from kontsevich witten to linear hodge integrals via virasoro operators we give a proof of alexandrov s conjecture on a formula connecting the kontsevich witten and hodge tau functions using only the virasoro operators this formula has been confirmed up to an unknown constant factor in this paper we show that this factor is indeed equal to one by investigating series expansions for the lambert w function on different points we give a proof of alexandrov s conjecture on a formula connecting the kontsevich witten and hodge tau functions using only the virasoro operators this formula has been confirmed up to an unknown constant factor in this paper we show that this factor is indeed equal to one by investigating series expansions for the lambert w function on different points

nonlinear mixed h 2 h control for robust tracking design of robotic systems a mixed h 2 h control design is proposed in this study for tracking of rigid 2 robotic systems under parameter perturbations and external disturbances the design objective is that under a prescribed disturbance attenuation level h attenuation constraint the optimal least square error h 2 optimal tracking 2 must be achieved additionally an explicit and global solution to this nonlinear time varying mixed h 2 h control problem is presented by combining nonlinear minimax nash game theory and lq optimal control techniques moreover by virtue of the skew symmetric property of robotic systems and adequate choice of state variable transformation only two linear algebraic instead of nonlinear timevarying riccati like equations are required to construct the proposed mixed h 2 h tracking control law furthermore these two linear algebraic equations 2 can be solved with a very simple method so that a closed form nonlinear mixed h 2 h tracking controller can be constructed finally extensive

givental formula in terms of virasoro operators we present a conjecture that the universal enveloping algebra of differential operators tk over c coincides in the origin with the universal enveloping algebra of the borel subalgebra of virasoro generators from the kontsevich model thus we can decompose any pseudo differential operator to a combination of the virasoro operators using this decomposition we present the right hand side of the givental formula math ag 0008067 as a constant part of the differential operator we introduce in the case of cp1 studied in the paper by song and song hep th 0103254 the left hand side of the givental formula is a unit which imposes certain constraints on this differential operator we explicitly check that these constraints are correct up to o q4 we also propose a conjecture of factorization modulo hirota equation of the differential operator introduced and check this conjecture with the same accuracy

kp hierarchy for hodge integrals abstract starting from the elsv formula we derive a number of new equations on the generating functions for hodge integrals over the moduli space of complex curves this gives a new simple and uniform treatment of certain known results on hodge integrals like witten s conjecture virasoro constrains faber s g conjecture etc among other results we show that a properly arranged generating function for hodge integrals satisfies the equations of the kp hierarchy

compactified picard stacks over the moduli stack of stable curves with marked points abstract in this paper we give a construction of algebraic artin stacks endowed with a modular map onto the moduli stack of stable curves of genus g with n marked points the stacks we construct are smooth irreducible and have dimension 4 g 3 n yielding a geometrically meaningful compactification of the universal picard stack parametrizing n pointed smooth curves together with a line bundle

tau functions as widom constants we define a tau function for a generic riemann hilbert problem posed on a union of non intersecting smooth closed curves with jump matrices analytic in their neighborhood the tau function depends on parameters of the jumps and is expressed as the fredholm determinant of an integral operator with block integrable kernel constructed in terms of elementary parametrices its logarithmic derivatives with respect to parameters are given by contour integrals involving these parametrices and the solution of the riemann hilbert problem in the case of one circle the tau function coincides with widom s determinant arising in the asymptotics of block toeplitz matrices our construction gives the jimbo miwa ueno tau function for riemann hilbert problems of isomonodromic origin painleve vi v iii garnier system etc and the sato segal wilson tau function for integrable hierarchies such as gelfand dickey and drinfeld sokolov

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

isomonodromic deformations of a rational differential system and reconstruction with the topological recursion the sl2 case in this paper we show that it is always possible to deform a differential equation x x l x x with l x sl2 c x by introducing a small formal parameter in such a way that it satisfies the topological type properties of bergere borot and eynard annales henri poincare 16 12 2713 2782 2015 this is obtained by including the former differential equation in an isomonodromic system and using some homogeneity conditions to introduce the topological recursion is then proved to provide a formal series expansion of the corresponding tau function whose coefficients can thus be expressed in terms of intersections of tautological classes in the deligne mumford compactification of the moduli space of surfaces we present a few examples including any fuchsian system of sl2 c x as well as some elements of painleve hierarchies

enumerative geometry tau functions and heisenberg virasoro algebra in this paper we establish relations between three enumerative geometry tau functions namely the kontsevich witten hurwitz and hodge tau functions the relations allow us to describe the tau functions in terms of matrix integrals virasoro constraints and kac schwarz operators all constructed operators belong to the algebra or group of symmetries of the kp hierarchy

on certain properties for some classes of starlike functions abstract let f z z a2z2 be analytic in the unit disk u z z 1 by using the method of differential subordinations we give a criterion for a function f z to be in a certain class l a b of starlike functions for functions f z l a b a subordination relation for f z z is also given

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

sufficient conditions for caratheodory functions the purpose of the present paper is to derive some sufficient conditions for caratheodory functions in the open unit disk by using miller and mocanu s lemma several special cases are considered as the corollaries of main results

brief trading robustness with optimality in nonlinear control the mixed h 2 h control problem is studied for a class of nonlinear systems a procedure providing a sub optimal solution to this problem is discussed and analyzed in detail the sub optimal state feedback control law can be obtained solving two hamilton jacobi equations and a simple one parameter static minimization problem a simple one dimensional example is discussed in detail and applications to the optimal robust stabilization of the angular velocities of a rigid body are included

connecting the kontsevich witten and hodge tau functions by the widehat gl infty operators in this paper we present an explicit formula that connects the kontsevich witten tau function and the hodge tau function by differential operators belonging to the widehat gl infty group indeed we show that the two tau functions can be connected using virasoro operators this proves a conjecture posted by alexandrov in from hurwitz numbers to kontsevich witten tau function a connection by virasoro operators letters in mathematical physics doi 10 1007 s11005 013 0655 0 2014

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

sufficient conditions for caratheodory functions by using the method of differential subordinations we derive certain sufficient conditions for caratheodory functions our results extend and improve some results due to 1 4

mixed h sub 2 h sub spl infin fuzzy output feedback control design for nonlinear dynamic systems an lmi approach this study introduces a mixed h sub 2 h sub spl infin fuzzy output feedback control design method for nonlinear systems with guaranteed control performance first the takagi sugeno fuzzy model is employed to approximate a nonlinear system next based on the fuzzy model a fuzzy observer based mixed h sub 2 h sub spl infin controller is developed to achieve the suboptimal h sub 2 control performance with a desired h sub spl infin disturbance rejection constraint a robust stabilization technique is also proposed to override the effect of approximation error in the fuzzy approximation procedure by the proposed decoupling technique and two stage procedure the outcome of the fuzzy observer based mixed h sub 2 h sub spl infin control problem is parametrized in terms of the two eigenvalue problems evps one for observer and the other for controller the evps can be solved very efficiently using the linear matrix inequality lmi optimization techniques a simulation example is given to illustrate the design procedures and performances of the proposed method

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

on quantum cohomology rings for hypersurfaces in cp n 1 using the torus action method we construct a one variable polynomial representation of quantum cohomology ring for degree k hypersurface in cpn 1 the results interpolate the well known result of cpn 2 model and the one of calabi yau hypersuface in cpn 1 we find in the k n 2 case the principal relation of this ring has a very simple form compatible with toric compactification of moduli space of holomorphic maps from cp1 to cpn 1

towards the geometry of double hurwitz numbers abstract double hurwitz numbers count branched covers of cp 1 with fixed branch points with simple branching required over all but two points 0 and and the branching over 0 and specified by partitions of the degree with m and n parts respectively single hurwitz numbers or more usually hurwitz numbers have a rich structure explored by many authors in fields as diverse as algebraic geometry symplectic geometry combinatorics representation theory and mathematical physics the remarkable elsv formula relates single hurwitz numbers to intersection theory on the moduli space of curves this connection has led to many consequences including okounkov and pandharipande s proof of witten s conjecture in this paper we determine the structure of double hurwitz numbers using techniques from geometry algebra and representation theory our motivation is geometric we give evidence that double hurwitz numbers are top intersections on a moduli space of curves with a line bundle a universal picard variety in particular we prove a piecewise polynomiality result analogous to that implied by the elsv formula in the case m 1 complete branching over one point and n is arbitrary we conjecture an elsv type formula and show it to be true in genus 0 and 1 the corresponding witten type correlation function has a richer structure than that for single hurwitz numbers and we show that it satisfies many geometric properties such as the string and dilaton equations and an itzykson zuber style genus expansion ansatz we give a symmetric function description of the double hurwitz generating series which leads to explicit formulae for double hurwitz numbers with given m and n as a function of genus in the case where m is fixed but not necessarily 1 we prove a topological recursion on the corresponding generating series which leads to closed form expressions for double hurwitz numbers and an analogue of the goulden jackson polynomiality conjecture an early conjectural variant of the elsv formula in a later paper faber s intersection number conjecture and genus 0 double hurwitz numbers 2005 in preparation the formulae in genus 0 will be shown to be equivalent to the formulae for top intersections on the moduli space of smooth curves m g for example three formulae we give there will imply faber s intersection number conjecture in moduli of curves and abelian varieties aspects of mathematics vol e33 vieweg braunschweig 1999 pp 109 129 in arbitrary genus with up to three points

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

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

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

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

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

analyticity of the total ancestor potential in singularity theory abstract k saito s theory of primitive forms gives a natural semi simple frobenius manifold structure on the space of miniversal deformations of an isolated singularity on the other hand givental introduced the notion of a total ancestor potential for every semi simple point of a frobenius manifold and conjectured that in the settings of singularity theory his definition extends analytically to non semisimple points as well in this paper we prove givental s conjecture by using the eynard orantin recursion

sufficient conditions for strongly caratheodory functions recently by applying the principle of subordination for analytic functions in the open unit disk u kim and cho i h kim n e cho sufficient conditions for caratheodory functions comput math appl 59 2010 2067 2073 considered several sufficient conditions for a family of caratheodory functions the main purpose of this paper is to investigate some presumably new sufficient conditions for the class of strongly caratheodory functions in u one illustrative example and several corollaries of the main results presented here are also considered

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

the partition function of the two matrix model as an isomonodromic τ function we consider the itzykson zuber eynard mehta two matrix model and prove that the partition function is an isomonodromic function in a sense that generalizes that of jimbo et al monodromy preserving deformation of linear ordinary differential equations with rational coefficients physica d 2 306 1981 in order to achieve the generalization we need to define a notion of function for isomonodromic systems where the adregularity of the leading coefficient is not a necessary requirement

the unstable integral homology of the mapping class groups of a surface with boundary we construct a graph complex calculating the integral homology of the bordered mapping class groups we compute the homology of the bordered mapping class groups of the surfaces s 1 1 s 1 2 and s 2 1 using the circle action on this graph complex we build a double complex and a spectral sequence converging to the homology of the unbordered mapping class groups we compute the homology of the punctured mapping class groups associated to the surfaces s 1 1 and s 2 1 finally we use miller s operad to get the first kudo araki and browder operations on our graph complex we also consider an unstable version of the higher kudo araki dyer lashoff operations

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

a formula of two partition hodge integrals jmg n where tpi ci l is the first chern class of l and xj cj e is the j th chern class of the hodge bundle the study of hodge integrals is an important part of intersection theory on mg n hodge integrals also naturally arise when one computes gromov witten invariants by localization techniques for example the following generating series of hodge integrals arises when one computes local invariants of a toric fano surface in a calabi yau 3 fold by virtual localization 33

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

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

advances in random matrix theory zeta functions and sphere packing over four hundred years ago sir walter raleigh asked his mathematical assistant to find formulas for the number of cannonballs in regularly stacked piles these investigations aroused the curiosity of the astronomer johannes kepler and led to a problem that has gone centuries without a solution why is the familiar cannonball stack the most efficient arrangement possible here we discuss the solution that hales found in 1998 almost every part of the 282 page proof relies on long computer verifications random matrix theory was developed by physicists to describe the spectra of complex nuclei in particular the statistical fluctuations of the eigenvalues the energy levels follow certain universal laws based on symmetry types we describe these and then discuss the remarkable appearance of these laws for zeros of the riemann zeta function which is the generating function for prime numbers and is the last special function from the last century that is not understood today explaining this phenomenon is a central problem these topics are distinct so we present them separately with their own introductory remarks

equations of the moduli of pointed curves in the infinite grassmannian the main result of this paper is the explicit computa tion of the equations defining the moduli space of triples c p where c is an integral and complete algebraic curve p a smooth rational point anda certain isomorphism this is achieved by introducing algebraically infinite grassmannians tau and baker ahkiezer functions and by proving an addition formula for tau functions

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