staggered lattice models some staggered models are studied along three different lines exact solvability yang baxter equations the existence of other exact solutions called disorder solutions their group of symmetries the analysis based on the commutation of row to row transfer matrices of arbitrary size n suggests that the exactly solvable models are either of non staggered type or of free fermion staggered type in contrast disorder solutions can easily be exhibited for staggered models the example of the ashkin teller model is analysed finally the symmetry group of these models is seen to be a straightforward generalisation of the one of homogeneous models

holonomic functions of several complex variables and singularities of anisotropic ising n fold integrals focusing on examples associated with holonomic functions we try to bring new ideas on how to look at phase transitions for which the critical manifolds are not points but curves depending on a spectral variable or even fill higher dimensional submanifolds lattice statistical mechanics often provides a natural holonomic framework to perform singularity analysis with several complex variables that would in the most general mathematical framework be too complex or simply could not be defined in a learn by example approach considering several picard fuchs systems of two variables above calabi yau odes associated with double hypergeometric series we show that d finite holonomic functions are actually a good framework for finding properly the singular manifolds the singular manifolds are found to be genus zero curves we then analyze the singular algebraic varieties of quite important holonomic functions of lattice statistical mechanics the n fold integrals n corresponding to the n particle decomposition of the magnetic susceptibility of the anisotropic square ising model in this anisotropic case we revisit a set of so called nickelian singularities that turns out to be a two parameter family of elliptic curves we then find the first set of non nickelian singularities for 3 and 4 that also turns out to be rational or elliptic curves we underline the fact that these singular curves depend on the anisotropy of the ising model or equivalently that they depend on the spectral parameter of the model this has important consequences on the physical nature of the anisotropic n s which appear to be highly composite objects we address from a birational viewpoint the emergence of families of elliptic curves and that of calabi yau manifolds on such problems we also address the question of singularities of non holonomic functions with a discussion on the accumulation of these singular curves for the non holonomic anisotropic full susceptibility this article is part of lattice models and integrability a special issue of journal of physics a mathematical and theoretical in honour of f y wu s 80th birthday

hyperbolic coxeter groups for triangular potts models the symmetry groups generated by the inversion relations of lattice models of statistical mechanics on triangular lattices are analysed for vertex models and for the standard scalar potts model with two and three site interactions these groups are generated by three inversion relations and are seen to be generically very large ones hyperbolic groups two situations for which the representations of these groups degenerate into smaller ones hopefully compatible with integrability are considered the first reduction for the vertex triangular model corresponds to the situation where the vertex of the triangular model coincides with the left or right hand side of a yang baxter relation in this case the representation of the group is isomorphic up to a semi direct product by a finite group to z z the second reduction for q state ports models occurs for particular values of q the so called tutte beraha numbers for this model algebraic varieties including the known ferromagnetic critical variety happen to be invariant under such large groups of symmetries as a byproduct this analysis provides nice birational representations of hyperbolic coxeter groups

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

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

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

baxter equations and the zamolodchikov model abstract a summary is given of the significance of the star triangle or yang baxter equations in the solution of two dimensional statistical mechanical models the extension to the three dimensional tetrahedron relations is quoted and it is shown how this can be used to obtain the partition function of the zamolodchikov model for an n lattice the n result differs from that recently obtained by bazhanov and stroganov

a cost optimal parallel algorithm for computing force field in n body simulations on a crew pram abstract we consider the following force field computation problem given a cluster of n particles in three dimensional space compute the force exerted on each particle by the other particles depending on different applications the pairwise interaction could be either gravitational or lennard jones in both cases the force between two particles vanishes as the distance between them approaches to infinity since there are n n 1 2 pairs direct method requires n 2 time for force evaluation which is very expensive for astronomical simulations in 1985 and 1986 two famous n log n time hierarchical tree algorithms were published by appel siam j sci statist comput 6 1985 85 103 and by barnes and hut nature 324 1980 446 449 respectively in a recent paper we presented a linear time algorithm which builds the oct tree bottom up and showed that appel s algorithm can be implemented in n sequential time in this paper we present an algorithm which computes the force field in log n time on a n log n processor crew pram a key to this optimal parallel algorithm is replacing a recursive top down force calculation procedure of appel by an equivalent non recursive bottom up procedure our parallel algorithm also yields a new n time sequential algorithm for force field computation

random matrix theory and higher genus integrability the quantum chiral potts model we perform a random matrix theory rmt analysis of the quantum fourstate chiral potts chain for different sizes of the chain up to size l 8 ou ra nalysis gives clear evidence of a gaussian orthogonal ensemble goe statistics suggesting the existence of a generalized time reversal invariance furthermore a change from the generic goe distribution to a poisson distribution occurs when the integrability conditions are met the chiral potts model is known to correspond to a star triangle integrability associated with curves of genus higher than zero or one therefore the rmt analysis can also be seen as a detector of higher genus integrability

phase diagram of a six state chiral potts model abstract we study the phase diagram of an isotropic six state chiral potts model on a square lattice by means of both exact and numerical methods the phase diagram of this model presents many similarities with the phase diagrams of the ashkin teller model or the models studied by zamolodchikov and monarstirskii a remarkable line globally invariant under a transformation generalizing the kramers wannier duality seems to correspond to a first order transition line up to a bifurcation point where this line splits into two second order lines all the numerical calculations are compared with exact results which can be performed using a canonical elliptic parametrization of this model the bifurcation point is found to correspond to the intersection of a generalized self dual line with an algebraic curve this curve corresponds to the set of points of the phase diagram for which a non trivial infinite symmetry group of the model degenerates into a finite group of order six the agreement between numerical and analytical results is very good

inversion relations and disorder solutions on potts models the inversion symmetry and the automorphy group generated by the latter when combined with the spatial symmetries are studied for the anisotropic potts models on the triangular and checkerboard lattices the exact expression of the partition function which is known on some particular disorder subvarieties of the triangular model is checked and a generalisation is proposed for the checkerboard lattice the automorphy group is then used to extend the disorder solutions to the infinity of transformed subvarieties

the ising model and special geometries we show that the globally nilpotent g operators corresponding to the factors of the linear differential operators annihilating the multifold integrals n of the magnetic susceptibility of the ising model n 6 are homomorphic to their adjoint this property of being self adjoint up to operator homomorphisms is equivalent to the feature of their symmetric squares or their exterior squares having rational solutions the differential galois groups are in the special orthogonal or symplectic groups this self adjoint up to operator equivalence property means that the factor operators that we already know to be derived from geometry are special globally nilpotent operators they correspond to special geometries beyond the small order factor operators occurring in the linear differential operators associated with 5 and 6 and in particular those associated with modular forms we focus on the quite large order 12 and order 23 operators we show that the order 12 operator has an exterior square which annihilates a rational solution then its differential galois group is in the symplectic group the order 23 operator is shown to factorize into an order 2 operator and an order 21 operator the symmetric square of this order 21 operator has a rational solution its differential galois group is thus in the orthogonal group

algebraic invariants in soluble models exhibits on the known solutions of the triangle relation for the exactly solved models some simplifying methods of recovering their parametrisation in terms of algebraic varieties the relation to the automorphy group generated by the inverse and spatial symmetries of the model is also analysed

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

inversion relations and symmetry groups for potts models on the triangular lattice inversion relations are obtained for the standard scalar q state triangular potts model with two and three spin interactions generalizing previously known results for two spin interaction models it is shown that these inversion relations generate a group of symmetries of the model which is naturally represented in terms of birational transformations in a four dimensional parameter space this group of birational transformations is generically a very large one namely a hyperbolic coxeter group in this framework of very large groups of symmetries a remarkable situation pops out the one for which q corresponds to tutte beraha numbers

selected non holonomic functions in lattice statistical mechanics and enumerative combinatorics we recall that the full susceptibility series of the ising model modulo powers of the prime reduce to algebraic functions we also recall the nonlinear polynomial differential equation obtained by tutte for the generating function of the q coloured rooted triangulations by vertices which is known to have algebraic solutions for all the numbers of the form the holonomic status of being unclear we focus on the analysis of the case showing that the corresponding series is quite certainly non holonomic along the line of a previous work on the susceptibility of the ising model we consider this series modulo the first eight primes and show that this probably non holonomic function reduces modulo these primes to algebraic functions we conjecture that this probably non holonomic function reduces to algebraic functions modulo almost every prime or power of prime numbers this raises the question of whether such remarkable non holonomic functions can be seen as a ratio of diagonals of rational functions or even algebraic functions of diagonals of rational functions

a decomposition of multidimensional point sets with applications to k nearest neighbors and n body potential fields we define the notion of a well separated pair decomposition of points in d dimensional space we then develop efficient sequential and parallel algorithms for computing such a decomposition we apply the resulting decomposition to the efficient computation of k nearest neighbors and n body potential fields

integrable mappings and polynomial growth abstract we describe birational representations of discrete groups generated by involutions having their origin in the theory of exactly solvable vertex models in lattice statistical mechanics these involutions correspond respectively to two kinds of transformations on q q matrices the inversion of the q q matrix and an involutive permutation of the entries of the matrix we concentrate on the case where these permutations are elementary transpositions of two entries in this case the birational transformations fall into six different classes for each class we analyze the factorization properties of the iteration of these transformations these factorization properties enable to define some canonical homogeneous polynomials associated with these factorization properties some mappings yield a polynomial growth of the complexity of the iterations for three classes the successive iterates for q 4 actually lie on elliptic curves this analysis also provides examples of integrable mappings in arbitrary dimension even infinite moreover for two classes the homogeneous polynomials are shown to satisfy non trivial non linear recurrences the relations between factorizations of the iterations the existence of recurrences on one or several variables as well as the integrability of the mappings are analyzed

from integrability to weak chaos abstract we analyze birational transformations obtained from very simple algebraic calculations namely taking the inverse of q q matrices and permuting some of the entries of these matrices we concentrate on 4 4 matrices and elementary transpositions of two entries this analysis brings out six classes of birational transformations three classes correspond to integrable mappings their iteration yielding elliptic curves the iterations corresponding to the three other classes are included in higher dimensional non trivial algebraic varieties for many initial conditions in the parameter space these orbits lie on transcendental curves and finally explode in these higher dimensional varieties these transformations act on fifteen or q 2 1 variables however one can associate to them remarkably simple non linear recurrences bearing on a single variable the study of these last recurrences gives a complementary understanding of these amazingly regular non integrable mappings

greengard s n body algorithm is not order n greengard s n body algorithm claims to compute the pairwise interactions in a system of n particles in o n time for a fixed precision in this paper we show that the choice of precision is not independent of n and has a lower bound of log n we use this result to show that greengard s algorithm is not o n

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

discrete symmetry groups of vertex models in statistical mechanics we analyze discrete symmetry groups of vertex models in lattice statistical mechanics represented as groups of birational transformations they can be seen as generated by involutions corresponding respectively to two kinds of transformations onq q matrices the inversion of theq q matrix and an involutive permutation of the entries of the matrix we show that the analysis of the factorizations of the iterations of these transformations is a precious tool in the study of lattice models in statistical mechanics this approach enables one to analyze two dimensionalq4 state vertex models as simply as three dimensional vertex models or higher dimensional vertex models various examples of birational symmetries of vertex models are analyzed a particular emphasis is devoted to a three dimensional vertex model the 64 state cubic vertex model which exhibits a polynomial growth of the complexity of the calculations a subcase of this general model is seen to yield integrable recursion relations we also concentrate on a specific two dimensional vertex model to see how the generic exponential growth of the calculations reduces to a polynomial growth when the model becomes yang baxter integrable it is also underlined that a polynomial growth of the complexity of these iterations can occur even for transformations yielding algebraic surfaces or higher dimensional algebraic varieties

necessary versus sufficient conditions for exact solubility of statistical models on lattices it is shown that under rather mild conditions the triangle relation represents a necessary condition for the existence of commuting transfer matrices of arbitrary size the cases of spin models and vertex models are treated separately

let s baxterise we recall the concept of baxterisation of an r matrix or of a monodromy matrix which corresponds to build from one point in the r matrix parameter space the algebraic variety where the spectral parameter s live we show that the baxterisation which amounts to studying the iteration of a birational transformation is a win win strategy it enables to discard efficiently the non integrable situations focusing directly on the two interesting cases where the algebraic varieties are of the so called general type finite order iteration or are abelian varieties infinite order iteration we emphasize the heuristic example of the sixteen vertex model and provide a complete description of the finite order iterations situations for the baxter model we show that the baxterisation procedure can be introduced in much larger frameworks where the existence of some underlying yang baxter structure is not used we baxterise l operators local quantum lax matrices and quantum hamiltonians

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

three state chiral potts models in two dimensions integrability and symmetry abstract symmetry and integrability relations for the three state chiral potts model on two dimensional lattices are reviewed our detailed and systematic analysis leads to new integrability varieties for specific chiral potts models including new integrable points for the standard antiferromagnetic potts model we also describe a three state vertex model representation of the chiral potts model and write down the associated fundamental invariants for three coordinated lattices

automorphisms of algebraic varieties and yang baxter equations it is shown that commuting transfer matrix models in statistical mechanics are parametrized by algebraic varieties having a set of automorphisms deduced from the so called inversion relation in general this set of automorphisms is infinite this shows that for algebraic varieties of dimension 1 the models are parametrized by algebraic curves of genus 0 or 1

an o n time hierarchical tree algorithm for computing force field in n body simulations abstract we consider the following force field computation problem given a cluster of n particles in three dimensional space compute the force exerted on each particle by the other particles depending on different applications the pairwise interaction could be either gravitational or lennard jones in both cases the force between two particles vanishes as the distance between them approaches to infinity since there are n n 1 2 pairs direct method requires n 2 time for force evaluation which is very expensive for astronomical simulations in 1985 and 1986 two famous o n log n time hierarchical tree algorithms were published by appel 1985 and by barnes and hut 1986 respectively in this paper we show that appel s algorithm can be implemented in o n time

new integrable cases of a cremona transformation a finite order orbits analysis we analyse the properties of a particular birational mapping of two variables cremona transformation depending on two free parameters e and associated with the action of a discrete group of non linear birational transformations on the entries of a q q matrix this mapping originates from the analysis of birational transformations obtained from very simple algebraic calculations namely taking the inverse of q q matrices and permuting some of the entries of these matrices it has been seen to yield weak chaos and integrability we have found new integrable cases of this cremona transformation corresponding to the values of 0 when e 12 13 1 besides the already known values e 0 and e 1 and also arbitrary when e 0 for these cases one has a foliation of the parameter space in elliptic curves we give the equations of these elliptic curves based on this very example we show how one can find these integrability cases of the cremona transformation and actually integrate it using a method based on the systematic study of the finite order conditions of the cremona transformation the method is shown to be efficient and straightforward the various integrability cases are revisited using many different representations of this very mapping birational transformations recursion in one variable

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

the ising model from elliptic curves to modular forms and calabi yau equations we show that almost all the linear differential operators factors obtained in the analysis of the n particle contributions s of the susceptibility of the ising model for n 6 are linear differential operators associated with elliptic curves beyond the simplest differential operators factors which are homomorphic to symmetric powers of the second order operator associated with the complete elliptic integral e the second and third order differential operators z2 f2 f3 can actually be interpreted as modular forms of the elliptic curve of the ising model a last order 4 globally nilpotent linear differential operator is not reducible to this elliptic curve modular form scheme this operator is shown to actually correspond to a natural generalization of this elliptic curve modular form scheme with the emergence of a calabi yau equation corresponding to a selected 4f3 hypergeometric function this hypergeometric function can also be seen as a hadamard product of the complete elliptic integral k with a remarkably simple algebraic pull back square root extension the corresponding calabi yau fourth order differential operator having a symplectic differential galois group the mirror maps and higher order schwarzian odes associated with this calabi yau ode present all the nice physical and mathematical ingredients we had with elliptic curves and modular forms in particular an exact isogenies representation of the generators of the renormalization group extending the modular group to a symmetry group