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Abstract: This is the second paper of a series (started by Braverman and Finkelberg, 2010 ) which describes a conjectural analog of the affine Grassmannian for affine Kac–Moody groups (also known as the double affine Grassmannian). The current paper is dedicated to describing a conjectural analog of the convolution diagram for the double affine Grassmannian. In the case when our conjectures can be derived from Nakajima (2009) . [Copyright &y& Elsevier]

Abstract: This paper derives an explicit formula for Branson''s Q-curvature in even-dimensional conformal geometry. The ingredients in the formula come from the Poincaré metric in one higher dimension; hence the formula is called holographic. When specialized to the conformally flat case, the holographic formula expresses the Q-curvature as a multiple of the Pfaffian and the divergence of a natural 1-form. The paper also outlines the relation between holographic formulae for Q-curvature and a new theory of conformally covariant families of differential operators due to the second author. [Copyright &y& Elsevier]

Abstract: Let be the set of all nonnegative integers, be a finite set, and 2A be the set of all numbers of the form for all a and b in A. In [G.A. Freiman, Foundations of a Structural Theory of Set Addition, Transl. Math. Monogr., vol. 37, American Mathematical Society, Providence, RI, 1973 (translated from the Russian)] the arithmetic structure of A was optimally characterized when . 2 In [G.A. Freiman, Foundations of a Structural Theory of Set Addition, Transl. Math. Monogr., vol. 37, American Mathematical Society, Providence, RI, 1973 (translated from the Russian)] the structure of A was also characterized without proof when . Since then the efforts have been made to generalize these results, see [V. Lev, P.Y. Smeliansky, On addition of two distinct sets of integers, Acta Arith. 70 (1) (1995) 85–91; V. Lev, On the structure of sets of integers with small doubling property, unpublished manuscripts, 1995; Y.O. Hamidoune, A. Plagne, A generalization of Freiman''s theorem, Acta Arith. 103 (2) (2002) 147–156] for example. However, no optimal characterization of the structure of A has been obtained without imposing extra conditions, until now, when . In this paper we optimally characterize, with the help of nonstandard analysis, the arithmetic structure of A when , where b is positive but not too large. Precisely, we prove that there is a real number and there is such that if and for , then A is either a subset of an arithmetic progression of length at most or a subset of a bi-arithmetic progression 3 of length at most . An application of this result to the inverse problem for upper asymptotic density is presented near the end of the paper. In the application we improve the most important part of the main theorem in [R. Jin, Solution to the inverse problem for upper asymptotic density, J. Reine Angew. Math. (Crelle''s J.) 595 (2006) 121–166]. [2] One can easily prove that is always true and implies that A is an arithmetic progression. [3] The definition of bi-arithmetic progression can be found in the beginning of Introduction. [Copyright &y& Elsevier]

Abstract: This paper deals with a “naive” way of generalizing Kazhdan''s property (T) to -algebras. Our approach differs from the approach of Connes and Jones, which has already demonstrated its utility. Nevertheless, it turns out that our approach is applicable to a rather subtle question in the theory of -Hilbert modules. Namely, we prove that a separable unital -algebra A has property MI (module infinite—i.e. any countably generated self-dual Hilbert module over A is finitely generated and projective) if and only if A does not satisfy our definition of property (T). The commutative case was studied in an earlier paper. [Copyright &y& Elsevier]

Abstract: The sub-elliptic obstacle problem arises in various branches of the applied sciences, e.g., in mechanical engineering and robotics, mathematical finance, image reconstruction and neurophysiology. In the recent paper [Donatella Danielli, Nicola Garofalo, Sandro Salsa, Variational inequalities with lack of ellipticity. I. Optimal interior regularity and non-degeneracy of the free boundary, Indiana Univ. Math. J. 52 (2) (2003) 361–398; MR1976081 (2004c:35424)] it was proved that weak solutions to the sub-elliptic obstacle problem in a Carnot group belong to the Folland–Stein (optimal) Lipschitz class (the analogue of the well-known interior local regularity for the classical obstacle problem). However, the regularity of the free boundary remained a challenging open problem. In this paper we prove that, in Carnot groups of step , the free boundary is (Euclidean) near points satisfying a certain thickness condition. This constitutes the sub-elliptic counterpart of a celebrated result due to Caffarelli [Luis A. Caffarelli, The regularity of free boundaries in higher dimensions, Acta Math. 139 (3–4) (1977) 155–184; MR0454350 (56 #12601)]. [Copyright &y& Elsevier]

Abstract: We construct a new equivariant cohomology theory for a certain class of differential vertex algebras, which we call the chiral equivariant cohomology. A principal example of a differential vertex algebra in this class is the chiral de Rham complex of Malikov–Schechtman–Vaintrob of a manifold with a group action. The main idea in this paper is to synthesize the algebraic approach to classical equivariant cohomology due to H. Cartan, 2 with [2] Cartan''s theory was further developed by Duflo–Kumar–Vergne [M. Duflo, S. Kumar, M. Vergne, Sur la cohomologie équivariante des variétés différentiables, Astérisque 215 (1993)] and Guillemin–Sternberg [V. Guillemin, S. Sternberg, Supersymmetry and Equivariant de Rham Theory, Springer, 1999]. This paper follows closely the latter approach. the theory of differential vertex algebras, by using an appropriate notion of invariant theory. We also construct the vertex algebra analogues of the Mathai–Quillen isomorphism, the Weil and the Cartan models for equivariant cohomology, and the Chern–Weil map. We give interesting cohomology classes in the new theory that have no classical analogues. [Copyright &y& Elsevier]

Abstract: This note is a follow-up on the paper [A. Borel, G. Harder, Existence of discrete cocompact subgroups of reductive groups over local fields, J. Reine Angew. Math. 298 (1978) 53–64] of A. Borel and G. Harder in which they proved the existence of a cocompact lattice in the group of rational points of a connected semi-simple algebraic group over a local field of characteristic zero by constructing an appropriate form of the semi-simple group over a number field and considering a suitable S-arithmetic subgroup. Some years ago A. Lubotzky initiated a program to study the subgroup growth of arithmetic subgroups, the current stage of which focuses on “counting” (more precisely, determining the asymptotics of) the number of lattices of bounded covolume (the finiteness of this number was established in [A. Borel, G. Prasad, Finiteness theorems for discrete subgroups of bounded covolume in semi-simple groups, Publ. Math. Inst. Hautes Études Sci. 69 (1989) 119–171; Addendum: Publ. Math. Inst. Hautes Études Sci. 71 (1990) 173–177] using the formula for the covolume developed in [G. Prasad, Volumes of S-arithmetic quotients of semi-simple groups, Publ. Math. Inst. Hautes Études Sci. 69 (1989) 91–117]). Work on this program led M. Belolipetsky and A. Lubotzky to ask questions about the existence of isotropic forms of semi-simple groups over number fields with prescribed local behavior. In this paper we will answer these questions. A question of similar nature also arose in the work [D. Morris, Real representations of semisimple Lie algebras have -forms, in: Proc. Internat. Conf. on Algebraic Groups and Arithmetic, December 17–22, 2001, TIFR, Mumbai, 2001, pp. 469–490] of D. Morris (Witte) on a completely different topic. We will answer that question too. [Copyright &y& Elsevier]

Abstract: This paper is a continuation of [S. Alexakis, The decomposition of global conformal invariants I, submitted for publication, see also math.DG/0509571], where we complete our partial proof of the Deser–Schwimmer conjecture on the structure of “global conformal invariants.” Our theorem deals with such invariants that locally depend only on the curvature tensor (without covariant derivatives). In [S. Alexakis, The decomposition of global conformal invariants I, Ann. of Math., in press] we developed a powerful tool, the “super divergence formula” which applies to any Riemannian operator that always integrates to zero on compact manifolds. In particular, it applies to the operator that measures the “non-conformally invariant part” of . This paper resolves the problem of using this information we have obtained on the structure of to understand the structure of . [Copyright &y& Elsevier]

Abstract: Double Hurwitz numbers count branched covers of 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 (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 . 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. [Copyright &y& Elsevier]

Abstract: We continue the study of the Hochschild structure of a smooth space that we began in our previous paper, examining implications of the Hochschild–Kostant–Rosenberg theorem. The main contributions of the present paper are:[] we introduce a generalization of the usual notions of Mukai vector and Mukai pairing on differential forms that applies to arbitrary manifolds; [] we give a proof of the fact that the natural Chern character map becomes, after the HKR isomorphism, the usual one ; and [] we present a conjecture that relates the Hochschild and harmonic structures of a smooth space, similar in spirit to the Tsygan formality conjecture. [Copyright &y& Elsevier]

The paper (Discrete Comput. Geom. 25 (2001) 629) of Solymosi and Tóth implicitly raised the following arithmetic problem. Consider 𝑛 pairwise disjoint s element sets and form all (s2)n sums of pairs of elements of the same set. What is the minimum number of distinct sums one can get this way? This paper proves that the number of distinct sums is at least 𝑛ds, where ds=1/c⌈s/2⌉ is defined in the paper and tends to e−1 as s goes to infinity. Here e is the base of the natural logarithm. As an application we improve the Solymosi–Tóth bound on an old Erdo&x030B;s problem: we prove that n distinct points in the plane determine Ω(n4e5e−1−ϵ) distinct distances, where ϵ>0 is arbitrary. Our bound also finds applications in other related results in discrete geometry. Our bounds are proven through an involved calculation of entropies of several random variables. [Copyright &y& Elsevier]

We present several contributions to the enumerative theory of wreath product representations developed in a previous paper by the first named author (Adv. in Math.153 (2000), 118–154). Theorem 3.1 of the present paper establishes an explicit formula for one of the key ingredients in the description of the corresponding generating functions given in Mu¨ller (2000) (the exterior function ΦΓ). Building on Theorem 1 in Mu¨ller (2000) and the latter result, we derive explicit formulae for the exponential generating function of the series {∣Hom (Γ,Rn)∣} in the case where Γ is dihedral or a finite abelian group, and the representation sequence {Rn} is any of {H≀Sn} or {H≀An} with a fixed finite group H, or the sequence {∣Hom(G, Wn)}. Moreover, we verify a conjecture concerning the asymptotic behaviour of the sequence \{∣Hom(G,Wn)∣\} for finite groups G made in Mu¨ller (2000) in the case when G is dihedral or abelian. [Copyright &y& Elsevier]

Inversive distance circle packing metric was introduced by P Bowers and K Stephenson [7] as a generalization of Thurston's circle packing metric [34] . They conjectured that the inversive distance circle packings are rigid. For nonnegative inversive distance, Guo [22] proved the infinitesimal rigidity and then Luo [27] proved the global rigidity. In this paper, based on an observation of Zhou [37] , we prove this conjecture for inversive distance in ( − 1 , + ∞ ) by variational principles. We also study the global rigidity of a combinatorial curvature introduced in [14,16,19] with respect to the inversive distance circle packing metrics where the inversive distance is in ( − 1 , + ∞ ) . [ABSTRACT FROM AUTHOR]

A common feature of many duality results is that the involved equivalence functors are liftings of hom-functors into the two-element space resp. lattice. Due to this fact, we can only expect dualities for categories cogenerated by the two-element set with an appropriate structure. A prime example of such a situation is Stone's duality theorem for Boolean algebras and Boolean spaces, the latter being precisely those compact Hausdorff spaces which are cogenerated by the two-element discrete space. In this paper we aim for a systematic way of extending this duality theorem to categories including all compact Hausdorff spaces. To achieve this goal, we combine duality theory and quantale-enriched category theory. Our main idea is that, when passing from the two-element discrete space to a cogenerator of the category of compact Hausdorff spaces, all other involved structures should be substituted by corresponding enriched versions. Accordingly, we work with the unit interval [ 0 , 1 ] and present duality theory for ordered and metric compact Hausdorff spaces and (suitably defined) finitely cocomplete categories enriched in [ 0 , 1 ] . [ABSTRACT FROM AUTHOR]

Solving a well-known problem of Maharam, Talagrand [18] constructed an exhaustive non uniformly exhaustive submeasure, thus also providing the first example of a Maharam algebra that is not a measure algebra. To each exhaustive submeasure one can canonically assign a certain countable ordinal, its exhaustivity rank. In this paper, we use carefully constructed Schreier families and norms derived from them to provide examples of exhaustive submeasures of arbitrary high exhaustivity rank. This gives rise to uncountably many non isomorphic separable atomless Maharam algebras. [ABSTRACT FROM AUTHOR]

In this paper we study the multiple ergodic averages 1 n ∑ k = 1 n φ ( x k , x k q , ⋯ , x k q ℓ − 1 ) , ( x n ) ∈ Σ m on the symbolic space Σ m = { 0 , 1 , ⋯ , m − 1 } N ⁎ where m ≥ 2 , ℓ ≥ 2 , q ≥ 2 are integers. We give a complete solution to the problem of multifractal analysis of the limit of the above multiple ergodic averages. Actually we develop a non-invariant and non-linear version of thermodynamic formalism that is of its own interest. We study a large class of measures (called telescopic product measures). The special case of telescopic product measures defined by the fixed points of some non-linear transfer operators plays a crucial role in studying the level sets of the limit, which are not shift-invariant. These measures share many properties with Gibbs measures in the classical thermodynamic formalism. Our work also concerns variational principle, pressure function and Legendre transform in this new setting. [ABSTRACT FROM AUTHOR]

This is the second of two papers, in which we study the problem of prescribing Webster scalar curvature on the CR sphere as a given function f . Using the Webster scalar curvature flow, we prove an existence result under suitable assumptions on the Morse indices of f . [ABSTRACT FROM AUTHOR]

Abstract: In this paper we obtain an inequality on the unit disk B in , which improves the classical Moser–Trudinger inequality and the classical Hardy inequality at the same time. Namely, there exists a constant such that where This inequality is a two-dimensional analog of the Hardy–Sobolev–Mazʼya inequality in higher dimensions, which has been intensively studied recently. We also prove that the supremum is achieved in a suitable function space, which is an analog of the celebrated result of Carleson–Chang for the Moser–Trudinger inequality. [Copyright &y& Elsevier]

Abstract: This paper is devoted to a study on closed geodesics on Finsler and Riemannian spheres. We call a prime closed geodesic on a Finsler manifold rational, if the basic normal form decomposition (cf. [Y. Long, Bott formula of the Maslov-type index theory, Pacific J. Math. 187 (1999) 113–149]) of its linearized Poincaré map contains no rotation matrix with rotation angle which is an irrational multiple of π, or irrational otherwise. We prove that if there exists only one prime closed geodesic on a d-dimensional irreversible Finsler sphere with , it cannot be rational. Then we further prove that there exist always at least two distinct prime closed geodesics on every irreversible Finsler 3-dimensional sphere. Our method yields also at least two geometrically distinct closed geodesics on every reversible Finsler as well as Riemannian 3-dimensional sphere. We prove also such results hold for all compact simply connected 3-dimensional manifolds with irreversible or reversible Finsler as well as Riemannian metrics. [Copyright &y& Elsevier]

Abstract: The well-known combinatorial lemma of Karpovsky, Milman and Alon and a very recent one of Kerr and Li are extended. The obtained lemmas are applied to study the maximal pattern entropy introduced in the paper. It turns out that the maximal pattern entropy is equal to the supremum of sequence entropies over all sequences both in topological and measure-theoretical settings. Moreover, it is shown the maximal pattern entropy of any topological system is for some with k the maximal length of intrinsic sequence entropy tuples; and a zero-dimensional system has zero sequence entropy for any sequence if and only if the maximal pattern with respect to any open cover is of polynomial order. [Copyright &y& Elsevier]

Abstract: The purpose of the paper is three-fold: (a) we prove that every sequence which is a multidimensional sum of a balanced hypergeometric term has an asymptotic expansion of Gevrey type-1 with rational exponents, (b) we construct a class of G-functions that come from enumerative combinatorics, and (c) we give a counterexample to a question of Zeilberger that asks whether holonomic sequences can be written as multisums of balanced hypergeometric terms. The proofs utilize the notion of a G-function, introduced by Siegel, and its analytic/arithmetic properties shown recently by André. [Copyright &y& Elsevier]

Abstract: For each classical symmetric pair , there is a naturally defined multi-graded algebra , called the branching algebra for , which encodes the branching rule from G to H. This algebra has a natural family of subalgebras, depending on integer parameters. For a certain range of the parameters, the subalgebras have a particularly simple structure and are called stable branching algebras. In this paper, we show that the stable branching algebras for eight out of the ten families of classical symmetric pairs are flat deformations of the semigroup algebras of explicitly described lattice cones. [Copyright &y& Elsevier]

Abstract: This paper studies the embeddings of a complex submanifold S inside a complex manifold M; in particular, we are interested in comparing the embedding of S in M with the embedding of S as the zero section in the total space of the normal bundle of S in M. We explicitly describe some cohomological classes allowing to measure the difference between the two embeddings, in the spirit of the work by Grauert, Griffiths, and Camacho, Movasati and Sad; we are also able to explain the geometrical meaning of the separate vanishing of these classes. Our results hold for any codimension, but even for curves in a surface we generalize previous results due to Laufert and Camacho, Movasati and Sad. [Copyright &y& Elsevier]

Abstract: In this paper we continue our studies of the one-dimensional conformal metric flows, which were introduced in [Y. Ni, M. Zhu, Extremal one-dimensional conformal metrics, Commun. Contemp. Math., in press]. We prove the global existence and convergence of the flows, as well as the exponential convergence of the metrics under these flows. [Copyright &y& Elsevier]

Abstract: In this paper we develop a general framework for verifying hyperbolicity of holomorphic dynamical systems in . Our framework in particular enables us to construct the first example of a hyperbolic Hénon map of which is non-planar, i.e. which is not topologically conjugate on its Julia set to a small perturbation of any expanding polynomial in one variable. The key ideas in its proof are: the Poincaré box, which is a building block to apply our criterion for hyperbolicity, an operation called fusion, to merge two polynomials in one variable to obtain essentially two-dimensional dynamics, and rigorous computation by using interval arithmetic. Some conjectures and problems are also presented. [Copyright &y& Elsevier]

Abstract: In this paper we extend categorically the notion of a finite nilpotent group to fusion categories. To this end, we first analyze the trivial component of the universal grading of a fusion category , and then introduce the upper central series of . For fusion categories with commutative Grothendieck rings (e.g., braided fusion categories) we also introduce the lower central series. We study arithmetic and structural properties of nilpotent fusion categories, and apply our theory to modular categories and to semisimple Hopf algebras. In particular, we show that in the modular case the two central series are centralizers of each other in the sense of M. Müger. [Copyright &y& Elsevier]