an infinite genus mapping class group and stable cohomology we exhibit a finitely generated group mathcal m whose rational homology is isomorphic to the rational stable homology of the mapping class group it is defined as a mapping class group associated to a surface mathcal s infty of infinite genus and contains all the pure mapping class groups of compact surfaces of genus g with n boundary components for any g 0 and n 0 we construct a representation of mathcal m into the restricted symplectic group rm sp rm res mathcal h r of the real hilbert space generated by the homology classes of non separating circles on mathcal s infty which generalizes the classical symplectic representation of the mapping class groups moreover we show that the first universal chern class in h 2 mathcal m mathbb z is the pull back of the pressley segal class on the restricted linear group rm gl rm res mathcal h via the inclusion rm sp rm res mathcal h r subset rm gl rm res mathcal h

a local index theorem for families of overline partial operators on punctured riemann surfaces and a new kahler metric on their moduli spaces we prove a local index theorem for families of bar partial operators on riemann surfaces of type g n i e of genusg withn 0 punctures we calculate the first chern form of the determinant line bundle on the teichmuller spacetg n endowed with quillen s metric where the role of the determinant of the laplace operators is played by the values of the selberg zeta function at integer points the result differs from the case of compact riemann surfaces by an additional term which turns out to be the kahler form of a new kahler metric on the moduli space of punctured riemann surfaces as a corollary of this result we derive for instance an analog of mumford s isomorphism in the case of the universal curve

generalized tate curve and integral teichmuller modular forms in this paper we establish a higher genus version of the tate elliptic curve i e for a given degenerate curve we construct its universal deformation defined over the ring of formal power series of certain deformation parameters whose coefficients are in a finitely generated z algebra described by the dual graph of the degenerate curve further we study automorphic forms on the moduli space of algebraic curves which we call teichmuller modular forms by evaluating these forms on the generalized tate curve in particular we show that integral teichmuller modular forms of given degree make a finitely generated ring and we describe the structure of this ring in degree 2 and 3 cases introduction the aim of this paper is to construct a higher genus version of the tate elliptic curve over a nonlocal base ring and to study integral automor phic forms on the moduli space of algebraic curves of genus 2 by evaluating these forms on the generalized tate curve as will be stated below our results extend earlier results of the author in 10 11 12 14 and are obtained especially by using excellent works of mumford faltings and chai 22 21 3 4 the periods and 1 forms of the generalized tate curve were used in 13 to construct universal and p adic solutions of the kp kadomtsev petvyashvili hierarchy in 10 section 2 the author obtained universal power series for multiplica tive periods of curves of fixed genus g which combined with the results of 21 and 4 chapter iii allowed us to deduce the existence of a universal jacobian semi abelian scheme j univ of relative dimension g over the ring

deligne products of line bundles over moduli spaces of curves we study deligne products for forgetful maps between moduli spaces of marked curves by offering a closed formula for tautological line bundles associated to marked points in particular we show that the deligne products for line bundles on the total spaces corresponding to forgotten marked points are positive integral multiples of the weil petersson bundles on the base moduli spaces

the chen ruan cohomology of moduli of curves of genus 2 with marked points in this work we describe the chen ruan cohomology of the moduli stack of smooth and stable genus 2 curves with marked points in the first half of the paper we compute the additive structure of the

geometry of curves with exceptional secant planes linear series along the general curve we study linear series on a general curve of genus g whose images are exceptional with regard to their secant planes working in the framework of an extension of brill noether theory to pairs of linear series we prove that a general curve has no linear series with exceptional secant planes in a very precise sense whenever the total number of series is finite next we partially solve the problem of computing the number of linear series with exceptional secant planes in a one parameter family in terms of tautological classes associated with the family by evaluating our hypothetical formula along judiciously chosen test families as an application we compute the number of linear series with exceptional secant planes on a general curve equipped with a one dimensional family of linear series we pay special attention to the extremal case of d secant d 2 planes to 2d 1 dimensional series which appears in the study of hilbert schemes of points on surfaces in that case our formula may be rewritten in terms of hypergeometric series which allows us both to prove that it is nonzero and to deduce its asymptotics in d

picard group of moduli of hyperelliptic curves the main subject of this work is the difference between the coarse moduli space and the stack of hyperelliptic curves we compute their picard groups giving explicit description of the generators we get an application to the non existence of a tautological family over the coarse moduli space

presentation and central extensions of mapping class groups we give a presentation of the mapping class groupm of a possibly bounded surface considering either all twists or just non separating twists as generators we also study certain central extensions of m one of them plays a key role in studying tqft functors namely the mapping class group of a p1 structure surface we give a presentation of this extension

towards a general theory of topological maps we present a general theory of topological maps whereby sensory input topological and local metrical information are combined to define the topological maps explaining such information topological maps correspond to the minimal models of an axiomatic theory describing the relationships between the different sources of information explained by a map we use a circumscriptive theory to specify the minimal models associated with this representation the theory here proposed is independent of the exploration strategy the agent follows when building a map we provide an algorithm to calculate the models of the theory this algorithm supports different exploration strategies and facilitates map disambiguation when perceptual aliasing arises

the second homology group of the mapping class group of an orientable surface in i 7 mumford shows that the picard group p ic is isomorphic to h2 f 2 and conjectures the latter is rank one g 3 we prove this below for g 5 another interpretation of this theorem may be obtained by identifying h2 f as bordism classes of fiber bundles f w4 t where t is a closed oriented surface sect 0 when f is closed every such bundle is bordant to f w t a bundle admitting a section s t w the theorem then says that

moduli of curves and spin structures via algebraic geometry gilberto bini and claudio fontanariabstract here we investigate some birational properties of two collec tions of moduli spaces namely moduli spaces of pointed stable curvesand of pointed spin curves in particular we focus on vanishings ofhodge numbers of type p 0 and on computations of the kodaira di mension our methods are purely algebro geometric and rely on aninduction argument on the number of marked points and the genus ofthe curves cf 3

nested abnormality theories we propose a new approach to the use of circumscription for representing knowledge nested abnormality theories are similar to simple abnormality theories introduced by mccarthy except that their axioms may have a nested structure with each level corresponding to another application of the circumscription operator the new style of applying circumscription sometimes leads to more economical and elegant formalizations mathematical properties of nested abnormality theories may be easier to investigate these advantages are demonstrated by recasting several familiar applications of circumscription in the new format including some examples of inheritance hierarchies the domain closure assumption and causal minimization nested abnormality theories provide also a convenient representation for the explanation closure approach to the frame problem developed by schubert

decision theoretic foundations of qualitative possibility theory abstract this paper presents a justification of two qualitative counterparts of the expected utility criterion for decision under uncertainty which only require bounded linearly ordered valuation sets for expressing uncertainty and preferences this is carried out in the style of savage starting with a set of acts equipped with a complete preordering relation conditions on acts are given that imply a possibilistic representation of the decision maker uncertainty in this framework pessimistic i e uncertainty averse as well as optimistic attitudes can be explicitly captured the approach thus proposes an operationally testable description of possibility theory

factorial property of a ring of automorphic forms a ring of automorphic forms is shown to be factorial under some conditions on the domain and on the picard group as an application we show that any divisor on the moduli space 2jf k of curves of genus g 3 is defined by a single element and that the satake compactification of u is is written as a projective spectrum of a factorial graded ring we find a single element which defines the closure of w4 in 5w4 where 3tf 4 is the moduli of curves of genus four whose canonical curves are exhibited as complete intersections of quadric cones and of cubics in p3 introduction let x be a normal quasi projective variety over c such that i x has a normal projective compactification x with codim a a 2 and ii x is a quotient space 2 t of a connected domain 2 with pic 0 by a discontinu ous group t of holomorphic automorphisms then the canonically determined homogeneous coordinate ring of x is factorial under some conditions on t or essentially on the picard group pic t in our notation theorem 1 when 2 is the siegel space and t is the modular group freitag 9 10 has shown that the homogeneous coordinate ring of x or more precisely the graded ring of siegel modular forms is factorial provided that the degree is greater than two our theorem is applicable to the case when x is the moduli space wlg of curves of genus g we need to know the first or second betti number of 3rg or of the teichmuller modular group tg which will almost determine the picard group in mumford 19 the first betti number ex 2rg of 3 would be one harer 12 proved it for g 5 and in this paper we show it for g 3 4 which seems more or less well known as for the case of 2 being a bounded symmetric domain the determination of the betti numbers is covered by the calculation of the stable real cohomology due to borel 2 for a fairly large class of 2 as an application of our method we show that the closure in 9ji4 of the moduli w4 of nonhyperelliptic curves of genus four exhibited as complete intersections of quadric cones and of cubics in ip3 is a subvariety defined by a principal ideal and gives the generator of the principal ideal this work was done while the author was staying at harvard university he

rank 1 phenomena for mapping class groups we prove that every element of the mapping class group g has linear growth confirming a conjecture of n ivanov and that g is not boundedly generated we also provide restrictions on linear representations of g and its finite index subgroups

decision making on the sole basis of statistical likelihood this paper presents a new axiomatic decision theory for choice under uncertainty unlike bayesian decision theory where uncertainty is represented by a probability function in our theory uncertainty is given in the form of a likelihood function extracted from statistical evidence the likelihood principle in statistics stipulates that likelihood functions encode all relevant information obtainable from experimental data in particular we do not assume any knowledge of prior probabilities consequently a bayesian conversion of likelihoods to posterior probabilities is not possible in our setting we make an assumption that defines the likelihood of a set of hypotheses as the maximum likelihood over the elements of the set we justify an axiomatic system similar to that used by von neumann and morgenstern for choice under risk our main result is a representation theorem using the new concept of binary utility we also discuss how ambiguity attitudes are handled applied to the statistical inference problem our theory suggests a novel solution the results in this paper could be useful for probabilistic model selection

bounded geometry for kleinian groups we show that a kleinian surface group or hyperbolic 3 manifold with a cusp preserving homotopy equivalence to a surface has bounded geometry if and only if there is an upper bound on an associated collection of coefficients that depend only on its end invariants bounded geometry is a positive lower bound on the lengths of closed geodesics when the surface is a once punctured torus the coefficients coincide with the continued fraction coefficients associated to the ending laminations

the picard group of the moduli space of r spin riemann surfaces abstract an r spin riemann surface is a riemann surface equipped with a choice of r th root of the co tangent bundle we give a careful construction of the moduli space orbifold of r spin riemann surfaces and explain how to establish a madsen weiss theorem for it this allows us to prove the mumford conjecture for these moduli spaces but more interestingly allows us to compute their algebraic picard groups for g 10 or g 9 in the 2 spin case we give a complete description of these picard groups in terms of explicitly constructed line bundles

centrally extended mapping class groups from quantum teichmuller theory abstract the central extension of the mapping class groups of punctured surfaces of finite type that arises in quantum teichmuller theory is 12 times the meyer class plus the euler classes of the punctures this is analogous to the result obtained in 12 for the thompson groups

an axiomatic treatment of three qualitative decision criteria the need for computationally efficient decision making techniques together with the desire to simplify the processes of knowledge acquisition and agent specification have led various researchers in artificial intelligence to examine qualitative decision tools however the adequacy of such tools is not clear this paper investigates the foundations of maximin minmax regret and competitive ratio three central qualitative decision criteria by characterizing those behaviors that could result from their use this characterizaton provides two important insights 1 under what conditions can we employ an agent model based on these basic qualitative decision criteria and 2 how rational are these decision procedures for the competitive ratio criterion in particular this latter issue is of central importance to our understanding of current work on on line algorithms our main result is a constructive representation theorem that uses two choice axioms to characterize maximin minmax regret and competitive ratio

the picard group of the moduli space of curves with level structures for 4 l and g large we calculate the integral picard groups of the moduli spaces of curves and principally polarized abelian varieties with level l structures in particular we determine the divisibility properties of the standard line bundles over these moduli spaces and we calculate the second integral cohomology group of the level l subgroup of the mapping class group in a previous paper the author determined this rationally this entails calculating the abelianization of the level l subgroup of the mapping class group generalizing previous results of perron sato and the author finally along the way we calculate the first homology group of the mod l symplectic group with coefficients in the adjoint representation

a quantum group structure in integrable conformal field theories we discuss a quantization prescription of the conformal algebras of a class ofd 2 conformal field theories which are integrable we first give a geometrical construction of certain extensions of the classical virasoro algebra known as classicalw algebras in which these algebras arise as the lie algebra of the second hamiltonian structure of a generalized korteweg de vries hierarchy this fact implies that thew algebras obtained as a reduction with respect to the nilpotent subalgebras of the kac moody algebra describe the integrability of a toda field theory subsequently we determine the coadjoint operators of thew algebras and relate these to classical yang baxter matrices the quantization of these algebras can be carried out using the concept of a so called quantum group we derive the condition under which the representations of these quantum groups admit a hilbert space completion by exploring the relation with the braid group then we consider a modification of the miura transformation which we use to define a quantumw algebra this leads to an alternative interpretation of the coset construction for kac moody algebras in terms of nonlinear integrable hierarchies subsequently we use the connection between the induced braid group representations and the representations of the mapping class group of riemann surfaces to identify an action of thew algebras on the moduli space of stable curves and we give the invariants of this action this provides a generalization of the situation for the virasoro algebra where such as invariant is given by the so called mumford form which describes the partition function of the bosonic string

the second rational homology group of the moduli space of curves with level structures abstract let be a finite index subgroup of the mapping class group of a closed genus g surface that contains the torelli group for instance can be the level l subgroup or the spin mapping class group we show that h 2 q q for g 5 a corollary of this is that the rational picard groups of the associated finite covers of the moduli space of curves are equal to q we also prove analogous results for surface with punctures and boundary components

stability of the homology of the mapping class groups of orientable surfaces the mapping class group of f fgs r is f rgs wo a where a is the topological group of orientation preserving diffeomorphisms of f which are the identity on df and fix the s punctures when r 0 and 2g s 3 an important feature of f is that it acts properly discontinuously on the teichmiller space us the quotient is the moduli space f of isometry classes of complete hyperbolic metrics of finite area on f since us is homeomorphic to euclidean space and f is virtually torsion free we have

subjective probability and expected utility without additivity comonotic acts are defined and an axiom of comonotomic independence 19 introduced that weakens the von newmann morgenstern axiom of independence if a nondegenerate continuous and monotonic state independent weak order over acts satisfies comonotonic independence then it induces a unique non necessarily additive probability and a von neumann morgenstern utility an extension of theexpected utility theory covers situations as the ellsberg paradox which are inconsistent with additive expected utility the concept of uncertainty aversion and interpretation of comonotonic independence in the context of social welfare functions are included

a presentation of mapping class groups in terms of artin groups and geometric monodromy of singularities from wajnryb s presentation we extract a simple presentation of the mapping class group of the genus g surface as a quotient of an artin group by simple relations among the centers of sub artin groups topological meanings are given by using deformation of simple singularities

two axiomatic approaches to decision making using possibility theory this paper proposes a utility theory for decision making under uncertainty that is described by possibility theory we show that our approach is a natural generalization of the two axiomatic systems that correspond to pessimistic and optimistic decision criteria proposed by dubois et al the generalization is achieved by removing axioms that are supposed to reflect attitudes toward uncertainty namely pessimism and optimism in their place we adopt an axiom that imposes an order on a class of canonical lotteries that realize either in the best or in the worst prize we prove an expected utility theorem for the generalized axiomatic system based on the newly introduced concept of binary utility

moduli spaces of curves and representation theory we establish a canonical isomorphism between the second cohomology of the lie algebra of regular differential operators on x of degree 1 and the second singular cohomology of the moduli space hat f g 1 of quintuples c p z l wherec is a smooth genusg riemann surface p a point onc z a local parameter atp l a degreeg 1 line bundle onc and a class of local trivializations ofl atp which differ by a non zero factor the construction uses an interplay between various infinite dimensional manifolds based on the topological spaceh of germs of holomorphic functions in a neighborhood of 0 in x and related topological spaces the basic tool is a canonical map from hat f g 1 to the infinite dimensional grassmannian of subspaces ofh which is the orbit of the subspaceh of holomorphic functions on x vanishing at under the group auth as an application we give a lie algebraic proof of the mumford formula n 6n2 6n 1 1 where n is the determinant line bundle of the vector bundle on the moduli space of curves of genusg whose fiber overc is the space of differentials of degreen onc

learning dynamics system identification for perceptually challenged agents from the perspective of an agent the input output behavior of the environment in which it is embedded can be described as a dynamical system inputs correspond to the actions executable by the agent in making transitions between states of the environment outputs correspond to the perceptual information available to the agent in particular states of the environment we view dynamical system identification as inference of deterministic finite state automata from sequences of input output pairs the agent can influence the sequence of input output pairs it is presented by pursuing a strategy for exploring the environment we identify two sorts of perceptual errors errors in perceiving the output of a state and errors in perceiving the inputs actually carried out in making a transition from one state to another we present efficient high probability learning algorithms for a number of system identification problems involving such errors we also present the results of empirical investigations applying these algorithms to learning spatial representations

every mapping class group is generated by 6 involutions abstract let mod g b denote the mapping class group of a surface of genus g with b punctures luo asked in torsion elements in the mapping class group of a surface math gt 0004048 v1 8apr2000 if there is a universal upper bound independent of genus for the number of torsion elements needed to generate mod g b we answer luo s question by proving that 3 torsion elements suffice to generate mod g 0 we also prove the more delicate result that there is an upper bound independent of genus not only for the number of torsion elements needed to generate mod g b but also for the order of those elements in particular our main result is that 6 involutions i e orientation preserving diffeomorphisms of order two suffice to generate mod g b for every genus g 3 b 0 and g 4 b 1

the stability space of compactified universal jacobians in this paper we describe compactified universal jacobians i e compactifications of the moduli space of line bundles on smooth curves obtained as moduli spaces of rank 1 torsion free sheaves on stable curves using an approach due to oda seshadri we focus on the combinatorics of the stability conditions used to define compactified universal jacobians we explicitly describe an affine space the stability space with a decomposition into polytopes such that each polytope corresponds to a proper deligne mumford stack that compactifies the moduli space of line bundles we apply this description to describe the set of isomorphism classes of compactified universal jacobians answering a question of melo and to resolve the indeterminacy of the abel jacobi sections addressing a problem raised by grushevsky zakharov

fuzzy sets as a basis for a theory of possibility the theory of possibility described in this paper is related to the theory of fuzzy sets by defining the concept of a possibility distribution as a fuzzy restriction which acts as an elastic constraint on the values that may be assigned to a variable more specifically if f is a fuzzy subset of a universe of discourse u u which is characterized by its membership function f then a proposition of the form x is f where x is a variable taking values in u induces a possibility distribution tx which equates the possibility of x taking the value u to f u the compatibility of u with f in this way x becomes a fuzzy variable which is associated with the possibility distribution tx in much the same way as a random variable is associated with a probability distribution in general a variable may be associated both with a possibility distribution and a probability distribution with the weak connection between the two expressed as the possibility probability consistency principle a thesis advanced in this paper is that the imprecision that is intrinsic in natural languages is in the main possibilistic rather than probabilistic in nature thus by employing the concept of a possibility distribution a proposition p in a natural language may be translated into a procedure which computes the probability distribution of a set of attributes which are implied by p several types of conditional translation rules are discussed and in particular a translation rule for propositions of the form x is f is possible where is a number in the interval 0 1 is formulated and illustrated by examples

qualitative possibilistic influence diagrams based on qualitative possibilistic utilities this paper proposes a new approach for decision making under uncertainty based on influence diagrams and possibility theory the so called qualitative possibilistic influence diagrams extend standard influence diagrams in order to avoid difficulties attached to the specification of both probability distributions relative to chance nodes and utilities relative to value nodes in fact generally it is easier for experts to quantify dependencies between chance nodes qualitatively via possibility distributions and to provide a preferential relation between different consequences in such a case the possibility theory offers a suitable modeling framework different combinations of the quantification between chance and utility nodes offer several kinds of possibilistic influence diagrams this paper focuses on qualitative ones and proposes an indirect evaluation method based on their transformation into possibilistic networks the proposed approach is implemented via a possibilistic influence diagram toolbox pidt

on injective homomorphisms between teichmuller modular groups i in this paper and its sequel we prove that injective homomorphisms between teichmuller modular groups of compact orientable surfaces are necessarily isomorphisms if an appropriately measured size of the surfaces in question differs by at most one in particular we establish the co hopfian property for modular groups of surfaces of positive genus

canonical extensions of the johnson homomorphisms to the torelli groupoid abstract we prove that every trivalent marked bordered fatgraph comes equipped with a canonical generalized magnus expansion in the sense of kawazumi this magnus expansion is used to give canonical extensions of the higher johnson homomorphisms m for m 1 to the torelli groupoid and we provide a recursive combinatorial formula for tensor representatives of these extensions in particular we give an explicit 1 cocycle in the dual fatgraph complex which extends 2 and thus answer affirmatively a question of morita and penner to illustrate our techniques for calculating higher johnson homomorphisms in general we give explicit examples calculating m for m 3

neron models and compactified picard schemes over the moduli stack of stable curves we construct modular deligne mumford stacks representable over parametrizing neron models of jacobians as follows let b be a smooth curve and k its function field let be a smooth genus g curve over k admitting stable minimal model over b the neron model b is then the base change of via the moduli map b of f i e b moreover is compactified by a deligne mumford stack over giving a completion of neron models naturally stratified in terms of neron models

spaces of graphs and surfaces on the work of soren galatius galatius most striking result is easy enough to state let n be the symmetric group on n letters and fn be the free non abelian group with n generators the symmetric group n acts naturally by permutation on the n generators of fn and every permutation gives thus rise to an automorphism of the free group galatius proves that the map n autfn in homology induces an isomorphism in degrees less than n 1 2 the homology of the symmetric groups in these ranges is well understood in particular in common with all finite groups it has no non trivial rational homology by galatius theorem in low degrees this then is also true for autfn h autfn q 0 for 0 2n 3

geometry of pseudodifferential algebra bundles and fourier integral operators we study the geometry and topology of filtered algebra bundles z over a smooth manifold x with typical fiber z z v the algebra of classical pseudodifferential operators acting on smooth sections of a vector bundle v over the compact manifold z and of integral order first a theorem of duistermaat and singer is generalized to the assertion that the group of projective invertible fourier integral operators pg fc z v is precisely the automorphism group of the filtered algebra of pseudodifferential operators we replace some of the arguments in their work by microlocal ones thereby removing the topological assumption we define a natural class of connections and b fields on the principal bundle to which z is associated and obtain a de rham representative of the dixmier douady class in terms of the outer derivation on the lie algebra and the residue trace of guillemin and wodzicki the resulting formula only depends on the formal symbol algebra z examples of pseudodifferential algebra bundles are given that are not associated to a finite dimensional fiber bundle over x

the virtual cohomological dimension of the mapping class group of an orientable surface let f f r be the mapping class group of a surface f of genus g with s punctures and r boundary components the purpose of this paper is to establish cohomology properties of f parallel to those of the arithmetic groups if g is a linear algebraic group defined over q x is the symmetric space associated to g and a is an arithmetic subgroup of g then a is virtually torsion free and acts properly discontinuously on x the rational homology of a is the same as that of x a furthermore there is a bordification of x bs to a manifold with corners 3 and an extension of the action of a to a properly discontinuous action on jf so that the quotient x a is compact the boundary of jf is homotopy equivalent to a wedge of spheres say of dimension d and the virtual cohomological dimension of a is n d 1 where n is the dimension of x in the case of the mapping class group there is no analog for g ivanov unpublished has proven that f is not arithmetic nevertheless f acts properly discontinuously on teichmiiller space r which is homeomorphic to euclidean space of dimension 6 g 6 2 s 3 will play the role of the symmetric space the quotient of 9by f is the moduli space of curves whose rational homology is then identified with that of e harvey har has constructed a borel serre bordification j of 3 by analytic methods in the case where f has punctures we will build g by a different combinatorial method in addition we will explicitly describe inside y a cell complex y of dimension 4 g 4 s onto which j may be f equivariantly retracted thus establishing an analog of the constructions for sl n by serre soul6 sol and ash a this complex will be of the lowest possible dimension because we will use j to prove our main result

lexicographic refinements in possibilistic decision trees and finite horizon markov decision processes possibilistic decision theory has been proposed twenty years ago and has had several extensions since then even though ap pealing for its ability to handle qualitative decision problems possibilisticdecision theory suffers from an important drawback qualitative possibilistic utility criteria compare acts through min and max operators which leads to a drowning effect to over come this lack of decision power of the theory several refinements have been proposed lexicographic refinements are particularly appealing since they allow to benefit from the expected utility background while remaining qualitative this article aims at extend ing lexicographic refinements to sequential decision problems i e to possibilistic decision trees and possibilistic markov decision processes when the horizon is finite we present two criteria that refine qualitative possibilistic utilities and provide dynamic programming algorithms for calculating lexicographically optimal policies

relative pro l completions of mapping class groups abstract fix a prime number l in this paper we develop the theory of relative pro l completion of discrete and profinite groups a natural generalization of the classical notion of pro l completion and show that the pro l completion of the torelli group does not inject into the relative pro l completion of the corresponding mapping class group when the genus is at least 2 see theorem 1 below as an application we prove that when g 2 the action of the pro l completion of the torelli group t g 1 on the pro l fundamental group of a pointed genus g surface is not faithful the choice of a first order deformation of a maximally degenerate stable curve of genus g determines an action of the absolute galois group g q on the relative pro l completion of the corresponding mapping class group we prove that for all g all such representations are unramified at all primes l when the first order deformation is suitably chosen this proof was communicated to us by mochizuki and tamagawa

compactified picard stacks over overline mathcal m _g we study algebraic artin stacks over overline mathcal m g giving a functorial way of compactifying the relative degree d picard variety for families of stable curves we also describe for every d the locus of genus g stable curves over which we get deligne mumford stacks strongly representable over overline mathcal m g

a simple axiomatization of nonadditive expected utility textabstractthis paper provides an extension of savage s subjective expected utility theory for decisions under uncertainty it includes in the set of events both unambiguous events for which probabilities are additive and ambiguous events for which probabilities are permitted to be nonadditive the main axiom is cumulative dominance which adapts stochastic dominance to decision making under uncertainty we derive a choquet expected utility representation and show that a modification of cumulative dominance leads to the classical expected utility representation the relationship of our approach with that of schmeidler who uses a two stage formulation to derive choquet expected utility is also explored our work may be viewed as a unification of schmeidler 1989 and gilboa 1987

subjective foundation of possibility theory this paper presents a new characterization of subjective possibility theory anscombe and aumann 1 pioneered a multiple variable approach to characterize the subjective probability on the basis of its relationship with objective probability we generalize the anscombe aumann approach to define subjective possibility theory to characterize the uncertainty of the variable of interest h we consider its relationship with a variable of probability and a variable of ignorance our axiomatization assumes that the preference over the acts contingent on the outcomes of the variables is a weak order and satisfies a monotonicity property the preference over probabilistic lotteries is represented by linear utility and the preference over acts under ignorance by a recently developed anchor utility theory we show that if h and the variable of ignorance are completely reversible in anscombe aumann s sense and h and the variable of probability are partially reversible then the uncertainty of h must obey the laws of possibility theory we outline a framework to manage heterogeneous uncertainty that includes probability possibility and ignorance in decision making

characteristic classes of surface bundles let m be a c manifold if one wants to classify all differentiable fibre bundles over a given manifold which have m as fibres then one must study the topology of the corresponding structure group diffm the group of all diffeomorphisms of m equipped with the c topology it is easy to see that the connected component diffo m of the identity of diff m is an open subgroup therefore the natural topology on the quotient group m diff m diffo m the diffeotopy group of m is the discrete topology hence we have a fibration

the noether lefschetz conjecture and generalizations we prove the noether lefschetz conjecture on the moduli space of quasi polarized k3 surfaces this is deduced as a particular case of a general theorem that states that low degree cohomology classes of arithmetic manifolds of orthogonal type are dual to the classes of special cycles i e sub arithmetic manifolds of the same type for compact manifolds this was proved in 3 here we extend the results of 3 to non compact manifolds this allows us to apply our results to the moduli spaces of quasi polarized k3 surfaces

the picard group of the universal moduli space of vector bundles on stable curves abstract we construct the moduli stack of properly balanced vector bundles on semistable curves and we determine explicitly its picard group as a consequence we obtain an explicit description of the picard groups of the universal moduli stack of vector bundles on smooth curves and of the schmitt s compactification over the stack of stable curves we prove some results about the gerbe structure of the universal moduli stack over its rigidification by the natural action of the multiplicative group in particular we give necessary and sufficient conditions for the existence of poincare bundles over the universal curve of an open substack of the rigidification generalizing a result of mestrano ramanan

planning routes through uncertain territory abstract planning routes and executing them requires both topological and metric information a natural implementation of a cognitive map might therefore consist of an assertional data base for topological information and a fuzzy map for the metric information a fuzzy map captures facts about objects by recording their relative positions orientations and scales in convenient frames of reference it is fuzzy in the sense that coordinates are specified to lie in a range rather than having fixed values the fuzzy map allows easy retrieval of information the same information is also represented in a discrimination tree which allows an object to be retrieved given its location and other attributes the problem of constructing a fuzzy map is more difficult we present a partial solution an algorithm that assimilates a fact first by imposing constraints on the fuzzy coordinates of the objects involved then by rearranging or growing the tree of frames of reference route planning is modelled as a process of finding the overall direction and topology of the path then filling in the details by deciding how to go around barriers it uses the retrieval algorithms our program spam carries out all these processes

the birman craggs johnson homomorphism and abelian cycles in the torelli group in the 1970s birman craggs johnson bcj trans ams 237 283 309 1978 trans ams 261 1 423 422 1980 used rochlin s invariant for homology 3 spheres to construct a remarkable surjective homomorphism sigma mathcal i g 1 to b 3 where mathcal i g 1 is the torelli group and b 3 is a certain bf f 2 vector space of boolean square free polynomials by pulling back cohomology classes and evaluating them on abelian cycles we construct 2g 4 o g 3 dimensions worth of nontrivial elements of h 2 mathcal i g 1 bf f 2 which cannot be detected rationally these classes in fact restrict to nontrivial classes in the cohomology of the subgroup mathcal k g 1 mathcal i g 1 generated by dehn twists about separating curves we also use the casson morita algebra and morita s integral lift of the bcj map restricted to mathcal k g 1 to give the same lower bound on h 2 mathcal k g 1 bf z

chow rings of moduli spaces of curves ii some results on the chow ring of overline mathscr m _4 ring of the moduli space of stable curves of genus 4 these results are not complete we find generators for the chow ring of 4 and for the chow groups in codimension 1 and 2 of w4 for a2 g 4 we find fourteen generators using test surfaces we prove that the dimension of a 2 4 4 is at least 13 and explicitly determine the single relation between the fourteen generators which still can exist finally we have two proofs that this relation does indeed hold so that the dimension of a2 4 4 equals 13 this enables us to determine the chow ring of 4 4 our original proof is based on a rather delicate argument the second proof uses a result of ran see r and is much simpler

stability of the homology of the moduli spaces of riemann surfaces with spin structure recently due largely to its importance in fermionic string theory there has been much interest in the moduli spaces gl e of riemann surfaces of genus g with spin structure of arf invariant e e z 2z algebraic geometers have long studied these spaces in their alternate guise as moduli spaces of pairs algebraic curve square root of the canonical bundle for topologists these spaces are rational classifying spaces for spin mapping class groups however despite the fact that r162 is a finite cover of the ordinary moduli space j 4g little is known about the topology of these spaces in this paper we begin a study of the homology of g e by proving that its homology groups are independent of g and e when g is adequately large theorem3 1 in a second paper h4 we will compute nl v g e and h2 r thereby calculating the picard group of jlg e putting this all together we know approximately the same amount about the homology of jt g e as we do about that of d g itself the techniques used here are an extension of those of h 2 which are in turn strongly related to those of c q v w and others we begin by constructing several simplicial complexes from configurations of simple closed curves and properly imbedded arcs in a surface of genus g the homology of the spin moduli space is identified with that of the spin mapping class group g which acts on these complexes in a natural way the borel construction is then applied to obtain a spectral sequence which describes the homology of g in terms the homology of the stabilizers of the cells of these complexes these turn out to be spin mapping class groups in an extended sense of smaller genus and the result is established inductively the complexes are exactly the same as those of h 2 however the spectral sequence arguments are more difficult because there are more orbits of cells under the action of g furthermore in sect 4 we apply an entirely different and much simpler version of the argument of h 2 to obtain stability in the case of a closed surface

moore and seiberg s equations and 3d topological field theory we prove that any solution to moore and seiberg s equations defines a projective rational topological field theory prtft using surgery presentations for decorated three manifolds and kirby s calculus