analytic torsion and holomorphic determinant bundles in this paper we prove that in the case of holomorphic locally kahler fibrations the analytic and algebraic geometry constructions of determinant bundles for direct images coincide we calculate the curvature of the holomorphic hermitian connection for the quillen metric on the determinant bundle we study the behavior of the quillen metric under change of metrics in the fibers and also on the twisting vector bundles we thus generalize the conformal anomaly formula to kahler manifolds of arbitrary dimension we also study the quillen metrics on determinants associated with exact sequences of vector bundles we prove that the quillen metric is smooth on the grothendieck knudsen mumford determinant for arbitrary holomorphic fibrations

edge scaling limits for a family of non hermitian random matrix ensembles a family of random matrix ensembles interpolating between the ginibre ensemble of n n matrices with iid centered complex gaussian entries and the gaussian unitary ensemble gue is considered the asymptotic spectral distribution in these models is uniform in an ellipse in the complex plane which collapses to an interval of the real line as the degree of non hermiticity diminishes scaling limit theorems are proven for the eigenvalue point process at the rightmost edge of the spectrum and it is shown that a non trivial transition occurs between poisson and airy point process statistics when the ratio of the axes of the supporting ellipse is of order n 1 3 in this regime the family of limiting probability distributions of the maximum of the real parts of the eigenvalues interpolates between the gumbel and tracy widom distributions

critical statistics for non hermitian matrices we introduce a generalized ensemble of non hermitian matrices interpolating between the gaussian unitary ensemble the ginibre ensemble and the poisson ensemble the joint eigenvalue distribution of this model is obtained by means of an extension of the itzykson zuber formula to general complex matrices its correlation functions are studied both in the case of weak non hermiticity and in the case of strong non hermiticity in the weak non hermiticity limit we show that the spectral correlations in the bulk of the spectrum display critical statistics the asymptotic linear behavior of the number variance is already approached for energy differences of the order of the eigenvalue spacing to lowest order its slope does not depend on the degree of non hermiticity close the edge the spectral correlations are similar to the hermitian case in the strong non hermiticity limit the crossover behavior from the ginibre ensemble to the poisson ensemble first appears close to the surface of the spectrum our model may be relevant for the description of the spectral correlations of an open disordered system close to an anderson transition

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

determinant bundles manifolds with boundary and surgery we define determinant bundles associated to the following data i a family of generalized dirac operators on even dimensional manifolds with boundary ii the choice of a spectral section for the family of dirac operators induced on the boundary under the assumption that the operators of the boundary family have null spaces of constant dimension we define through the notion ofb zeta function a quillen metric we also introduce the analogue of the bismut freed connection we prove that the curvature of a natural perturbation of the bismut freed connection equals the 2 form piece in the right hand side of the family index formula thus extending to manifolds with boundary results of quillen bismut and freed given a closed fibration we investigate the behaviour of the quillen metric and of the bismut freed connection under the operation of surgery along a fibering hypersurface we prove in particular additivity formulae for the curvature and for the logarithm of the holonomy

analytic torsion and holomorphic determinant bundles i bott chern forms and analytic torsion we attach secondary invariants to any acyclic complex of holomorphic hermitian vector bundles on a complex manifold these were first introduced by bott and chern bot c our new definition uses quillen s superconnections we also give an axiomatic characterization of these classes these results will be used in bgs2 and bgs3 to study the determinant of the cohomology of a holomorphic vector bundle

determinants of laplacians the determinant of the laplacian on spinor fields on a riemann surface is evaluated in terms of the value of the selberg zeta function at the middle of the critical strip a key role in deriving this relation is played by the barnes double gamma function

omega admissible theory ii deligne pairings over moduli spaces of punctured riemann surfaces in part i deligne riemann roch isometry is generalized for punctured riemann surfaces equipped with quasi hyperbolic metrics this is achieved by proving the mean value lemmas which explicitly explain how metrized deligne pairings for omega admissible metrized line bundles depend on omega in part ii we first introduce several line bundles over knudsen deligne mumford compactification of the moduli space or rather the algebraic stack of stable n pointed algebraic curves of genus g which are rather natural and include weil petersson takhtajan zograf and logarithmic mumford line bundles then we use deligne riemann roch isomorphism and its metrized version proved in part i to establish some fundamental relations among these line bundles finally we compute first chern forms of the metrized weil petersson takhtajan zograf and logarithmic mumford line bundles by using results of wolpert and takhtajan zograf and show that the so called takhtajan zograf metric on the moduli space is algebraic

analytic torsion for surfaces with cusps i compact perturbation theorem and anomaly formula we define the analytic torsion associated with a riemann surface endowed with a metric having poincare type singularities in the neighborhood of a finite number of points and a hermitian vector bundle with at most logarithmic singularities at those points coming from the metric on the negative power of the canonical line bundle twisted by the divisor of the points then we provide a relation between this analytic torsion and the ray singer analytic torsion of the compactified surface from this relation we then establish the anomaly formula which describes how the analytic torsion changes under the change of the metric on the surface and on the vector bundle

pseudodifferential operators on manifolds with foliated boundaries abstract let x be a smooth compact manifold with boundary for smooth foliations on the boundary of x admitting a resolution in terms of a fibration we construct a pseudodifferential calculus generalizing the fibred cusp calculus of mazzeo and melrose in particular we introduce certain symbols leading to a simple description of the fredholm operators inside the calculus when the leaves of the fibration resolving the foliation are compact we also obtain an index formula for fredholm perturbations of dirac type operators along the way we obtain a formula for the adiabatic limit of the eta invariant for invertible perturbations of dirac type operators a result of independent interest generalizing the well known formula of bismut and cheeger

analytic torsion for surfaces with cusps ii regularity asymptotics and curvature theorem abstract we study the quillen metric on the determinant line bundle associated with a family of complex singular curves with hyperbolic cusp singularities more precisely we fix a family of complex curves which admit at most double point singularities we endow the fibers of this family with kahler metrics which are defined away from a finite set of points a divisor on the total space of the family in the vicinity of which the metric has poincare type singularities we fix a holomorphic vector bundle over the total space of the family and endow it with a hermitian metric with at most logarithmic singularities on the divisor coming from the power of the relative canonical line bundle twisted by the divisor an important example is the family of stable surfaces with marked points endowed with constant scalar curvature metric the associated determinant line bundle is naturally endowed with the quillen metric defined using the analytic torsion from the first paper of this series generalizing the classical definition of ray singer we study the regularity of this quillen metric near the locus of singular curves the singularities turn out to be reasonable enough so that the curvature of the chern connection of the determinant line bundle endowed with the quillen norm is well defined as a current we derive the explicit formula for this current which gives a refinement of riemann roch grothendieck theorem at the level of currents this generalizes the curvature theorems of takhtajan zograf and bismut bost

transition between hermitian and non hermitian gaussian ensembles the transition between hermitian and non hermitian matrices of the gaussian unitary ensemble is revisited an expression for the kernel of the rescaled hermite polynomials is derived which expresses the sum in terms of the highest order polynomials from this christoffel darboux like formula some results are derived including an extension to the complex plane of the airy kernel

a local families index formula for overline partial operators on punctured riemann surfaces using heat kernel methods developed by vaillant a local index formula is obtained for families of overline partial operators on the teichmuller universal curve of riemann surfaces of genus g with n punctures the formula also holds on the moduli space mathcal m g n in the sense of orbifolds where it can be written in terms of mumford morita miller classes the degree two part of the formula gives the curvature of the corresponding determinant line bundle equipped with the quillen connection a result originally obtained by takhtajan and zograf

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

on the distribution of the length of the longest increasing subsequence of random permutations let sn be the group of permutations of 1 2 n if 7r e sn we say that 7 i1 7f ik is an increasing subsequence in 7r if il i2 ik and 7r ii 7r i2 7r ik let 1n r be the length of the longest increasing subsequence for example if n 5 and 7r is the permutation 5 1 3 2 4 in one line notation thus 7r 1 5 7r 2 1 then the longest increasing subsequences are 1 2 4 and 1 3 4 and n 3 equip sn with uniform distribution

a limit theorem at the edge of a non hermitian random matrix ensemble the study of the edge behaviour in the classical ensembles of gaussian hermitian matrices has led to the celebrated distributions of tracy widom here we take up a similar line of inquiry in the non hermitian setting we focus on the family of n n random matrices with all entries independent and distributed as complex gaussian of mean zero and variance 1 n this is a fundamental non hermitian ensemble for which the eigenvalue density is known using this density our main result is a limit law for the scaled spectral radius as n as a corollary we get the analogous statement for the case where the complex gaussians are replaced by quaternion gaussians

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

discrete polynuclear growth and determinantal processes we consider a discrete polynuclear growth png process and prove a functional limit theorem for its convergence to the airy process this generalizes previous results by prahofer and spohn the result enables us to express the f1 goe tracy widom distribution in terms of the airy process we also show some results and give a conjecture about the transversal fluctuations in a point to line last passage percolation problem furthermore we discuss a rather general class of measures given by products of determinants and show that these measures have determinantal correlation functions

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

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

from gumbel to tracy widom the tracy widom distribution that has been much studied in recent years can be thought of as an extreme value distribution we discuss interpolation between the classical extreme value distribution exp exp x the gumbel distribution and the tracy widom distribution there is a family of determinantal processes whose edge behaviour interpolates between a poisson process with density exp x and the airy kernel point process this process can be obtained as a scaling limit of a grand canonical version of a random matrix model introduced by moshe neuberger and shapiro we also consider the deformed gue ensemble m m 0 sqrt 2s v with m 0 diagonal with independent elements and v from gue here we do not see a transition from tracy widom to gumbel but rather a transition from tracy widom to gaussian

the analysis of elliptic families i metrics and connections on determinant bundles in this paper we construct the quillen metric on the determinant bundle associated with a family of elliptic first order differential operators we also introduce a unitary connection on and calculate its curvature our results will be applied to the case of dirac operators in a forthcoming paper

the spectral edge of some random band matrices we study the asymptotic distribution of the eigenvalues of random hermitian periodic band matrices focusing on the spectral edges the eigenvalues close to the edges converge in distribution to the airy point process if and only if the band is sufficiently wide w n 5 6 otherwise a different limiting distribution appears

discrete orthogonal polynomial ensembles and the plancherel measure we consider discrete orthogonal polynomial ensembles which are discrete analogues of the orthogonal polynomial ensembles in random matrix theory these ensembles occur in certain problems in combinatorial probability and can be thought of as probability measures on partitions the meixner ensemble is related to a two dimensional directed growth model and the charlier ensemble is related to the lengths of weakly increasing subsequences in random words the krawtchouk ensemble occurs in connection with zig zag paths in random domino tilings of the aztec diamond and also in a certain simplifled directed flrst passage percolation model we use the charlier ensemble to investigate the asymptotics of weakly increasing subsequences in random words and to prove a conjecture of tracy and widom as a limit of the meixner ensemble or the charlier ensemble we obtain the plancherel measure on partitions and using this we prove a conjecture of baik deift and johansson that under the plancherel measure the distribution of the lengths of the flrst k rows in the partition appropriately scaled converges to the asymptotic joint distribution for the k largest eigenvalues of a random matrix from the gaussian unitary ensemble in this problem a certain discrete kernel which we call the discrete bessel kernel plays an important role

the analysis of elliptic families ii dirac operators eta invariants and the holonomy theorem in this paper we specialize the results obtained in bf1 to the case of a family of dirac operators we first calculate the curvature of the unitary connection on the determinant bundle which we introduced in bf1

large scale correlations in normal non hermitian matrix ensembles we compute the large scale macroscopic correlations in ensembles of normal random matrices with a general non gaussian measure and in ensembles of general non hermitian matrices with a class of non gaussian measures in both cases the eigenvalues are complex and in the large n limit they occupy a domain in the complex plane for the case when the support of eigenvalues is a connected compact domain we compute two three and four point connected correlation functions in the first non vanishing order in 1 n in a manner that the algorithm of computing higher correlations becomes clear the correlation functions are expressed through the solution of the dirichlet boundary problem in the domain complementary to the support of eigenvalues

determinants of laplacians on surfaces of finite volume determinants of the laplace and other elliptic operators on compact manifolds have been an object of study for many years see mp rs vor up until now however the theory of determinants has not been extended to non compact situations since these typically involve a mixture of discrete and continuous spectra recent advances in this theory which are partially motivated by developments in mathematical physics have led to a connection in the compact riemann surface case between determinants of laplacians on spinors and the selberg zeta function of the underlying surface see dp kie sar vor

eigenvalues of large chiral non hermitian random matrices we study a non hermitian chiral random matrix of which the eigenvalues are complex random variables the empirical distributions and the radius of the eigenvalues are investigated the limit of the empirical distributions is a new probability distribution defined on the complex plane the graphs of the density functions are plotted the surfaces formed by the density functions are understood through their convexity and their gaussian curvatures the limit of the radius is a gumbel distribution the main observation is that the joint density function of the eigenvalues of the chiral ensemble after a transformation becomes a rotation invariant determinantal point process on the complex plane then the eigenvalues are studied by the tools developed by jiang and qi j theor probab 30 326 2017 32 353 2019 most efforts are devoted to deriving the central limit theorems for distributions defined by the bessel functions via the method of steepest descent and the estimates of the zero of a non trivial equation as the saddle point

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

eta invariants and their adiabatic limits 0 introduction 1 algebraic preliminaries 2 the finite dimensional case superconnections a odd dimensional base spaces b even dimensional base spaces appendix 1 coupling to general superconnections 3 auxiliary grassman variables and auxiliary adiabatic limits 4 the infinite dimensional case fibrations a elementary geometry of fibrations b odd dimensional base spaces c even dimensional base spaces d the case in which the dirac operator on the fiber is not invertible appendix 2 multifibrations

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

on determinants of laplacians on riemann surfaces determinants of laplacians on tensors and spinors of arbitrary weights on compact hyperbolic riemann surfaces are computed in terms of values of selberg zeta functions at half integer points

interpolation between airy and poisson statistics for unitary chiral non hermitian random matrix ensembles we consider a family of chiral non hermitian gaussian random matrices in the unitarily invariant symmetry class the eigenvalue distribution in this model is expressed in terms of laguerre polynomials in the complex plane these are orthogonal with respect to a non gaussian weight including a modified bessel function of the second kind and we give an elementary proof for this in the large n limit the eigenvalue statistics at the spectral edge close to the real axis are described by the same family of kernels interpolating between airy and poisson that was recently found by one of the authors for the elliptic ginibre ensemble we conclude that this scaling limit is universal appearing for two different non hermitian random matrix ensembles with unitary symmetry as a second result we give an equivalent form for the interpolating airy kernel in terms of a single real integral similar to representations for the asymptotic kernel in the bulk and at the hard edge of the spectrum this makes its structure as a one parameter deformation of the airy kernel more transparent