer r such that no principally polarized abelian variety a k has full level p r structure to this end we use a result of zuo to prove that for each closed subvariety x in the moduli space a g of principally polarized abelian varieties of dimension g there exists a level m x such that the irreducible components of the preimage of x in a g m are of general type for m m x

fuzzy random variables abstract in this paper we define the concepts of fuzzy random variable and the expectation of a fuzzy random variable the new definition of expectation generalizes the integral of a set valued function we derive some properties of these new concepts by considering a suitable generalization of the hausdorff metric we derive the lebesgue dominated convergence type theorem