on the lower central quotients of the free groups of finite rank of a certain variety abstract let f n be a free group of rank n and k f n the k th term of the lower central series of f n for l 1 we denote by f n q l the quotient group of f n by a normal subgroup 2 3 f n l 2 2 f n in this paper we show that each of the graded quotients of the lower central series of the group f n q l for any l 1 is a free abelian group and give a basis of it by using a generalized chen s integration in free groups

yet another algorithm which can generate topography map this paper presents an algorithm to form a topographic map resembling to the self organizing map the idea stems on defining an energy function which reveals the local correlation between neighboring neurons the larger the value of the energy function the higher the correlation of the neighborhood neurons on this account the proposed algorithm is defined as the gradient ascent of this energy function simulations on two dimensional maps are illustrated

principal bundles whose restriction to curves are trivial let k be an algebraically closed field and x a smooth projective variety defined over k let e g be a principal g bundle over x where g is an algebraic group defined over k with the property that for every smooth curve c in x the restriction of e g to c is the trivial g bundle we prove that the principal g bundle e g over x is trivial we also give examples of nontrivial principal bundle over a quasi projective variety y whose restriction to every smooth curve in y is trivial

a superquadratic indefinite elliptic system and its morse conley floer homology we study critical points of the indefinite functional f h u v int nabla u cdot nabla v h x u v mathrm d x by applying floer s homology construction to the ordinary gradient flow of the functional f on a suitable sobolev space one of our main observations is that even though this flow is well posed in both time directions and lacks any kind of smoothing property one can still obtain compactness of connecting orbit spaces and thus define the floer homology for f h

moduli of sheaves on surfaces and action of the oscillator algebra this paper gives a generalization of some results on hilbert schemes of points on surfaces let m g r n resp m u r n be the gieseker resp uhlenbeck compactification of the moduli spaces of stable bundles on a smooth projective surface we show that for surfaces satisfying some technical condition

dirac operators coupled to instantons on positive definite four manifolds we study the moduli space of instantons on a simply connected positive definite closed four manifold by analyzing the classifying map of the index bundle of a family of dirac operators parameterized by the moduli space as an application we compute the cohomology rings for the charge 1 and 2 moduli spaces in the rank stable limit

extremal kahler metrics on projectivized vector bundles we prove the existence of extremal nonconstant scalar curvature kahler metrics on certain unstable projectivized vector bundles p e m over a compact constant scalar curvature kahler manifold m with discrete holomorphic automorphism group in certain adiabatic kahler classes in particular the vector bundles e m are assumed to split as a direct sum of stable subbundles e el circle plus circle plus e s all having different mumford takemoto slope for example mu e 1 mu e s

semistable sheaves in positive characteristic we prove maruyama s conjecture on the boundedness of slope semistable sheaves on a projective variety defined over a noetherian ring our approach also gives a new proof of the boundedness for varieties defined over a characteristic zero field this result implies that in mixed characteristic the moduli spaces of gieseker semistable sheaves are projective schemes of finite type the proof uses a new inequality bounding slopes of the restriction of a sheaf to a hypersurface in terms of its slope and the discriminant this inequality also leads to effective restriction theorems in all characteristics improving earlier results in characteristic zero

generalized johnson homomorphisms for extended n series abstract the johnson filtration of the mapping class group of a compact oriented surface is the descending series consisting of the kernels of the actions on the nilpotent quotients of the fundamental group of the surface each term of the johnson filtration admits a johnson homomorphism whose kernel is the next term in the filtration in this paper we consider a general situation where a group acts on a group with a filtration called an extended n series we develop a theory of johnson homomorphisms in this general setting including many known variants of the original johnson homomorphisms as well as several new variants

the cokernel of the johnson homomorphisms of the automorphism group of a free metabelian group in this paper we determine the cokernel of the k th johnson homomorphisms of the automorphism group of a free metabelian group for k 2 and n 4 as a corollary we obtain a lower bound on the rank of the graded quotient of the johnson filtration of the automorphism group of a free group furthermore by using the second johnson homomorphism we determine the image of the cup product map in the rational second cohomology group of the ia automorphism group of a free metabelian group and show that it is isomorphic to that of the ia automorphism group of a free group which is already determined by pettet finally by considering the kernel of the magnus representations of the automorphism group of a free group and a free metabelian group we show that there are non trivial rational second cohomology classes of the ia automorphism group of a free metabelian group which are not in the image of the cup product map

model based clustering by probabilistic self organizing maps in this paper we consider the learning process of a probabilistic self organizing map pbsom as a model based data clustering procedure that preserves the topological relationships between data clusters in a neural network based on this concept we develop a coupling likelihood mixture model for the pbsom that extends the reference vectors in kohonen s self organizing map som to multivariate gaussian distributions we also derive three expectation maximization em type algorithms called the socem soem and sodaem algorithms for learning the model pbsom based on the maximum likelihood criterion socem is derived by using the classification em cem algorithm to maximize the classification likelihood soem is derived by using the em algorithm to maximize the mixture likelihood and sodaem is a deterministic annealing da variant of socem and soem moreover by shrinking the neighborhood size socem and soem can be interpreted respectively as da variants of the cem and em algorithms for gaussian model based clustering the experimental results show that the proposed pbsom learning algorithms achieve comparable data clustering performance to that of the deterministic annealing em daem approach while maintaining the topology preserving property

the role of the green s function in a non linear elliptic equation involving the critical sobolev exponent this paper is concerned with non linear elliptic problems of the type pe u u n 2 n 2 eu u 0 on u 0 on where is a smooth and bounded domain in rn n 4 and e 0 we show that if the ue are solutions of pe which concentrate around a point as e 0 then this point cannot be on the boundary of and is a critical point of the regular part of the green s function conversely we show that for n 5 and any non degenerate critical point x0 of the regular part of the green s function there exist solutions of pe concentrating around x0 as e 0

controlling the magnification factor of self organizing feature maps the magnification exponents occurring in adaptive map formation algorithms like kohonen s self organizing feature map deviate for the information theoretically optimal value 1 as well as from the values that optimize e g the mean square distortion error 1 3 for one dimensional maps at the same time models for categorical perception such as the perceptual magnet effect which are based on topographic maps require negative magnification exponents 0 we present an extension of the self organizing feature map algorithm which utilizes adaptive local learning step sizes to actually control the magnification properties of the map by change of a single parameter maps with optimal information transfer with various minimal reconstruction errors or with an inverted magnification can be generated analytic results on this new algorithm are complemented by numerical simulations

einstein hermitian connections on polystable principal bundles over a compact kahler manifold given a compact kahler manifold m and a connected reductive algebraic group g over inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 01i a prinlcipal g bundle over m admits an einstein hermitian connection if and only if the principal bundle is polystable if m is a projective manifold a higgs g bundle over m admits an einstein hermitian connection if and only if the higgs bundle is polystable

enumeration of genus two curves with a fixed complex structure in ℙ 2 and ℙ 3 the main concrete result of this paper is enumeration of genus two curves with complex structure fixed in p and p along the way rational curves with certain simple singularities are counted as well while the methods described can be used to count positive genus curves in some other cases the most powerful direct applications of the machinery developed are to enumeration of rational curves with a very large class of singularities in projective spaces

an obstruction bundle relating gromov witten invariants of curves and kahler surfaces in a previous paper the authors defined symplectic local gromov witten invariants associated to spin curves and showed that the gw invariants of a k ahler surface x with p g 0 are a sum of such local gw invariants this paper describes how the local gw invariants arise from an obstruction bundle in the sense of taubes over the space of stable maps into curves together with the results of our earlier paper this reduces the calculation of the gw invariants of elliptic and general type complex surfaces to computations in the gw theory of curves with additional classes the euler classes of the real obstruction bundles

flat holomorphic connections on principal bundles over a projective manifold let g be a connected complex linear algebraic group and r u g its unipotent radical a principal g bundle e g over a projective manifold m will be called polystable if the associated principal g r u g bundle is so a g bundle e g over m is polystable with vanishing characteristic classes of degrees one and two if and only if e g admits a at holomorphic connection with the property that the image in g r u g of the monodromy of the connection is contained in a maximal compact subgroup of g r u g

mapping class group action versus galois action on profinite fundamental groups let x be an algebraic curve of genus g n punctured defined over a number field k then the profinite or the pro l completion of the topological fundamental group of x admits two actions the action of the profinite completion of the mapping class group of the orientable surface of topological type g n and the action of the absolute galois group of k this paper compares these two in the profinite case it is shown that the intersection of the images of these two actions is trivial if x is affine and its fundamental group is nonabelian on the contrary in the pro l case there are many curves such that the image of the galois action contains the image of the mapping class group action it is proved that the set of points corresponding to such curves is dense in the moduli space of g n curves

instanton moduli as a novel map from tori to k3 surfaces a map is constructed from the moduli of hyper kahler tori to hyper kahler k3 surfaces which does not coincide with the kummer map the map takes a torus to the moduli space of so 3 connections on a bundle with nontrivial first stiefel whitney class and first pontrjagin class equal to 4 this map is shown to intersect the kummer moduli and also certain subvarieties of singular k3 surfaces our map is shown to satisfy the local torelli theorem and the k3 surfaces in its image are shown to carry a natural metric which is calabi yau

equivalence relations for homology cylinders and the core of the casson invariant let r be a compact oriented surface of genus g with one boundary component homology cylinders over r form a monoid ic into which the torelli group i of r embeds by the mapping cylinder construction two homology cylinders m and m are said to be y k equivalent if m is obtained from m by twisting an arbitrary surface s in m with a homeomorphim belonging to the k th term of the lower central series of the torelli group of s the j k equivalence relation on ic is defined in a similar way using the k th term of the johnson filtration in this paper we characterize the y 3 equivalence with three classical invariants 1 the action on the third nilpotent quotient of the fundamental group of r 2 the quadratic part of the relative alexander polynomial and 3 a by product of the casson invariant similarly we show that the j 3 equivalence is classified by 1 and 2 we also prove that the core of the casson invariant originally defined by morita on the second term of the johnson filtration of i has a unique extension to the corresponding submonoid of ic that is preserved by y 3 equivalence and the mapping class group action

on the fourth johnson homomorphism of the automorphism group of a free group abstract in this paper we consider the johnson homomorphism of the automorphism group of a free group with respect to the lower central series of the ia automorphism group of a free group in particular we determine the rational cokernel of the fourth johnson homomorphism and show that there appears a new obstruction for the surjectivity of the johnson homomorphism furthermore we characterize this obstruction using trace maps

stable vector bundles on algebraic surfaces we prove an existence result for stable vector bundles with arbitrary rank on an algebraic surface and determine the birational structure of a certain moduli space of stable bundles on a rational ruled surface

extensions verselles et automorphismes des groupes nilpotents libres abstract nous montrons que si l est un groupe libre de rang 2 n superieur ou egal a 2 alors la surjection canonique aut l gl 2 n z n est pas scindable

moduli of high rank vector bundles over surfaces the purpose of this work is to apply the degeneration theory developed in gl to study the moduli space of stable vector bundles of arbitrary rank on any smooth algebraic surface over c we will show that most of the recent progress in understanding moduli of rank two vector bundles can be carried over to high rank cases after introducing the notion of stable vector bundles the first author constructed the moduli schemes of vector bundles on surfaces he showed that for any smooth algebraic surface x with ample divisor h and line bundle i on x there is a coarse moduli scheme m x i h parameterizing modulo equivalence relation the set of all h semistable rank r torsion free sheaves e on x with dete i and c2 e d since then many mathematicians have studied the geometry of this moduli space especially for the rank two case to cite a few maruyama taubes and the first author showed that the moduli space m x m r d x i h is non empty when d is large moduli spaces of vector bundles of some special surfaces have been studied also the deep understanding of m x for arbitrary x and r 2 begins with donaldson s generic smoothness result roughly speaking donaldson do later generalized by friedman fr and k zhu zh showed that when d is large enough then the singular locus sing m x of m x is a proper subset of m 2 d x and its codimension in m x increases linearly in d this theorem indicates that the moduli m x behaves as expected when the second chern class d is large later using general deformation theory the second author proved that m x is normal and has local complete intersection l c i singularities at stable sheaves provided d is large l2 he also showed that when x is a surface of general type satisfying some mild technical conditions then m x is of general type for d 0 l2 in our paper gl we also proved that m x is irreducible if d is large in this and subsequent papers we shall show that the geometry of m x and the geometry of m x r 3 is rather similar the main obstacle in doing so is the lack of an analogy of the generic smoothness result in high rank case in this paper we will use the degeneration of moduli developed in gl to establish the following main technical theorem

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

restriction theorems for principal bundles let e be a semistable or stable principal bundle over a smooth complex projective variety x and let d x be a complete intersection we study the semi stability of the restriction e d some of the results known for vector bundles such as grauert mulich flenner and mehta ramanathan theorems are generalized to principal bundles

p johnson homomorphisms and pro p groups abstract we propose an approach to study non abelian iwasawa theory using the idea of johnson homomorphisms in low dimensional topology we introduce arithmetic analogues of johnson homomorphisms maps called the p johnson homomorphisms maps associated to the zassenhaus filtration of a pro p galois group over a z p extension of a number field we give their cohomological interpretation in terms of massey products in galois cohomology

instabilities in kohonen s self organizing feature map the topology preserving representation of a rectangular part of space onto a square network of formal neurons is studied using the kohonen algorithm linear stability analysis shows that there is a critical ratio for the sides of the rectangle for larger ratios the map becomes unstable the value of the critical ratio depends on the actual shape of the adjustment function the problems cannot be scaled away in the case of inhomogeneously sampled input space the results of the analysis are compared with computer simulations

a reduction of the target of the johnson homomorphisms of the automorphism group of a free group let f n be a free group of rank n and f n n the quotient group of f n by a subgroup n 3 n 3 n 2 n 2 n 2 where n k denotes the k th subgroup of the lower central series of the free group f n in this paper we determine the group structure of the graded quotients of the lower central series of the group f n n by using a generalized chen s integration in free groups then we apply it to the study of the johnson homomorphisms of the automorphism group of f n in particular under taking a reduction of the target of the johnson homomorphism induced from a quotient map f n f n n we see that there appear only two irreducible components the morita obstruction s k h q and the schur weyl module of type h k 2 12 q in the cokernel of the rational johnson homomorphism k q k id q for k 5 and n k 2

semistable and numerically effective principal higgs bundles we study miyaoka type semistability criteria for principal higgs g bundles e on complex projective manifolds of any dimension we prove that e has the property of being semistable after pullback to any projective curve if and only if certain line bundles obtained from some characters of the parabolic subgroups of g are numerically effective one also proves that these conditions are met for semistable principal higgs bundles whose adjoint bundle has vanishing second chern class in a second part of the paper we introduce notions of numerical effectiveness and numerical flatness for principal higgs bundles discussing their main properties for non higgs principal bundles we show that a numerically flat principal bundle admits a reduction to a levi factor which has a flat hermitian yang mills connection and as a consequence that the cohomology ring of a numerically flat principal bundle with coefficients in r is trivial to our knowledge this notion of numerical effectiveness is new even in the case of non higgs principal bundles

an error control scheme for transmission of vector quantization data over noisy channels this paper presents an error control scheme for transmitting vector quantized data over noisy channels first we review a self organizing feature map sofm based approach to construct the mapping from the codebook of the quantizer to the channel signal set of the communication system this approach is robust with respect to channel noise even if we do not use any error control scheme in the communication system afterwards a new trellis type quantizer namely the trellis coded kohonen 1984 map tckm is presented the design process of the tckm which is based on the neighborhood structure of sofms is simpler than that of conventional trellis coded vector quantizers tcvqs simulation results show that the performance of tckms is comparable with that of tcvqs last we introduce the error control scheme based on the concepts of the above two applications of sofms simulation results show that the proposed error control scheme is more robust with respect to channel noise the advantage of our approaches mentioned above is that the design processes of transmission systems are predefined before the construction of the codebook thus different codebooks with the same neighborhood structure can share the same design

topological maps on ising spin networks abstract we study the formation of a topologically corrected representation of input signals we use a network of ising spins and update the external synapses after network relaxation according to local rules there is no unphysiological winner takes all mechanism involved for a two dimensional input we demonstrate the formation of a topological map

kernel based topographic map formation by local density modeling we introduce a new learning algorithm for kernel based topographic map formation the algorithm generates a gaussian mixture density model by individually adapting the gaussian kernels centers and radii to the assumed gaussian local input densities

finite type 3 manifold invariants and the structure of the torelli group i using the recently developed theory of finite type invariants of integral homology 3 spheres we study the structure of the torelli group of a closed surface explicitly we construct a natural cocycles of the torelli group with coefficients in a space of trivalent graphs and cohomology classes of the abelianized torelli group b group homomorphisms that detect rationally the nontriviality of the lower central series of the torelli group our results are motivated by the appearance of trivalent graphs in topology and in representation theory and the dual role played by the casson invariant in the theory of finite type invariants of integral homology 3 spheres and in morita s study mo2 mo3 of the structure of the torelli group our results generalize those of s morita mo2 mo3 and complement the recent calculation due to r hain ha2 of the i adic completion of the rational group ring of the torelli group we also give analogous results for two other subgroups of the mapping class group

a sum formula for the casson walker invariant we study the following question how does the casson walker invariant of a rational homology 3 sphere obtained by gluing two pieces along a surface depend on the two pieces our partial answer may be stated as follows for a compact oriented 3 manifold a with boundary a the kernel l a of the map from h 1 a q to h 1 a q induced by the inclusion is called the lagrangian of a let be a closed oriented surface and let a a b and b be four rational homology handlebodies such that a a b and b are identified via orientation preserving homeomorphisms with assume that l a l a and l b l b inside h 1 q and also assume that l a and l b are transverse then we express in terms of the form induced on 3 l a by the algebraic intersection on h 2 a a paired to the analogous form on 3 l b via the intersection form of the simple formula that we obtain naturally extends to the extension of the casson walker invariant of the author it also extends to gluings along non connected surfaces

structure of symplectic invariant lie subalgebras of symplectic derivation lie algebras abstract we study the structure of the symplectic invariant part h g 1 sp of the lie algebra h g 1 consisting of symplectic derivations of the free lie algebra generated by the rational homology group of a closed oriented surface g of genus g first we describe the orthogonal direct sum decomposition of this space which is induced by the canonical metric on it and compute it explicitly up to degree 20 in this framework we give a general constraint which is imposed on the sp invariant component of the bracket of two elements in h g 1 second we clarify the relations among h g 1 and the other two related lie algebras h g and h g which correspond to the cases of a closed surface g with and without base point g in particular based on a theorem of labute we formulate a method of determining these differences and describe them explicitly up to degree 20 third by giving a general method of constructing elements of h g 1 sp we reveal a considerable difference between two particular submodules of it one is the sp invariant part of a certain ideal j g 1 and the other is that of the johnson image finally we combine these results to determine the structure of h g 1 completely up to degree 6 including the unstable cases where the genus 1 case has an independent meaning in particular we see a glimpse of the galois obstructions explicitly from our point of view

phase diagrams of self organizing maps we present a method which allows the analytic determination of phase diagrams in the self organizing map a model for the formation of topographic projection patterns in the brain and in signal processing applications the method only requires an ansatz for the tesselation of the data space induced by the map not for the explicit state of the map we analytically obtain phase diagrams for various examples including models for the development of orientation and ocular dominance maps the latter phase diagram exhibits transitions to broadening ocular dominance patterns as observed in a recent experiment s1063 651x 96 00109 2

automorphisms of descending mod p central series abstract in this paper we study the automorphism group of the quotient n k of a free group of finite rank n by the k 1 term k 1 of the zassenhaus z and the stallings s mod p central series with p a prime number just like in the already known case for the usual terms of the lower central series these automorphism groups fit naturally into a short exact sequence containing the automorphism groups of two consecutive quotients n k and n k 1 more precisely if we denote by l k the quotient k k 1 we prove that there is a non central extension of groups 0 h o m n 1 l k 1 a u t n k 1 a u t n k 1 which splits if and only if k 1 and p is odd if z or k 1 and p n 3 2 2 2 if s moreover we also prove that the restriction of this extension to the subgroup i a p n k 1 of a u t n k 1 formed by the automorphisms that act trivially on n 1 give us a non split central extension of groups 0 h o m n 1 l k 1 i a p n k 1 i a p n k 1

the so 3 monopole cobordism and superconformal simple type abstract we show that the so 3 monopole cobordism formula from 15 implies that all closed oriented smooth four manifolds with b 1 0 and b 3 and odd with seiberg witten simple type satisfy the superconformal simple type condition defined by marino moore and peradze 28 27 this implies the lower bound conjectured by fintushel and stern 18 on the number of seiberg witten basic classes in terms of topological data

joint entropy maximization in kernel based topographic maps a new learning algorithm for kernel based topographic map formation is introduced the kernel parameters are adjusted individually so as to maximize the joint entropy of the kernel outputs this is done by maximizing the differential entropies of the individual kernel outputs given that the map s output redundancy due to the kernel overlap needs to be minimized the latter is achieved by minimizing the mutual information between the kernel outputs as a kernel the radial incomplete gamma distribution is taken since for a gaussian input density the differential entropy of the kernel output will be maximal since the theoretically optimal joint entropy performance can be derived for the case of nonoverlapping gaussian mixture densities a new clustering algorithm is suggested that uses this optimum as its null distribution finally it is shown that the learning algorithm is similar to one that performs stochastic gradient descent on the kullback leibler divergence for a heteroskedastic gaussian mixture density model

integral homology 3 spheres and the johnson filtration the mapping class group of an oriented surface g 1 of genus g with one boundary component has a natural decreasing filtration mg 1 mg i 3 me i d m g 1 3 where m g 1 k is the kernel of the action of m g 1 on the k th nilpotent quotient of 1 g 1 using a tree lie algebra approximating the graded lie algebra k m g 1 k m g 1 k 1 we prove that any integral homology sphere of dimension 3 has for some g a heegaard decomposition of the form m hg lg h g where g 1 3 and lg is such that h g lg h g s 3 this proves a conjecture due to s morita and shows that the core of the casson invariant is indeed the casson invariant

principal bundles on projective varieties and the donaldson uhlenbeck compactification let h be a semisimple algebraic group we prove the semistable reduction theorem for semistable principal h bundles over a smooth projective variety x defined over the field c when x is a smooth projective surface and h is simple we construct the algebro geometric donaldson uhlenbeck compactification of the moduli space of semistable principal h bundles with fixed characteristic classes and describe its points for large characteristic classes we show that the moduli space of stable principal h bundles is non empty

on the graded quotients of the rings of fricke characters of free groups abstract for a free group f n on n generators we consider the aut f n invariant ideal j generated by tr x 2 for all x f n in the ring x q f n of fricke characters of f n this ideal defines a descending filtration j j 2 j 3 of x q f n for each k 1 we introduce the normal subgroup e n k consisting of all automorphisms of f n which act on j j k 1 trivially these normal subgroups define a central filtration of aut f n this is a fricke character analogue of the andreadakis johnson filtration a n k of aut f n the main purpose of the paper is to show that e n 1 coincides with inn f n a n 2 where inn f n is the inner automorphism group of the free group f n and that a n 2 k e n k for any k 1

on a generalised fourier transform of instantons over flat tori recently p braam pointed out that nahm s adaption of the adhm procedure to the case of monopoles equally well applies to instantons over flat tori relating them to instantons over the first brillouin zone we show that this construction has an inverse hence the nahm transform actually is a duality transform

anti self dual yang mills connections over complex algebraic surfaces and stable vector bundles on presente une correspondance entre la geometrie algebrique et la geometrie differentielle des fibres vectoriels soit une surface algebrique projective x qui a un plongement donne x cp n et soit une metrique de kahler sur x dont la classe de cohomologie associee est duale a la classe de section d hyperplan h un fibre sur x est stable par rapport au plongement projectif si et seulement si il admet une connexion irreductible d hermite einstein par rapport a la metrique cette connexion est alors unique

moduli spaces for principal bundles in arbitrary characteristic in this article we solve the problem of constructing moduli spaces of semistable principal bundles and singular versions of them over smooth projective varieties over algebraically closed ground fields of positive characteristic