perron type theorems for functional differential equations with infinite delay in a banach space for functional differential equations in a banach space we obtain some results on the asymptotic behavior of solutions which correspond to perron type theorems for differential equations without delay or with finite delay

an algebro geometric proof of witten s conjecture we present a new proof of witten s conjecture the proof is based on the analysis of the relationship between intersection indices on moduli spaces of complex curves and hurwitz numbers enumerating ramified coverings of the 2 sphere

uniform bounds for weil petersson curvatures we find bounds for weil petersson holomorphic sectional curvature and the weil petersson curvature operator in several regimes that do not depend on the topology of the underlying surface among other results we show that the minimal weil petersson holomorphic sectional curvature of a sufficiently thick hyperbolic surface is comparable to 1 independently of the genus this provides a counterexample to some suggestions that the weil petersson metric becomes asymptotically flat as the genus g goes to infinity in some loci in teichmuller space adopting a different perspective on curvature we also show that the minimal most negative eigenvalue of the curvature operator at any point in the teichmuller space teich sg of a closed surface sg of genus g is uniformly bounded away from zero restricting to a thick part of teich sg we show that the minimal eigenvalue is uniformly bounded below by an explicit constant which does not depend on the topology of the surface but only on the given bound on injectivity radius

a perron type theorem for functional differential equations a perron type theorem about the existence of the strict lyapunov exponents of the solutions of retarded functional differential equations is established

numerical computation of minimal polynomial bases a generalized resultant approach we propose a new algorithm for the computation of a minimal polynomial basis of the left kernel of a given polynomial matrix f s the proposed method exploits the structure of the left null space of generalized wolovich or sylvester resultants to compute row polynomial vectors that form a minimal polynomial basis of left kernel of the given polynomial matrix the entire procedure can be implemented using only orthogonal transformations of constant matrices and results to a minimal basis with orthonormal coefficients

asymptotic behavior of solutions of functional difference equations for linear functional difference equations we obtain some results on the asymptotic behavior of solutions which correspond to a perron type theorem for linear ordinary difference equations we also apply our results to volterra difference equations with infinite delay

tau functions and virasoro symmetries for drinfeld sokolov hierarchies abstract for each drinfeld sokolov integrable hierarchy associated to affine kac moody algebra we obtain a uniform construction of tau function by using tau symmetric hamiltonian densities moreover we represent its virasoro symmetries as linear nonlinear actions on the tau function the relations between the tau function constructed in this paper and those defined for particular cases of drinfeld sokolov hierarchies in the literature are clarified we also show that whenever the affine kac moody algebra is simply laced or twisted the tau function of the drinfeld sokolov hierarchy coincides with the solution of the corresponding kac wakimoto hierarchy constructed from the principal vertex operator realization of the affine algebra

intersection numbers and rank one cohomological field theories in genus one we obtain a simple recursive presentation of the tautological and classes on the moduli space of curves in genus 0 and 1 in terms of boundary strata graphs we derive differential equations for the generating functions for their intersection numbers which allow us to prove a simple relationship between the genus zero and genus one potentials as an application we describe the moduli space of normalized restricted rank one cohomological field theories in genus one in coordinates which are additive under taking tensor products our results simplify and generalize those of kaufmann manin and zagier

spaces of rna secondary structures abstract we prove two topological theorems in physical chemistry namely we introduce a hybrid of transverse and tangential measures on train tracks to prove sphericity of various simplicial complexes which arise from certain idealized models of physical chemistry these complexes are at once identified with thurston s space of projective geodesic laminations on an ideal polygon and with the analogue of a compactification described elsewhere of the moduli space of a punctured riemann surface the physical structures we study are various sub collections of the set of all possible planar chemical bonds among the sites of a linear macromolecule each such collection we consider has a natural partial ordering and the geometric realizations of appropriate posets are shown to be topological spheres such a topological statement encodes a wealth of combinatorial data as we briefly discuss in fact our primary motivation here is to study secondary structures on rna this imposes the further restriction that there can be at most one base pair supported at a given site of underlying linear macromolecule and imposing this restriction leads to the class of binary macromolecules our main results here assert the sphericity of certain topological spaces of both arbitrary and binary macromolecules and it is the latter which we hope may have applications to rna our techniques are largely elaborations of elementary topological techniques from techmuller theory and the theory of train tracks

zygmund vector fields hilbert transform and fourier coefficients in shear coordinates we parametrize the space cal z of zygmund vector fields on the unit circle in terms of infinitesimal shear functions on the farey tesselation then we express the hilbert transform and the fourier coefficients of the zygmund vector fields in terms of the above parametrization by infinitesimal shear functions finally we compute the weil petersson metric on the teichm uller space of a punctured surface in terms of shears

the weil petersson and thurston symplectic forms we consider the weil petersson form on the teichmuller space mathscr t s of a surface s of genus at least 2 and we compute it in terms of the shearing coordinates for mathscr t s associated to a geodesic lamination on s in the corresponding expression the weil petersson form coincides with thurston s intersection form on the space of transverse cocycles for

on the symplectic geometry of deformations of a hyperbolic surface let r be a riemann surface in this manuscript we consider a geometry on the moduli space x r for r which we regard as the space of equivalence classes of constant curvature metrics on the underlying smooth manifold of r classically the space of flat metrics for a torus is the locally symmetric space 0 2 sl 2 r sl 2 z we shall describe a symplectic geometry for the space of hyperbolic metrics on a surface of negative euler characteristic the teichmiiller space t r a covering of the moduli space is a complex kihler manifold a kuhler metric for t r defined in terms of the petersson product for automorphic forms was introduced by weil 1 the weil petersson metric is invariant under the covering transformations and so projects to the moduli space x r the metric provides a link between the function theory of r and the geometry of x r in the fenchel nielsen manuscript 8 a deformation based on an amalgamation construction for fuchsian groups is introduced the deformation is defined geometrically by cutting the surface along a simple closed geodesic rotating one side of the cut relative to the other and attaching the sides in their new position the hyperbolic metric in the complement of the cut extends to a hyperbolic metric on the new surface choose a free homotopy class a on the surface r then for each marked surface r realize a by the closed geodesic ar the fenchel nielsen deformations for the athen define a 1 parameter group of diffeomorphisms of t r whose infinitesimal generator by definition is the fenchel nielsen vector field ta in 21 the fenchel nielsen deformation was described in terms of quasiconformal mappings and an investigation of the vector fields t was begun the fenchel nielsen vector fields were found to be related to the geodesic length functions 1 introduced by fricke klein to provide coordinates for t r

the decorated teichmuller space of punctured surfaces a principal 5 bundle over the usual teichmuller space of ans times punctured surface is introduced the bundle is mapping class group equivariant and admits an invariant foliation several coordinatizations of the total space of the bundle are developed there is furthermore a natural cell decomposition of the bundle finally we compute the coordinate action of the mapping class group on the total space the total space is found to have a rich equivariant geometric structure we sketch some connections with arithmetic groups diophantine approximations and certain problems in plane euclidean geometry furthermore these investigations lead to an explicit scheme of integration over the moduli spaces and to the construction of a universal teichmuller space which we hope will provide a formalism for understanding some connections between the teichmuller theory the kp hierarchy and the virasoro algebra these latter applications are pursued elsewhere

lambert w solves the noncommutative varphi 4 φ 4 model the closed dyson schwinger equation for the 2 point function of the noncommutative lambda phi 4 2 model is rearranged into the boundary value problem for a sectionally holomorphic function in two variables we prove an exact formula for a solution in terms of lambert s w function this solution is holomorphic in lambda inside a domain which contains 1 log 4 infty our methods include the hilbert transform perturbation series and lagrange burmann resummation

simple geodesics and a series constant over teichmuller space we investigate the birman series set in a neighborhood of a cusp on a punctured surface showing that it is homeomorphic to a cantor set union countably many isolated points cross a line the local topology of the cantor set is shown to be related in a simple way to the global behavior of simple geodesics from this we deduce that a certain series is constant across the teichmuller space

a picture of moduli space in this paper we construct an explicit fundamental domain for the action of the teichm uller modular group on the teichm uller space of closed surfaces of genus g 2 fundamental domains for the modular groups of surfaces of low signature have been obtained by keen k rosenberger r and others including this author m1 m2 fundamental domains for modular groups of hyperbolic surfaces of nite area with punctures have been obtained by penner p his technique makes essential use of the punctures while our technique seems to work only for closed surfaces there is also a related fundamental domain for closed surfaces of genus 2 obtained by semmler s his method uses dividing geodesics while we use non dividing geodesics throughout this paper the word geodesic refers to a closed geodesic which we will usually regard as being unoriented the author is indebted to the referee for pointing out that there is some related work due to thurston unpublished in which he attempts to give a natural cell decomposition of the teichm uller space of closed surfaces using sequences of disjoint shortest geodesics there is also some related work by n a at anen and penner n p attempting to use dirichlet domains for the same purpose while we do not know the topology of our fundamental domain in particular we do not know if it is a cell one can regard this work as a step in the direction of nding a cell decomposition of teichm uller space given a discrete group g of isometries on a manifold x the classical literature regards a fundamental domain d as an object one can integrate over in place of x g in particular d satis es the following three properties i no two points of the interior of d are equivalent under g ii every point of x is equivalent under g to some point of the closure of d iii the boundary of d has measure zero

all loop topological string amplitudes from chern simons theory we demonstrate the equivalence of all loop closed topological string amplitudes on toric local calabi yau threefolds with computations of certain knot invariants for chern simons theory we use this equivalence to compute the topological string amplitudes in certain cases to very high degree and to all genera in particular we explicitly compute the topological string amplitudes for open image in new window 2 up to degree 12 and open image in new window 1 open image in new window 1 up to total degree 10 to all genera this also leads to certain novel large n dualities in the context of ordinary superstrings involving duals of type ii superstrings on local calabi yau three folds without any fluxes

topological recursion in enumerative geometry and random matrices we review the method of symplectic invariants recently introduced to solve matrix models loop equations in the so called topological expansion and further extended beyond the context of matrix models for any given spectral curve one defines a sequence of differential forms and a sequence of complex numbers fg called symplectic invariants we recall the definition of fg s and we explain their main properties in particular symplectic invariance integrability modularity as well as their limits and their deformations then we give several examples of applications in particular matrix models enumeration of discrete surfaces maps algebraic geometry and topological strings and non intersecting brownian motions

a limit boundary value problem for a nonlinear difference equation we give an asymptotic description of the monotone increasing solutions of a limit boundary value problem for a class of nonlinear difference equations with continuous arguments and rationally non related shifts

ergodic theory on moduli spaces let m be a compact surface with x m 0 and let g be a compact lie group whose levi factor is a product of groups locally isomorphic to su 2 for example su 2 itself then the mapping class group rm of m acts on the moduli space x m of flat g bundles over m possibly twisted by a fixed central element of g when m is closed then fm preserves a symplectic structure on x m which has finite total volume on m more generally the subspace of x m corresponding to flat bundles with fixed behavior over am carries a rm invariant symplectic structure the main result is that rm acts ergodically on x m with respect to the measure induced by the symplectic structure

higher weil petersson volumes of moduli spaces of stable n pointed curves moduli spaces of compact stablen pointed curves carry a hierarchy of cohomology classes of top dimension which generalize the weil petersson volume forms and constitute a version of mumford classes we give various new formulas for the integrals of these forms and their generating functions

triangulated riemann surfaces with boundary and the weil petersson poisson structure given a hyperbolic surface with geodesic boundary s the lengths of a maximal system of disjoint simple geodesic arcs on s that start and end at s perpendicularly are coordinates on the teichmuller space t s we express the weil petersson poisson structure of t s in this system of coordinates and we prove that it limits pointwise to the piecewise linear poisson structure defined by kontsevich on the arc complex of s at the same time we obtain a formula for the first order variation of the distance between two closed geodesics under fenchel nielsen deformation

new results of intersection numbers on moduli spaces of curves we present a series of results we obtained recently about the intersection numbers of tautological classes on moduli spaces of curves including a simple formula of the n point functions for witten s classes an effective recursion formula to compute higher weil petersson volumes several new recursion formulae of intersection numbers and our proof of a conjecture of itzykson and zuber concerning denominators of intersection numbers we also present virasoro and kdv properties of generating functions of general mixed and intersections

intersection theory on the moduli space of curves and the matrix airy function we show that two natural approaches to quantum gravity coincide this identity is nontrivial and relies on the equivalence of each approach to kdv equations we also investigate related mathematical problems

growth of weil petersson volumes and random hyperbolic surfaces of large genus in this paper we investigate the geometric properties of random hyperbolic surfaces of large genus we describe the relationship between the behavior of lengths of simple closed geodesics on a hyperbolic surface and properties of the moduli space of such surfaces first we study the asymptotic behavior of weil petersson volume vg n of the moduli spaces of hyperbolic surfaces of genus g with n punctures as g then we discuss basic geometric properties of a random hyperbolic surface of genus g with respect to the weil petersson measure as g

traveling waves for a nonlocal anisotropic dispersal equation with monostable nonlinearity abstract this paper is concerned with traveling wave solutions of the equation u t j u u f u on r 0 where the dispersion kernel j is nonnegative and the nonlinearity f is monostable type we show that there exists c r such that for any c c there is a nonincreasing traveling wave solution with 1 and lim e 1 where 1 c is the smallest positive solution to c r j z e z d z 1 f 0 furthermore the existence of traveling wave solutions with c c is also established for c 0 we further prove that the traveling wave solution is unique up to translation and is globally asymptotically stable in certain sense

the homology of the mapping class group 1 2 h btg a h mg q in this paper we will show that m g btg and b i l sg get more and more complicated as the genus g tends to infinity more precisely we will prove theorem 1 1 let q z2 z4 z6 denote the polynomial algebra of generators z2n in dimension in n 1 2 3 there are classes y2 y4 y2n with y2n in the 2nth cohomology group h 2n b i sg z such that the homomorphism of algebras sending z2n toy2n

how covariant closed string theory solves a minimal area problem for a given genusg riemann surface withn 0 punctures n 3 forg 0 we consider the problem of finding the metric of minimal area under the condition that the length of any nontrivial closed curve be greater or equal to 2 the minimal area metrics are found for the case of all punctured genus zero surfaces and for many of the higher genus surfaces both with and without punctures these metrics are induced by jenkins strebel quadratic differentials they arise from the string diagrams corresponding to restricted feynman graphs of a closed string field theory action containing classical and quantum restricted polyhedra

weil petersson volumes and intersection theory on the moduli space of curves in this paper we establish a relationship between the weil petersson volume vgin b of the moduli space mg n b of hyperbolic riemann surfaces with geodesic boundary components of lengths b bn and the intersection numbers of tauto logical classes on the moduli space mg n of stable curves as a result by using the recursive formula for vg n b obtained in 22 we derive a new proof of the virasoro constraints for a point this result is equivalent to the witten kontsevich formula 14

the symplectic nature of fundamental groups of surfaces soit 7b g la categorie des groupes de lie g avec une forme bilineaire symetrique non singuliere b sur l algebre de lie soit le groupe fondamental d une surface orientee close de genre 1 et soit 7b s la categorie des out espaces symplectiques alors la correspondance g b hom g g b definit un foncteur 7b g 7b s ou b est la structure symplectique sur hom g g definie par b

volume optimization normal surfaces and thurston s equation on triangulated 3 manifolds we propose a finite dimensional variational principle on triangulated 3 manifolds so that its critical points are related to solutions to thurston s gluing equation and haken s normal surface equation the action functional is the volume

correlation functions of the kdv hierarchy and applications to intersection numbers over m g n we derive an explicit generating function of correlations functions of an arbitrary tau function of the kdv hierarchy in particular applications our formulation gives closed formulae of a new type for the generating series of intersection numbers of classes as well as of mixed and classes in full genera

perturbative series and the moduli space of riemann surfaces on utilise des techniques de la theorie quantique des champs pour calculer des quantites reliees aux groupes de symetrie de paires f g ou f est une surface et g une epine de f

matrix models and a proof of the open analog of witten s conjecture in a recent work r pandharipande j p solomon and the second author have initiated a study of the intersection theory on the moduli space of riemann surfaces with boundary they conjectured that the generating series of the intersection numbers satisfies the open kdv equations in this paper we prove this conjecture our proof goes through a matrix model and is based on a kontsevich type combinatorial formula for the intersection numbers that was found by the second author

block minimal bases l ifications of matrix polynomials abstract the standard way of solving a polynomial eigenvalue problem associated with a matrix polynomial starts by embedding the matrix coefficients of the polynomial into a matrix pencil known as a strong linearization this process transforms the problem into an equivalent generalized eigenvalue problem however there are some situations in which is more convenient to replace linearizations by other low degree matrix polynomials this has motivated the idea of a strong l ification of a matrix polynomial which is a matrix polynomial of degree at most l having the same finite and infinite elementary divisors and the same numbers of left and right minimal indices as the original matrix polynomial we present in this work a novel method for constructing strong l ifications of matrix polynomials of size m n and grade d when l d and l divides nd or md this method is based on a family called strong block minimal bases matrix polynomials and relies heavily on properties of dual minimal bases we show how strong block minimal bases l ifications can be constructed from the coefficients of a given matrix polynomial p we also show that these l ifications satisfy many desirable properties for numerical applications they are strong l ifications regardless of whether p is regular or singular the minimal indices of the l ifications are related to those of p via constant uniform shifts and eigenvectors and minimal bases of p can be recovered from those of any of the strong block minimal bases l ifications in the special case where l divides d we introduce a subfamily of strong block minimal bases matrix polynomials named block kronecker matrix polynomials which is shown to be a fruitful source of companion l ifications

random matrix singularities and open close intersection numbers the s point correlation function of a gaussian hermitian random matrix model with an external source tuned to generate a multi critical singularity provides the intersection numbers of the moduli space for the p th spin curves through a duality for one marked point the intersection numbers are expressed to all orders in the genus by bessel functions the matrix models for the lie algebras of and provide the intersection numbers of non orientable surfaces the kontsevich penner model and the higher p th airy matrix model with a logarithmic potential are investigated for the open intersection numbers which describe the topological invariants of non orientable surfaces with boundaries string equations for the open closed riemann surface are derived from the structure of the s point correlation functions the gromov witten invariants of the model are evaluated for one marked point as an application of the present method

masur veech volumes and intersection theory on moduli spaces of abelian differentials we show that the masur veech volumes and area siegel veech constants can be obtained using intersection theory on strata of abelian differentials with prescribed orders of zeros as applications we evaluate their large genus limits and compute the saddle connection siegel veech constants for all strata we also show that the same results hold for the spin and hyperelliptic components of the strata

block kronecker linearizations of matrix polynomials and their backward errors we introduce a new family of strong linearizations of matrix polynomials which we call block kronecker pencils and perform a backward stability analysis of complete polynomial eigenproblems these problems are solved by applying any backward stable algorithm to a block kronecker pencil such as the staircase algorithm for singular pencils or the qz algorithm for regular pencils this analysis allows us to identify those block kronecker pencils that yield a computed complete eigenstructure which is exactly that of a slightly perturbed matrix polynomial these favorable pencils include the famous fiedler linearizations which are just a very particular case of block kronecker pencils thus our analysis offers the first proof available in the literature of global backward stability for fiedler pencils in addition the theory developed for block kronecker pencils is much simpler than the theory available for fiedler pencils especially in the case of rectangular matrix polynomials the global backward error analysis in this work presents for the first time the following key properties it is a rigorous analysis valid for finite perturbations i e it is not a first order analysis it provides precise bounds it is valid simultaneously for a large class of linearizations and it establishes a framework that may be generalized to other classes of linearizations these features are related to the fact that block kronecker pencils are a particular case of the new family of strong block minimal bases pencils which include certain perturbations of block kronecker pencils this allows us to extend the results in this paper to other contexts

matrix integration and combinatorics of modular groups integration over gaussian matrix ensembles was used to obtain the virtual euler characteristics of mapping class groups we present some simplifications in the combinatorial part of the calculation

an explicit upper bound for weil petersson volumes of the moduli spaces of punctured riemann surfaces an explicit upper bound for the weil petersson volumes of punctured riemann surfaces is obtained using penner s combinatorial integration scheme from 4 it is shown that for a fixed number of punctures n and for genus g increasing

gromov witten theory of orbicurves the space of tri polynomials and symplectic field theory of seifert fibrations we compute with symplectic field theory sft techniques the gromov witten theory of mathbb p 1 alpha 1 ldots alpha a i e the complex projective line with a orbifold points a natural subclass of these orbifolds the ones with polynomial quantum cohomology gives rise to a family of polynomial frobenius manifolds and integrable systems of hamiltonian pdes which extend the dispersionless bigraded toda hierarchy carlet the extended bigraded toda hierarchy arxiv preprint arxiv math ph 0604024 we then define a frobenius structure on the spaces of polynomials in three complex variables of the form f x y z xyz p1 x p2 y p3 z which contains as special cases the ones constructed on the space of laurent polynomials dubrovin geometry of 2d topologica field theories integrable systems and quantum groups springer lecture notes in mathematics 1620 120 348 1996 milanov and tseng the space of laurent polynomials mathbb p 1 orbifolds and integrable hierarchies preprint arxiv math 0607012v3 math ag we prove a mirror theorem stating that these frobenius structures are isomorphic to the ones found before for polynomial mathbb p 1 orbifolds finally we link rational sft of seifert fibrations over mathbb p 1 a b c with orbifold gromov witten invariants of the base extending a known result bourgeois a morse bott approach to contact homology ph d dissertation stanford university 2002 valid in the smooth case

edge contraction on dual ribbon graphs and 2d tqft abstract we present a new set of axioms for 2d tqft formulated on the category of cell graphs with edge contraction operations as morphisms we construct a functor from this category to the endofunctor category consisting of frobenius algebras edge contraction operations correspond to natural transformations of endofunctors which are compatible with the frobenius algebra structure given a frobenius algebra a every cell graph determines an element of the symmetric tensor algebra defined over the dual space a we show that the edge contraction axioms make this assignment depending only on the topological type of the cell graph but not on the graph itself thus the functor generates the tqft corresponding to a

necessary and sufficient condition for oscillation of a neutral differential system with several delays abstract in this paper we prove that a general neutral differential system with several delays has a nonoscillatory solution if and only if its characteristic equations has a real root

polynomial zigzag matrices dual minimal bases and the realization of completely singular polynomials minimal bases of rational vector spaces are a well known and important tool in systems theory if minimal bases for two subspaces of rational n space are displayed as the rows of polynomial matrices z 1 lambda k times n and z 2 lambda m times n respectively then z 1 and z 2 are said to be dual minimal bases if the subspaces have complementary dimension i e k m n and z 1 lambda z 2 t lambda 0 in other words each z j lambda provides a minimal basis for the nullspace of the other it has long been known that for any dual minimal bases z 1 lambda and z 2 lambda the row degree sums of z 1 and z 2 are the same in this paper we show that this is the only constraint on the row degrees thus characterizing the possible row degrees of dual minimal bases the proof is constructive making extensive use of a new class of sparse structured polynomial matrices that we have baptized zigzag matrices another application of these polynomial zigzag matrices is the constructive solution of the following inverse problem for minimal indices given a list of left and right minimal indices and a desired degree d does there exist a completely singular matrix polynomial i e a matrix polynomial with no elementary divisors whatsoever of degree d having exactly the prescribed minimal indices we show that such a matrix polynomial exists if and only if d divides the sum of the minimal indices the constructed realization is simple and explicitly displays the desired minimal indices in a fashion analogous to the classical kronecker canonical form of singular pencils

robustness and perturbations of minimal bases ii the case with given row degrees abstract this paper studies generic and perturbation properties inside the linear space of m m n polynomial matrices whose rows have degrees bounded by a given list d 1 d m of natural numbers which in the particular case d 1 d m d is just the set of m m n polynomial matrices with degree at most d thus the results in this paper extend to a much more general setting the results recently obtained in 29 only for polynomial matrices with degree at most d surprisingly most of the properties proved in 29 as well as their proofs remain to a large extent unchanged in this general setting of row degrees bounded by a list that can be arbitrarily inhomogeneous provided the well known sylvester matrices of polynomial matrices are replaced by the new trimmed sylvester matrices introduced in this paper the following results are presented among many others in this work 1 generically the polynomial matrices in the considered set are minimal bases with their row degrees exactly equal to d 1 d m and with right minimal indices differing at most by one and having a sum equal to i 1 m d i and 2 under perturbations these generic minimal bases are robust and their dual minimal bases can be chosen to vary smoothly

the weil petersson symplectic structure at thurston s boundary the weil petersson kahler structure on the teichmuller space f of a punctured surface is shown to extend in an appropriate sense to thurston s symplectic structure on the space m f 0 of measured foliations of compact support on the surface we introduce a space m f 0 of decorated measured foliations whose relationship to m f 0 is analogous to the relationship between the decorated teichmuller space f and f m f 0 is parametrized by a vector space and there is a naturl piecewise linear embedding of m f 0 in m f 0 which pulls back a global differential form to thurston s symplectic form we exhibit a homeomorphism between f and m f 0 which preseres the natural two forms on these spaces following thurston we finally consider the space y of all suitable classes of metrics of constant gaussian curvature on the surface form a natural completion y of y and identify y y with m f 0

topological recursion for the poincare polynomial of the combinatorial moduli space of curves abstract we show that the poincare polynomial associated with the orbifold cell decomposition of the moduli space of smooth algebraic curves with distinct marked points satisfies a topological recursion formula of the eynard orantin type the recursion uniquely determines the poincare polynomials from the initial data our key discovery is that the poincare polynomial is the laplace transform of the number of grothendieck s dessins d enfants

robustness and perturbations of minimal bases abstract polynomial minimal bases of rational vector subspaces are a classical concept that plays an important role in control theory linear systems theory and coding theory it is a common practice to arrange the vectors of any minimal basis as the rows of a polynomial matrix and to call such matrix simply a minimal basis very recently minimal bases as well as the closely related pairs of dual minimal bases have been applied to a number of problems that include the solution of general inverse eigenstructure problems for polynomial matrices the development of new classes of linearizations and l ifications of polynomial matrices and backward error analyses of complete polynomial eigenstructure problems solved via a wide class of strong linearizations these new applications have revealed that although the algebraic properties of minimal bases are rather well understood their robustness and the behavior of the corresponding dual minimal bases under perturbations have not yet been explored in the literature as far as we know therefore the main purpose of this paper is to study in detail when a minimal basis m is robust under perturbations i e when all the polynomial matrices in a neighborhood of m are minimal bases and in this case how perturbations of m change its dual minimal bases in order to study such problems a new characterization of whether or not a polynomial matrix is a minimal basis in terms of a finite number of rank conditions is introduced and based on it we prove that polynomial matrices are generically minimal bases with very specific properties in addition some applications of the results of this paper are discussed

spectral equivalence of matrix polynomials and the index sum theorem abstract the concept of linearization is fundamental for theory applications and spectral computations related to matrix polynomials however recent research on several important classes of structured matrix polynomials arising in applications has revealed that the strategy of using linearizations to develop structure preserving numerical algorithms that compute the eigenvalues of structured matrix polynomials can be too restrictive because some structured polynomials do not have any linearization with the same structure this phenomenon strongly suggests that linearizations should sometimes be replaced by other low degree matrix polynomials in applied numerical computations motivated by this fact we introduce equivalence relations that allow the possibility of matrix polynomials with coefficients in an arbitrary field to be equivalent with the same spectral structure but have different sizes and degrees these equivalence relations are directly modeled on the notion of linearization and consequently inherit the simplicity applicability and most relevant properties of linearizations simultaneously though they are much more flexible in the possible degrees of equivalent polynomials this flexibility allows us to define in a unified way the notions of quadratification and l ification to introduce the concept of companion form of arbitrary degree and to provide concrete and simple examples of these notions that generalize in a natural and smooth way the classical first and second frobenius companion forms the properties of l ifications are studied in depth in this process a fundamental result on matrix polynomials the index sum theorem is recovered and extended to arbitrary fields although this result is known in the systems theory literature for real matrix polynomials it has remained unnoticed by many researchers it establishes that the sum of the finite and infinite partial multiplicities together with the left and right minimal indices of any matrix polynomial is equal to the rank times the degree of the polynomial the index sum theorem turns out to be a key tool for obtaining a number of significant results on the possible sizes and degrees of l ifications and companion forms on the minimal index preservation properties of companion forms of arbitrary degree as well as on obstructions to the existence of structured companion forms for structured matrix polynomials of even degree this paper presents many new results blended together with results already known in the literature but extended here to the most general setting of matrix polynomials of arbitrary sizes and degrees over arbitrary fields therefore we have written the paper in an expository and self contained style that makes it accessible to a wide variety of readers

the euler characteristic of the moduli space of curves let fg 1 g 1 be the mapping class group consisting of all isotopy classes of base point and orientation preserving homeomorphisms of a closed oriented surface f of genus g let 1 be its euler characteristic in the sense of wall that is z f i fgl f l z e f where f is any torsion free subgroup of finite index in f 1 and e is a contractible space on which f acts freely and properly discontinuously an example of such a space is the teichmiiller space 1 and g f can be interpreted as the orbifold euler characteristic of f i join the moduli space of curves of genus g with base point the purpose of this paper is to prove the following formula for f i

constructing strong l ifications from dual minimal bases abstract we provide an algorithm for constructing strong l ifications of a given matrix polynomial p of degree d and size m n using only the coefficients of the polynomial and the solution of linear systems of equations a strong l ification of p is a matrix polynomial of degree l having the same finite and infinite elementary divisors and the same numbers of left and right minimal indices as the original matrix polynomial p all explicit constructions of strong l ifications introduced so far in the literature have been limited to the case where l divides d though recent results on the inverse eigenstructure problem for matrix polynomials show that more general constructions are possible based on recent developments on dual polynomial minimal bases we present a general construction of strong l ifications for wider choices of the degree l namely when l divides one of nd or md and d l in the case where l divides nd respectively md the strong l ifications we construct allow us to easily recover the minimal indices of p in particular we show that they preserve the left resp right minimal indices of p and the right resp left minimal indices of the l ification are the ones of p increased by d l each moreover in the particular case l divides d the new method provides a companion l ification that resembles very much the companion l ifications already known in the literature

the witten equation mirror symmetry and quantum singularity theory for any nondegenerate quasi homogeneous hypersurface singularity we describe a family of moduli spaces a virtual cycle and a corresponding cohomological eld theory associated to the singularity this theory is analogous to gromov witten theory and generalizes the theory of r spin curves which corresponds to the simple singularity ar 1 we also resolve two outstanding conjectures of witten the rst conjecture is that ade singularities are self dual and the second conjecture is that the total potential functions of ade singularities satisfy corresponding ade integrable hierarchies other cases of integrable hierarchies are also discussed

masur veech volumes frequencies of simple closed geodesics and intersection numbers of moduli spaces of curves we express the masur veech volume and the area siegel veech constant of the moduli space of meromorphic quadratic differential with simple poles as polynomials in the intersection numbers of psi classes supported on the boundary cycles of the deligne mumford compactification of the moduli space of curves our formulae are derived from lattice point count involving the kontsevich volume polynomials that also appear in mirzakhani s recursion for the weil petersson volumes of the moduli space of bordered hyperbolic riemann surfaces a similar formula for the masur veech volume though without explicit evaluation was obtained earlier by mirzakhani through completely different approach we prove further result up to an explicit normalization factor depending only on the genus and on the number of cusps the density of the orbit of any simple closed multicurve computed by mirzakhani coincides with the density of square tiled surfaces having horizontal cylinder decomposition associated to the simple closed multicurve we study the resulting densities in more detail in the special case when there are no cusps in particular we compute explicitly the asymptotic frequencies of separating and non separating simple closed geodesics on a closed hyperbolic surface of genus g for all small genera g and we show that in large genera the separating closed geodesics are exponentially less frequent we conclude with detailed conjectural description of combinatorial geometry of a random simple closed multicurve on a surface of large genus and of a random square tiled surface of large genus this description is conditional to the conjectural asymptotic formula for the masur veech volume in large genera and to the conjectural uniform asymptotic formula for certain sums of intersection numbers of psi classes in large genera

linearizations of matrix polynomials in newton bases we discuss matrix polynomials expressed in a newton basis and the associated polynomial eigenvalue problems properties of the generalized ansatz spaces associated with such polynomials are proved directly by utilizing a novel representation of pencils in these spaces also we show how the family of fiedler pencils can be adapted to matrix polynomials expressed in a newton basis these new newton fiedler pencils are shown to be strong linearizations and some computational aspects related to them are discussed

grafting hyperbolic metrics and eisenstein series the family hyperbolic metric for the plumbing variety zw t and the non holomorphic eisenstein series e zeta 2 are combined to provide an explicit expansion for the hyperbolic metrics for degenerating families of riemann surfaces applications include an asymptotic expansion for the weil petersson metric and a local form of symplectic reduction

on certain infinite dimensional aspects of arakelov intersection theory letk y x be an embedding of compact complex manifolds bismut and lebeau have calculated the quillen norm of the canonical isomorphism identifying the determinant of the cohomology of a holomorphic vector bundle overy and the determinant of the cohomology of a resolution by a complex of holomorhic vector bundles overx the purpose of this paper is to show that the formula of bismut lebeau can be viewed as an equivariant intersection formula over the loop space of the considered manifolds in the presence of an infinite dimensional excess normal bundle this excess normal bundle is responsible for the appearance of the additive genusr of gillet and soule in the formula of bismut and lebeau