algebraic volume density property of affine algebraic manifolds we introduce the notion of algebraic volume density property for affine algebraic manifolds and prove some important basic facts about it in particular that it implies the volume density property the main results of the paper are producing two big classes of examples of stein manifolds with volume density property one class consists of certain affine modifications of n equipped with a canonical volume form the other is the class of all linear algebraic groups equipped with the left invariant volume form

duality of hardy and bmo spaces associated with operators with heat kernel bounds the introduction and development of hardy and bmo spaces on euclidean spaces r in the 1960s and 1970s played an important role in modern harmonic analysis and applications in partial differential equations these spaces were studied extensively in 32 22 18 19 31 and many others an l function f on r is in the hardy spaceh r if the area integral function of the poisson integral e t f satisfies

the potts model and the tutte polynomial this is an invited survey on the relation between the partition function of the potts model and the tutte polynomial on the assumption that the potts model is more familiar we have concentrated on the latter and its interpretations in particular we highlight the connections with abelian sandpiles counting problems on random graphs error correcting codes and the ehrhart polynomial of a zonotope where possible we use the mean field and square lattice as illustrations we also discuss in some detail the complexity issues involved

weak type p p estimates for riesz transforms we consider the problem of l p boundedness of higher order riesz transforms associated to elliptic operators l of order 2m on as an application of the recently solved kato conjecture we show for all this generalizes the result of auscher and tchamitchian restricted to the case d 2m

large discrete sets in stein manifolds we construct infinite discrete subsets of stein manifolds with remarkable properties these generalize results of rosay and rudin on discrete subsets of mathbb c n

counting coloured planar maps differential equations we address the enumeration of q coloured planar maps counted by the number of edges and the number of monochromatic edges we prove that the associated generating function is differentially algebraic that is satisfies a non trivial polynomial differential equation with respect to the edge variable we give explicitly a differential system that characterizes this series we then prove a similar result for planar triangulations thus generalizing a result of tutte dealing with their proper q colourings in statistical physics terms we solve the q state potts model on random planar lattices this work follows a first paper by the same authors where the generating function was proved to be algebraic for certain values of q including q 1 2 and 3 it is known to be transcendental in general in contrast our differential system holds for an indeterminate q for certain special cases of combinatorial interest four colours proper q colourings maps equipped with a spanning forest we derive from this system in the case of triangulations an explicit differential equation of order 2 defining the generating function for general planar maps we also obtain a differential equation of order 3 for the four colour case and for the self dual potts model

counting cycles in permutations by group characters with an application to a topological problem the character theory of the symmetric group is used to derive properties of the number of permutations with k cycles which are expressible as the product of a full cycle with an element of an arbitrary but fixed conjugacy class for the conjugacy class of fixed point free involutions this problem has application to the analysis of singularities in surfaces 1 introduction for nonnegative integers k and n and a partition p of n let ef denote the number of permutations w on n symbols such that it has exactly k cycles and such that it can be expressed as a product of an arbitrary but fixed cycle of length n and a permutation in the conjugacy class indexed by p the purpose of this paper is to derive the generating function for these numbers and to obtain some of their properties the method makes direct use of combinatorial and algebraic properties of the group algebra of the symmetric group a special case of this problem is of particular interest let e p n denote the number e when p indexes the conjugacy class of permutations on pn symbols with n cycles of length p the matter of calculating ek2 n arose in connection with work by harris and morrison 4 on singularities in surfaces it has also occurred indirectly in the work of gross 2 on graph embeddings harer and zagier 3 have shown by an independent method that the sequence e k2 n for k n 1 satisfies a three term linear recurrence equation with coefficients which are polynomials in n to fix ideas note that for n 2 the permissible permutations are

new orlicz hardy spaces associated with divergence form elliptic operators abstract let l be the divergence form elliptic operator with complex bounded measurable coefficients the positive concave function on 0 of strictly critical lower type p 0 1 and t t 1 1 t 1 for t 0 in this paper the authors study the orlicz hardy space h l r n and its dual space bmo l r n where l denotes the adjoint operator of l in l 2 r n several characterizations of h l r n including the molecular characterization the lusin area function characterization and the maximal function characterization are established the carleson measure characterization and the john nirenberg inequality for the space bmo l r n are also given as applications the authors show that the riesz transform l 1 2 and the littlewood paley g function g l map h l r n continuously into l the authors further show that the riesz transform l 1 2 maps h l r n into the classical orlicz hardy space h r n for p n n 1 1 and the corresponding fractional integral l for certain 0 maps h l r n continuously into h l r n where is determined by and and satisfies the same property as all these results are new even when t t p for all t 0 and p 0 1

blocked edges on eulerian maps and mobiles application to spanning trees hard particles and the ising model we introduce eulerian maps with blocked edges as a general way to implement statistical matter models on random maps by a modification of intrinsic distances we show how to code these dressed maps by means of mobiles i e decorated trees with labelled vertices leading to a closed system of recursion relations for their generating functions we discuss particular solvable cases in detail as well as various applications of our method to several statistical systems such as spanning trees on quadrangulations mutually excluding particles on eulerian triangulations or the ising model on quadrangulations

factorial algebraic group actions and categorical quotients abstract given an action of an affine algebraic group with only trivial characters on a factorial variety we ask for categorical quotients we characterize existence in the category of algebraic varieties moreover allowing constructible sets as quotients we obtain a more general existence result which for example settles the case of a finitely generated algebra of invariants as an application we provide a combinatorial git type construction of categorical quotients for actions of not necessarily reductive groups on e g complete varieties with finitely generated cox ring via lifting to the characteristic space

maximal function characterizations for hardy spaces on spaces of homogeneous type with finite measure and applications abstract we prove nontangential and radial maximal function characterizations for hardy spaces associated to a non negative self adjoint operator satisfying gaussian estimates on a space of homogeneous type with finite measure this not only addresses an open point in the literature but also gives a complete answer to the question posed by coifman and weiss in the case of finite measure we then apply our results to give maximal function characterizations for hardy spaces associated to second order elliptic operators with neumann and dirichlet boundary conditions schrodinger operators with dirichlet boundary conditions and fourier bessel operators

a new notion of tameness abstract we generalize the notion of tame discrete sets introduced by rosay and rudin from complex euclidean space to arbitrary complex manifolds and establish their basic properties we show that complex linear algebraic groups different from the complex line or the punctured complex line contain tame discrete sets

the solution of the kato square root problem for second order elliptic operators on rn we prove the kato conjecture for elliptic operators on jfin more precisely we establish that the domain of the square root of a uniformly complex elliptic operator l div av with bounded measurable coefficients in ietn is the

the neumann problem for elliptic equations with non smooth coefficients in this paper we study the neumann and regularity problems for divergence form symmetric elliptic operators l div av where a aifix is a matrix of bounded measurable functions and ellipticity is the existence of a 0 such that 2 114 2 ais ir 21412 for all i r we assume that l is defined in the unit ball b but the methods of proof are sufficiently general to allow b to be a bounded starlike lipschitz domain the neumann problem lu 0 in b avun q f

invariant rings and quasiaffine quotients we study hilbert s fourteenth problem from a geometric point of view nagata s celebrated counterexample demonstrates that for an arbitrary group action on a variety the ring of invariant functions need not be isomorphic to the ring of functions of an affine variety in this paper we will show that nevertheless it is always isomorphic to the ring of functions on a quasi affine variety

a character theoretic approach to embeddings of rooted maps in an orientable surface of given genus the group algebra of the symmetric group and properties of the irreducible characters are used to derive combinatorial properties of embeddings of rooted maps in orientable surfaces of arbitrary genus in particular we show that there exists for each genus a correspondence between the set of rooted quadrangulations and a set of rooted maps of all lower genera with a distinguished subset of vertices

hardy and bmo spaces associated to divergence form elliptic operators consider a second order divergence form elliptic operator l with complex bounded measurable coefficients in general operators based on l such as the riesz transform or square function may lie beyond the scope of the calderon zygmund theory they need not be bounded in the classical hardy bmo and even some l p spaces in this work we develop a theory of hardy and bmo spaces associated to l which includes in particular a molecular decomposition maximal and square function characterizations duality of hardy and bmo spaces and a john nirenberg inequality

loop models on random maps via nested loops the case of domain symmetry breaking and application to the potts model we use the nested loop approach to investigate loop models on random planar maps where the domains delimited by the loops are given two alternating colors which can be assigned different local weights hence allowing for an explicit z2 domain symmetry breaking each loop receives a non local weight n as well as a local bending energy which controls loop turns by a standard cluster construction that we review the q n2 potts model on general random maps is mapped to a particular instance of this problem with domain non symmetric weights we derive in full generality a set of coupled functional relations for a pair of generating series which encode the enumeration of loop configurations on maps with a boundary of a given color and solve it by extending well known complex analytic techniques in the case where loops are fully packed we analyze in detail the phase diagram of the model and derive exact equations for the position of its non generic critical points in particular we underline that the critical potts model on general random maps is not self dual whenever q 1 in a model with domain symmetric weights we also show the possibility of a spontaneous domain symmetry breaking driven by the bending energy 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

asymptotic enumeration and limit laws of planar graphs a graph is planar if it can be embedded in the plane or in the sphere so that no two edges cross at an interior point a planar graph together with a particular embedding is called a map there is a rich theory of counting maps started by tutte in the 1960 s however in this paper we are interested in counting graphs as combinatorial objects regardless of how many nonequivalent topological embeddings they may have as we are going to see this makes the counting considerably more difficult

hardy spaces and divergence operators on strongly lipschitz domains of rn abstract let be a strongly lipschitz domain of r n consider an elliptic second order divergence operator l including a boundary condition on and define a hardy space by imposing the non tangential maximal function of the extension of a function f via the poisson semigroup for l to be in l1 under suitable assumptions on l we identify this maximal hardy space with h 1 r n if r n with h r 1 under the dirichlet boundary condition and with h z 1 under the neumann boundary condition

the self dual point of the two dimensional random cluster model is critical for q 1 we prove a long standing conjecture on random cluster models namely that the critical point for such models with parameter q 1 on the square lattice is equal to the self dual point p sd q sqrt q 1 sqrt q this gives a proof that the critical temperature of the q state potts model is equal to log 1 sqrt q for all q 2 we further prove that the transition is sharp meaning that there is exponential decay of correlations in the sub critical phase the techniques of this paper are rigorous and valid for all q 1 in contrast to earlier methods valid only for certain given q the proof extends to the triangular and the hexagonal lattices as well

some new function spaces and their applications to harmonic analysis abstract in this paper a family of spaces is introduced which seems well adapted for the study of a variety of questions related to harmonic analysis and its applications these spaces are the tent spaces they provide the natural setting for the study of such things as maximal functions the relevant space here is t p and also square functions where the space t2p is relevant as such these spaces lead to unifications and simplifications of some basic techniques in harmonic analysis thus they are closely related to lp and hardy spaces important parts of whose theory become corollaries of the description of tent spaces also as proc conf harmonic analysis cortona lect notes in math vol 992 springer verlag berlin new york 1983 already indicated where these spaces first appeared explicitly the tent spaces can be used to simplify some of the results related to the cauchy integral on lipschitz curves and multilinear analysis in retrospect one can recognize that various ideas important for tent spaces had been used if only implicitly for quite some time here one should mention carleson s inequality its simplifications and extensions the theory of hardy spaces and atomic decompositions

tent space boundedness via extrapolation we study the action of operators on tent spaces such as maximal operators calderon zygmund operators riesz potentials we also consider singular non integral operators we obtain boundedness as an application of extrapolation methods in the banach range in the non banach range boundedness results for calderon zygmund operators follow by using an appropriate atomic theory we end with some consequences on amalgam spaces

the kp hierarchy branched covers and triangulations the kp hierarchy is a completely integrable system of quadratic partial differential equations that generalizes the kdv hierarchy a linear combination of schur functions is a solution to the kp hierarchy if and only if its coefficients satisfy the plucker relations from geometry we give a solution to the plucker relations involving products of variables marking contents for a partition and thus give a new proof of a content product solution to the kp hierarchy previously given by orlov and shcherbin in our main result we specialize this content product solution to prove that the generating series for a general class of transitive ordered factorizations in the symmetric group satisfies the kp hierarchy these factorizations appear in geometry as encodings of branched covers and thus by specializing our transitive factorization result we are able to prove that the generating series for two classes of branched covers satisfies the kp hierarchy for the first of these the double hurwitz series this result has been previously given by okounkov the second of these that we call the m hypermap series contains the double hurwitz series polynomially as the leading coefficient in m the m hypermap series also specializes further first to the series for hypermaps and then to the series for maps both in an orientable surface for the latter series we apply one of the kp equations to obtain a new and remarkably simple recurrence for triangulations in a surface of given genus with a given number of faces this recurrence leads to explicit asymptotics for the number of triangulations with given genus and number of faces in recent work by bender gao and richmond 9 on the low lying zeros of hasse weil l functions for elliptic curves

flexible varieties and automorphism groups given an irreducible affine algebraic variety x of dimension n 2 we let saut x denote the special automorphism group of x that is the subgroup of the full automorphism group aut x generated by all one parameter unipotent subgroups we show that if saut x is transitive on the smooth locus x reg then it is infinitely transitive on x reg in turn the transitivity is equivalent to the flexibility of x the latter means that for every smooth point x x reg the tangent space t x x is spanned by the velocity vectors at x of one parameter unipotent subgroups of aut x we also provide various modifications and applications

new function spaces of bmo type the john nirenberg inequality interpolation and applications in this paper we introduce and develop some new function spaces of bmo bounded mean oscillation type on spaces of homogeneous type or measurable subsets of spaces of homogeneous type the new function spaces are defined by variants of maximal functions associated with generalized approximations to the identity and they generalize the classical bmo space we show that the john nirenberg inequality holds on these spaces and they interpolate with lp spaces by the complex interpolation method we then give applications on lp boundedness of singular integrals whose kernels do not satisfy the hormander condition 2005 wiley periodicals inc