the continuum random tree iii let mathscr r k k geq 1 be random trees with k leaves satisfying a consistency condition removing a random leaf from mathscr r k gives mathscr r k 1 then under an extra condition this family determines a random continuum tree mathscr l which it is convenient to represent as a random subset of l 1 this leads to an abstract notion of convergence in distribution as n rightarrow infty of rescaled random trees mathscr j n on n vertices to a limit continuum random tree mathscr l the notion is based upon the assumption that for fixed k the subtrees of mathscr j n determined by k randomly chosen vertices converge to mathscr r k as our main example under mild conditions on the offspring distribution the family tree of a galton watson branching process conditioned on total population size equal to n can be rescaled to converge to a limit continuum random tree which can be constructed from brownian excursion

orbifold quasimap theory we extend to orbifolds the quasimap theory of ciocan fontanine and kim adv math 225 6 3022 3051 2010 j geom phys 75 17 47 2014 as well as the genus zero wall crossing results from algebr geom 1 4 400 448 2014 proceedings of the conference on the occasion of mukai s 60th birthday 2015 as a consequence we obtain generalizations of orbifold mirror theorems in particular of the mirror theorem for toric orbifolds recently proved independently by coates et al a mirror theorem for toric stacks

the λ coalescent speed of coming down from infinity consider a coalescent that comes down from infinity meani ng that it starts from a configuration containing infinitely many blocks at time 0 yet it has a finite number nt of blocks at any positive time t 0 we exhibit a deterministic function v 0 0 such that nt v t 1 almost surely and in l p for any p 1 as t 0 our approach relies on a novel martingale technique

a countable representation of the fleming viot measure valued diffusion the fleming viot measure valued diffusion arises as the infinite population limit of various discrete genetic models with general type space the paper gives a countable construction of the process as the empirical measure carried by a certain interactive particle system this explicit representation facilitates the study of various properties of the fleming viot process the construction also carries versions of the familiar genealogical processes from population genetics in particular kingman s coalescent thus unifying the genealogical and measure valued approaches to the subject

particle representations for measure valued population models models of populations in which a type or location represented by a point in a metric space e is associated with each individual in the population are considered a population process is neutral if the chances of an individual replicating or dying do not depend on its type measurevalued processes are obtained as infinite population limits for a large class of neutral population models and it is shown that these measure valued processes can be represented in terms of the total mass of the population and the de finetti measures associated with an e valued particle model

stochastic flows associated to coalescent processes we study a class of stochastic flows connected to the coalescent processes that have been studied recently by mohle pitman sagitov and schweinsberg in connection with asymptotic models for the genealogy of populations with a large fixed size we define a bridge to be a right continuous process b r r 0 1 with nondecreasing paths and exchangeable increments such that b 0 0 and b 1 1 we show that flows of bridges are in one to one correspondence with the so called exchangeable coalescents this yields an infinite dimensional version of the classical kingman representation for exchangeable partitions of we then propose a poissonian construction of a general class of flows of bridges and identify the associated coalescents we also discuss an important auxiliary measure valued process which is closely related to the genealogical structure coded by the coalescent and can be viewed as a generalized fleming viot process

the orbifold chow ring of toric deligne mumford stacks the orbifold chow ring of a deligne mumford stack defined by abramovich graber and vistoli 2 is the algebraic version of the orbifold cohomology ring in troduced by w chen and ruan 7 8 by design this ring incorporates numerical invariants such as the orbifold euler characteristic and the orbifold hodge num bers of the underlying variety the product structure is induced by the degree zero part of the quantum product in particular it involves gromov witten invariants inspired by string theory and results in batyrev 3 and yasuda 28 one expects that in nice situations the orbifold chow ring coincides with the chow ring of a resolution of singularities fantechi and g ttsche 14 and uribe 25 verify this conjecture when the orbifold is symn 5 where 5 is a smooth projective surface with ks 0 and the resolution is hilbn the initial motivation for this project was to compare the orbifold chow ring of a simplicial toric variety with the chow ring of a cr pant resolution to achieve this goal we first develop the theory of toric deligne mumford stacks modeled on simplicial toric varieties a toric deligne mumford stack corresponds to a combinatorial object called a stacky fan as a first approximation this object is a simplicial fan with a distinguished lattice point on each ray in the fan more precisely a stacky fan s is a triple consisting of a finitely generated abelian group n a simplicial fan e in q z n with n rays and a map zn n where the image of the standard basis in zn generates the rays in e a rational simplicial fan e produces a canonical stacky fan s n e where n is the distinguished lattice and is the map defined by the minimal lattice points on the rays hence there is a natural toric deligne mumford stack associated to every simplicial toric variety a stacky fan encodes a group action on a quasi affine variety and the toric deligne mumford stack is the quotient if e corresponds to a smooth toric variety x e and s is the canonical stacky fan associated to e then we simply have x z we show that many of the basic concepts such as open and closed toric substacks line bundles and maps between toric deligne mumford stacks correspond to combinatorial notions we expect that many more results about toric varieties lift to the realm of stacks and hope that toric deligne mumford stacks will serve as a useful testing ground for general theories

kodaira spencer theory of gravity and exact results for quantum string amplitudes we develop techniques to compute higher loop string amplitudes for twistedn 2 theories with 3 i e the critical case an important ingredient is the discovery of an anomaly at every genus in decoupling of brst trivial states captured to all orders by a master anomaly equation in a particular realization of then 2 theories the resulting string field theory is equivalent to a topological theory in six dimensions the kodaira spencer theory which may be viewed as the closed string analog of the chern simons theory using the mirror map this leads to computation of the number of holomorphic curves of higher genus curves in calabi yau manifolds it is shown that topological amplitudes can also be reinterpreted as computing corrections to superpotential terms appearing in the effective 4d theory resulting from compactification of standard 10d superstrings on the correspondingn 2 theory relations withc 1 strings are also pointed out

branching processes in levy processes laplace functionals of snakes and superprocesses we use the exploration process introduced in a previous work to develop a new construction of superprocesses with a general branching mechanism this construction depends on a path valued process called the levy snake which is of independent interest our method of proof involves a calculation of the laplace functional of the occupation field of the levy snake this calculation relies on an evaluation of the corresponding moment functionals which requires precise information about the underlying genealogical structure

the reduced genus 1 gromov witten invariants of calabi yau hypersurfaces 0 introduction 691 0 1 mirror symmetry predictions for a quintic threefold 691 0 2 computing gw invariants of hypersurfaces 693 0 3 mirror symmetry formulas for projective cy hypersurfaces 695 1 equivariant cohomology and stable maps 698 1 1 definitions and notation 698 1 2 setup for localization computation on m1 1 p d 701 1 3 contributions from fixed loci i 704 1 4 contributions from fixed loci ii 708 2 localization computations 712 2 1 regularizable power series in rational functions 712 2 2 regularizability of gw generating functions 714 2 3 proofs of propositions 1 1 and 1 2 717 3 algebraic computations 721 3 1 linear independence in symmetric rational functions 722 3 2 the genus 0 generating functions 723 3 3 proof of theorem 3 725 appendix a some combinatorics 730 appendix b comparison of mirror symmetry formulations 733 acknowledgments 736 references 736

remodeling the b model we propose a complete new formalism to compute unambiguously b model open and closed amplitudes in local calabi yau geometries including the mirrors of toric manifolds the formalism is based on the recursive solution of matrix models recently proposed by eynard and orantin the resulting amplitudes are non perturbative in both the closed and the open moduli the formalism can then be used to study stringy phase transitions in the open closed moduli space at large radius this formalism may be seen as a mirror formalism to the topological vertex but it is also valid in other phases in the moduli space we develop the formalism in general and provide an extensive number of checks including a test at the orbifold point of ap fibrations where the amplitudes compute the t hooft expansion of vevs of wilson loops in chern simons theory on lens spaces we also use our formalism to predict the disk amplitude for the orbifold mathbb c 3 mathbb z 3

smaller population size at the mrca time for stationary branching processes we consider an elementary model of random size varying population governed by a stationary continuous state branching process we compute the distributions of various variables related to the most recent common ancestor mrca the time to the mrca the size of the current population and the size of the population just before the mrca in particular we observe a natural mild bottleneck effect as the size of the population just before the mrca is stochastically smaller than the size of the current population we also compute the number of individuals involved in the last coalescent event of the genealogical tree that is the number of individuals at the time of the mrca who have descendants in the current population by studying more precisely the genealogical structure of the population we get asymptotics for the number of ancestors just before the current time we give explicit computations in the case of the quadratic branching mechanism in this case the size of the population at the mrca is in mean 2 3 of the size of the current population we also provide in this case the fluctuations for the renormalized number of ancestors

branching processes in levy processes the exploration process the main idea of the present work is to associate with a general continuous branching process an exploration process that contains the desirable information about the genealogical structure the exploration process appears as a simple local time functional of a levy process with no negative jumps whose laplace exponent coincides with the branching mechanism function this new relation between spectrally positive levy processes and continuous branching processes provides a unified perspective on both theories in particular we derive the adequate formulation of the classical ray knight theorem for such levy processes as a consequence of this theorem we show that the path continuity of the exploration process is equivalent to the almost sure extinction of the branching process

topological open strings on orbifolds we use the remodeling approach to the b model topological string in terms of recursion relations to study open string amplitudes at orbifold points to this end we clarify modular properties of the open amplitudes and rewrite them in a form that makes their transformation properties under the modular group manifest we exemplify this procedure for the mathbb c 3 mathbb z 3 orbifold point of local mathbb p 2 where we present results for topological string amplitudes for genus zero and up to three holes and for the one holed torus these amplitudes can be understood as generating functions for either open orbifold gromov witten invariants of mathbb c 3 mathbb z 3 or correlation functions in the orbifold cft involving insertions of both bulk and boundary operators

topological strings and almost modular forms the b model topological string theory on a calabi yau threefold x has a symmetry group generated by monodromies of the periods of x this acts on the topological string wave function in a natural way governed by the quantum mechanics of the phase space h3 x we show that depending on the choice of polarization the genus g topological string amplitude is either a holomorphic quasi modular form or an almost holomorphic modular form of weight 0 under moreover at each genus certain combinations of genus g amplitudes are both modular and holomorphic we illustrate this for the local calabi yau manifolds giving rise to seiberg witten gauge theories in four dimensions and local ip2 and ip1 ip1 as a byproduct we also obtain a simple way of relating the topological string amplitudes near different points in the moduli space which we use to give predictions for gromov witten invariants of the orbifold mathbb c 3 mathbb z 3

an integral structure in quantum cohomology and mirror symmetry for toric orbifolds abstract we introduce an integral structure in orbifold quantum cohomology associated to the k group and the class in the case of compact toric orbifolds we show that this integral structure matches with the natural integral structure for the landau ginzburg model under mirror symmetry by assuming the existence of an integral structure we give a natural explanation for the specialization to a root of unity in y ruan s crepant resolution conjecture yongbin ruan the cohomology ring of crepant resolutions of orbifolds in contemp math vol 403 amer math soc providence ri 2006 pp 117 126

the topological vertex we construct a cubic field theory which provides all genus amplitudes of the topological a model for all non compact toric calabi yau threefolds the topology of a given feynman diagram encodes the topology of a fixed calabi yau with schwinger parameters playing the role of kahler classes of the threefold we interpret this result as an operatorial computation of the amplitudes in the b model mirror which is the quantum kodaira spencer theory the only degree of freedom of this theory is an unconventional chiral scalar on a riemann surface in this setup we identify the b branes on the mirror riemann surface as fermions related to the chiral boson by bosonization

real algebraic curves the moment map and amoebas in this paper we prove the topological uniqueness of maximal arrangements of a real plane algebraic curve with respect to three lines more generally we prove the topological uniqueness of a maximally arranged algebraic curve on a real toric surface we use the moment map as a tool for studying the topology of real algebraic curves and their complexiflcations

localization and gluing of topological amplitudes we develop a gluing algorithm for gromov witten invariants of toric calabi yau threefolds based on localization and gluing graphs the main building block of this algorithm is a generating function of cubic hodge integrals of special form we conjecture a precise relation between this generating function and the topological vertex at fractional framing

probabilistic and fractal aspects of levy trees we investigate the random continuous trees called levy trees which are obtained as scaling limits of discrete galton watson trees we give a mathematically precise definition of these random trees as random variables taking values in the set of equivalence classes of compact rooted trees which is equipped with the gromov hausdorff distance to construct levy trees we make use of the coding by the height process which was studied in detail in previous work we then investigate various probabilistic properties of levy trees in particular we establish a branching property analogous to the well known property for galton watson trees conditionally given the tree below level a the subtrees originating from that level are distributed as the atoms of a poisson point measure whose intensity involves a local time measure supported on the vertices at distance a from the root we study regularity properties of local times in the space variable and prove that the support of local time is the full level set except for certain exceptional values of a corresponding to local extinctions we also compute several fractal dimensions of levy trees including hausdorff and packing dimensions in terms of lower and upper indices for the branching mechanism function which characterizes the distribution of the tree we finally discuss some applications to super brownian motion with a general branching mechanism

gromov witten theory of deligne mumford stacks given a smooth complex deligne mumford stack cal x with a projective coarse moduli space we introduce gromov witten invariants of cal x and prove some of their basic properties including the wdvv equation

a new cohomology theory of orbifold based on the orbifold string theory model in physics we construct a new cohomology ring for any almost complex orbifold the key theorem is the associativity of this new ring some examples are computed

enumerative meaning of mirror maps for toric calabi yau manifolds abstract we prove that the inverse of a mirror map for a toric calabi yau manifold of the form k y where y is a compact toric fano manifold can be expressed in terms of generating functions of genus 0 open gromov witten invariants defined by fukaya oh ohta ono 2010 15 such a relation between mirror maps and disk counting invariants was first conjectured by gross and siebert 2011 24 conjecture 0 2 and remark 5 1 as part of their program and was later formulated in terms of fukaya oh ohta ono s invariants in the toric calabi yau case in chan et al 2012 8 conjecture 1 1

a note on superprocesses subject to a mild restriction ona generator of the one particle motion we show thea fleming viot superprocess can be obtained from thea dawson watanabe superprocess by conditioning the latter to have constant total mass

pitch of iterated rippled noise iterated rippled noise irn stimuli are generated by a cascade of delay d gain g and add networks the matched pitch of irn stimuli was determined as a function of the type of network used to generate irn stimuli the delay d 2 4 and 8 ms the number of iterations n 1 2 3 4 6 8 of the delay and add networks and whether the delayed noise was added g 1 to or subtracted g 1 from the undelayed noise the matched pitch of irn stimuli did not depend on the type of network when g 1 addition the matched pitch did not depend on the number of stages of iterations n and was always equal to the reciprocal of the delay 1 d when g 1 subtraction the match pitch depended on n when n 1 the matched pitch of irn stimuli was ambiguous having two pitches in the region of 1 d 10 when n 8 the match pitch was equal to 1 2d this change in the pitch occurred between n 2 and 6 the results are discussed in terms of spectral and temporal models of complex pitch processing a temporal model based on autocorrelation was able to provide quantitative predictions for the pitch of these irn stimuli however a spectral type of model can also account for the pitch of these irn stimuli

smooth toric deligne mumford stacks we give a geometric definition of smooth toric deligne mumford stacks using the action of a torus we show that our definition is equivalent to the one of borisov chen and smith in terms of stacky fans in particular we give a geometric interpretation of the combinatorial data contained in a stacky fan we also give a bottom up classification in terms of simplicial toric varieties and fiber products of root stacks

gromov witten donaldson thomas correspondence for toric 3 folds we prove the equivariant gromov witten theory of a nonsingular toric 3 fold x with primary insertions is equivalent to the equivariant donaldson thomas theory of x as a corollary the topological vertex calculations by agangic klemm marino and vafa of the gromov witten theory of local calabi yau toric 3 folds are proven to be correct in the full 3 leg setting

a continuum tree valued markov process we present a construction of a levy continuum random tree crt associated with a super critical continuous state branching process using the so called exploration process and a girsanov theorem we also extend the pruning procedure to this super critical case let be a critical branching mechanism we set let or be the set of values of for which is a conservative branching mechanism the pruning procedure allows to construct a decreasing levy crt valued markov process mathcal t theta theta in theta such that mathcal t theta has branching mechanism it is sub critical if 0 and super critical if 0 we then consider the explosion time a of the crt the smallest negative time for which the continuous state branching process cb associated with mathcal t theta has finite total mass i e the length of the excursion of the exploration process that codes the crt is finite we describe the law of a as well as the distribution of the crt just after this explosion time the crt just after explosion can be seen as a crt conditioned not to be extinct which is pruned with an independent intensity related to a we also study the evolution of the crt valued process after the explosion time this extends results from aldous and pitman on galton watson trees for the particular case of the quadratic branching mechanism we show that after explosion the total mass of the cb behaves like the inverse of a stable subordinator with index 1 2 this result is related to the size of the tagged fragment for the fragmentation of aldous s crt

identification of the givental formula with the spectral curve topological recursion procedure we identify the givental formula for the ancestor formal gromov witten potential with a version of the topological recursion procedure for a collection of isolated local germs of the spectral curve as an application we prove a conjecture of norbury and scott on the reconstruction of the stationary sector of the gromov witten potential of mathbb c rm p 1 via a particular spectral curve

two representations of a conditioned superprocess we consider a class of measure valued markov processes constructed by taking a superprocess over some underlying markov process and conditioning it to stay alive forever we obtain two representations of such a process the first representation is in terms of an immortal particle that moves around according to the underlying markov process and throws off pieces of mass which then proceed to evolve in the same way that mass evolves for the unconditioned superprocess as a consequence of this representation we show that the tail field of the conditioned superprocess is trivial if the tail field of the underlying process is trivial the second representation is analogous to one obtained by legall in the unconditioned case it represents the conditioned superprocess in terms of a certain process taking values in the path space of the underlying process this representation is useful for studying the transience and recurrence properties of the closed support process

on the gromov witten donaldson thomas correspondence and ruan s conjecture for calabi yau 3 orbifolds for any toric calabi yau 3 orbifold with transverse a singularities we prove the gromov witten donaldson thomas correspondence and ruan s crepant resolution conjecture in all genera

topological orbifold models and quantum cohomology rings we discuss the topological sigma model on an orbifold target space we describe the moduli space of classical minima for computing correlation functions involving twisted operators and show through a detailed computation of an orbifold ofcp1 by the dihedral groupd4 how to compute the complete ring of observables through this procedure we compute all the rings of dihedralcp1 orbifolds we then considercp2 d4 and show how the techniques of topologicalanti topological fusion might be used to compute twist field correlation functions for nonabelian orbifolds