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In a 1992 paper, Witten gave a formula for the intersection pairings of the moduli space of flat $G$-bundles over an oriented surface, possibly with markings. In this paper, we give a general proof of Witten's formula, for arbitrary compact, simple groups, and any markings for which the moduli space has at most orbifold singularities., 46 pages

This paper is a continuation of [A. Martinez, S. Nakamura, V. Sordoni, Analytic smoothing effect for the Schrödinger equation with long-range perturbation, Comm. Pure Appl. Math. LIX (2006) 1330–1351], where an analytic smoothing effect was proved for long-range type perturbations of the Laplacian H0 on Rn. In this paper, we consider short-range type perturbations H of the Laplacian on Rn, and we characterize the analytic wave front set of the solution to the Schrödinger equation: e−itHf, in terms of that of the free solution: e−itH0f, for t0 in the backward non-trapping region. This result is an analytic analogue of results by Hassel and Wunsch [A. Hassel, J. Wunsch, The Schrödinger propagator for scattering metrics, Ann. of Math. 162 (2005) 487–523] and Nakamura [S. Nakamura, Wave front set for solutions to Schrödinger equations, J. Funct. Anal. 256 (2009) 1299–1309].

Metric Embedding plays an important role in a vast range of application areas such as computer vision, computational biology, machine learning, networking, statistics, and mathematical psychology, to name a few. The mathematical theory of metric embedding is well studied in both pure and applied analysis and has more recently been a source of interest for computer scientists as well. Most of this work is focused on the development of bi-Lipschitz mappings between metric spaces. In this paper we present new concepts in metric embeddings as well as new embedding methods for metric spaces. We focus on finite metric spaces, however some of the concepts and methods are applicable in other settings as well.One of the main cornerstones in finite metric embedding theory is a celebrated theorem of Bourgain which states that every finite metric space on n points embeds in Euclidean space with O(logn) distortion. Bourgainʼs result is best possible when considering the worst case distortion over all pairs of points in the metric space. Yet, it is natural to ask: can an embedding do much better in terms of the average distortion? Indeed, in most practical applications of metric embedding the main criteria for the quality of an embedding is its average distortion over all pairs.In this paper we provide an embedding with constant average distortion for arbitrary metric spaces, while maintaining the same worst case bound provided by Bourgainʼs theorem. In fact, our embedding possesses a much stronger property. We define the ℓq-distortion of a uniformly distributed pair of points. Our embedding achieves the best possible ℓq-distortion for all 1⩽q⩽∞ simultaneously.The results are based on novel embedding methods which improve on previous methods in another important aspect: the dimension of the host space. The dimension of an embedding is of very high importance in particular in applications and much effort has been invested in analyzing it. However, no previous result improved the bound on the dimension which can be derived from Bourgainʼs embedding. Our embedding methods achieve better dimension, and in fact, shed new light on another fundamental question in metric embedding, which is: whether the embedding dimension of a metric space is related to its intrinsic dimension? I.e., whether the dimension in which it can be embedded in some real normed space is related to the intrinsic dimension which is reflected by the inherent geometry of the space, measured by the spaceʼs doubling dimension. The existence of such an embedding was conjectured by Assouad,44Assouad (1983) [10] conjectured that the distortion would be a function only of the doubling dimension as well, but this stronger conjecture was disproved (Semmes, 1996) [86].and was later posed as an open problem in several papers. Our embeddings give the first positive result of this type showing any finite metric space obtains a low distortion (and constant average distortion) embedding in Euclidean space in dimension proportional to its doubling dimension.Underlying our results is a novel embedding method. Probabilistic metric decomposition techniques have played a central role in the field of finite metric embedding in recent years. Here we introduce a novel notion of probabilistic metric decompositions which comes particularly natural in the context of embedding. Our new methodology provides a unified approach to all known results on embedding of arbitrary finite metric spaces. Moreover, as described above, with some additional ideas they allow to get far stronger results.The results presented in this paper55This paper is a full version based on the conference papers: (Abraham et al., 2005, 2006, 2008) [1,2,5].have been the basis for further developments both within the field of metric embedding and in other areas such as graph theory, distributed computing and algorithms. We present a comprehensive study of the notions and concepts introduced here and provide additional extensions, related results and some examples of algorithmic applications.

This is the second paper in a series. In part I we developed deformation theory of objects in homotopy and derived categories of DG categories. Here we extend these (derived) deformation functors to an appropriate bicategory of artinian DG algebras and prove that these extended functors are pro-representable in a strong sense., Comment: Alexander Efimov is a new co-author of this paper. New material was added: A_{\infty}-structures, Maurer-Cartan theory for A_{\infty}-algebras. This allows us to strengthen our main results on the pro-representability of pseudo-functors coDEF_{-} and DEF_{-}. We also obtain an equivalence between homotopy and derived deformation functors under weaker hypotheses

The classical Eckmann–Hilton argument shows that two monoid structures on a set, such that one is a homomorphism for the other, coincide and, moreover, the resulting monoid is commutative. This argument immediately gives a proof of the commutativity of the higher homotopy groups. A reformulation of this argument in the language of higher categories is: suppose we have a one object, one arrow 2-category, then its Hom-set is a commutative monoid. A similar argument due to A. Joyal and R. Street shows that a one object, one arrow tricategory is ‘the same’ as a braided monoidal category. In this paper we begin to investigate how one can extend this argument to arbitrary dimension. We provide a simple categorical scheme which allows us to formalise the Eckmann–Hilton type argument in terms of the calculation of left Kan extensions in an appropriate 2-category. Then we apply this scheme to the case of n-operads in the author's sense and classical symmetric operads. We demonstrate that there exists a functor of symmetrisation Sym n from a certain subcategory of n-operads to the category of symmetric operads such that the category of one object, one arrow, … , one ( n − 1 ) -arrow algebras of A is isomorphic to the category of algebras of Sym n ( A ) . Under some mild conditions, we present an explicit formula for Sym n ( A ) which involves taking the colimit over a remarkable categorical symmetric operad. We will consider some applications of the methods developed to the theory of n-fold loop spaces in the second paper of this series.

This is the second paper of a series (started by Braverman and Finkelberg, 2010 [2] ) 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 G = SL ( n ) our conjectures can be derived from Nakajima (2009) [12] .

Andrews' spt-function can be written as the difference between the second symmetrized crank and rank moment functions. Using the machinery of Bailey pairs a combinatorial interpretation is given for the difference between higher order symmetrized crank and rank moment functions. This implies an inequality between crank and rank moments that was only know previously for sufficiently large n and fixed order. This combinatorial interpretation is in terms of a weighted sum of partitions. A number of congruences for higher order spt-functions are derived., Comment: 21 pages (previous version was 19 pages), added reference to Andrews and Rose's recent paper, MacMahon's paper and OEIS, changed some wording

This is the third paper in a series. In part I we developed a deformation theory of objects in homotopy and derived categories of DG categories. Here we show how this theory can be used to study deformations of objects in homotopy and derived categories of abelian categories. Then we consider examples from (noncommutative) algebraic geometry. In particular, we study noncommutative Grassmanians that are true noncommutative moduli spaces of structure sheaves of projective subspaces in projective spaces., Alexander Efimov is a new co-author of this paper. Besides some minor changes, a new part (part 3) about noncommutative Grassmanians was added

Let M be a smooth and compact moduli space of stable coherent sheaves on a projective surface S with an effective (or trivial) anti-canonical line bundle. We find generators for the cohomology ring of M, with integral coefficients. When S is simply connected and a universal sheaf E exists over SxM, then its class [E] admits a Kunneth decomposition as a class in the tensor product of the topological K-rings K(S) and K(M). The generators are the Chern classes of the Kunneth factors of [E] in K(M). The general case is similar, Comment: v3: Latex, 27 pages. Final version, to appear in Advances in Math. The proof of Lemma 21 is corrected and several other minor changes have been made. v2: Latex, 26 pages. The paper was split. The new version is a rewrite of the first three sections of version 1. The omitted results, about the monodromy of Hilbert schemes of point on a K3 surface, constitute part of the new paper arXiv:math.AG/0601304. v1: Latex, 53 pages

In this paper, the first of a series of two, we continue the study of higher index theory for expanders. We prove that if a sequence of graphs is an expander and the girth of the graphs tends to infinity, then the coarse Baum-Connes assembly map is injective, but not surjective, for the associated metric space $X$. Expanders with this girth property are a necessary ingredient in the construction of the so-called `Gromov monster' groups that (coarsely) contain expanders in their Cayley graphs. We use this connection to show that the Baum-Connes assembly map with certain coefficients is injective but not surjective for these groups. Using the results of the second paper in this series, we also show that the maximal Baum-Cones assembly map with these coefficients is an isomorphism., Comment: 38 pages

In this paper, we interpret the Gindikin–Karpelevich formula and the Casselman–Shalika formula as sums over Kashiwara–Lusztigʼs canonical bases, generalizing the results of Bump and Nakasuji (2010) [7] to arbitrary split reductive groups. We also rewrite formulas for spherical vectors and zonal spherical functions in terms of canonical bases. In a subsequent paper Kim and Lee (preprint) [14], we will generalize these formulas to p-adic affine Kac–Moody groups.

We extend the calculus of relations to embed a regular category A into a family of pseudo-abelian tensor categories T(A,d) depending on a degree function d. Under the condition that all objects of A have only finitely many subobjects, our main results are as follows: 1. Let N be the maximal proper tensor ideal of T(A,d). We show that T(A,d)/N is semisimple provided that A is exact and Mal'cev. Thereby, we produce many new semisimple, hence abelian, tensor categories. 2. Using lattice theory, we give a simple numerical criterion for the vanishing of N. 3. We determine all degree functions for which T(A,d) is Tannakian. As a result, we are able to interpolate the representation categories of many series of profinite groups such as the symmetric groups S_n, the hyperoctahedral groups S_n\semidir Z_2^n, or the general linear groups GL(n,F_q) over a fixed finite field. This paper generalizes work of Deligne, who first constructed the interpolating category for the symmetric groups S_n. It also extends (and provides proofs for) a previous paper math.CT/0605126 on the special case of abelian categories., Comment: v1: 52 pages; v2: 52 pages, proof of Lemma 7.2 fixed, otherwise minor changes

On page 335 in his lost notebook, Ramanujan recorded without proofs two identities involving finite trigonometric sums and doubly infinite series of Bessel functions. In each case, there are three possible interpretations for the double series. In an earlier paper, two of the present authors proved the first identity under one possible interpretation. In the present paper, the second identity is proved under a similar interpretation, with one additional assumption. Moreover, under a second interpretation, entirely different proofs of both identities, depending on weighted (or twisted) divisor sums, are offered. The two identities are intimately connected with the classical circle and divisor problems, respectively.

The aim of this paper is to introduce and study Lie algebras and Lie groups over noncommutative rings. For any Lie algebra $\gg$ sitting inside an associative algebra $A$ and any associative algebra $\FF$ we introduce and study the algebra $(\gg,A)(\FF)$, which is the Lie subalgebra of $\FF \otimes A$ generated by $\FF \otimes \gg$. In many examples $A$ is the universal enveloping algebra of $\gg$. Our description of the algebra $(\gg,A)(\FF)$ has a striking resemblance to the commutator expansions of $\FF$ used by M. Kapranov in his approach to noncommutative geometry. To each algebra $(\gg, A)(\FF)$ we associate a ``noncommutative algebraic'' group which naturally acts on $(\gg,A)(\FF)$ by conjugations and conclude the paper with some examples of such groups., Comment: Introduction is improved and some typos corrected. To appear in "Advances"

This paper deals with a “naive” way of generalizing Kazhdan's property (T) to C∗-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 C∗-Hilbert modules. Namely, we prove that a separable unital C∗-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.

Katzarkov has proposed a generalization of Kontsevich's mirror symmetry conjecture, covering some varieties of general type. Seidel \cite{Se} has proved a version of this conjecture in the simplest case of the genus two curve. Basing on the paper of Seidel, we prove the conjecture (in the same version) for curves of genus $g\geq 3,$ relating the Fukaya category of a genus $g$ curve to the category of Landau-Ginzburg branes on a certain singular surface. We also prove a kind of reconstruction theorem for hypersurface singularities. Namely, formal type of hypersurface singularity (i.e. a formal power series up to a formal change of variables) can be reconstructed, with some technical assumptions, from its D$(\Z/2)$-G category of Landau-Ginzburg branes. The precise statement is Theorem 1.2., Comment: 42 pages, 6 figures; Appendix and references added, minor corrections; Introduction slightly rewritten; v4: the exposition of Fukaya categories is rewritten, the paper is reorganised

Theorem 3.2 (on page 773) states that the set (Xτ )Q of characters which satisfy the Bowditch Q-conditions is an open subset of the relative character variety Xτ , and Theorem 2.3 (on page 768) states that the mapping class group MCG acts properly discontinuously on (Xτ )Q. We claimed on page 768 (immediately after Proposition 2.4) that it is easy to deduce from Proposition 2.4 that (Xτ )Q (resp. XQ) is the largest open subset of Xτ (resp. X ) on which MCG acts properly discontinuously. This claim was also stated in the abstract and introduction, and the announcement [1]. However, Proposition 2.4 does not imply what we have claimed. Hence the question of what the largest such open subsets can be remains open. Since this claim was not used in the proof of any lemma, proposition or theorem in the paper, this error does not affect the validity of the results in the paper stated in the lemmas, propositions and theorems. The authors would like to thank Yair Minsky and Bill Goldman for drawing their attention to the error.

We study an operator norm localization property and its applications to the coarse Novikov conjecture in operator K-theory. A metric space X is said to have operator norm localization property if there exists a positive number c such that for every r>0, there is R>0 for which, if m is a positive locally finite Borel measure on X, H is a separable infinite dimensional Hilbert space and T is a bounded linear operator acting on L^2(X,m) with propagation r, then there exists an unit vector v satisfying with support of diameter at most R and such that |Tv| is larger or equal than c|T|. If X has finite asymptotic dimension, then X has operator norm localization property. In this paper, we introduce a sufficient geometric condition for the operator norm localization property. This is used to give many examples of finitely generated groups with infinite asymptotic dimension and the operator norm localization property. We also show that any sequence of expanding graphs does not possess the operator norm localization property.

A subgroup of a Kac-Moody group is called bounded if it is contained in the intersection of two finite type parabolic subgroups of opposite signs. In this paper, we study the isomorphisms between Kac-Moody groups over arbitrary fields of cardinality at least 4, which preserve the set of bounded subgroups. We show that such an isomorphism between two such Kac-Moody groups induces an isomorphism between the respective twin root data of these groups. As a consequence, we obtain the solution of the isomorphism problem for Kac-Moody groups over finite fields of cardinality at least 4., 25 pages, 0 figure

In this paper we discuss the “Factorization phenomenon” which occurs when a representation of a Lie algebra is restricted to a subalgebra, and the result factors into a tensor product of smaller representations of the subalgebra. We analyze this phenomenon for symmetrizable Kac–Moody algebras (including finite-dimensional, semi-simple Lie algebras). We present a few factorization results for a general embedding of a symmetrizable Kac–Moody algebra into another and provide an algebraic explanation for such a phenomenon using Spin construction. We also give some application of these results for semi-simple, finite-dimensional Lie algebras. We extend the notion of Spin functor from finite-dimensional to symmetrizable Kac–Moody algebras, which requires a very delicate treatment. We introduce a certain category of orthogonal g -representations for which, surprisingly, the Spin functor gives a g -representation in Bernstein–Gelfand–Gelfand category O . Also, for an integrable representation, Spin produces an integrable representation. We give the formula for the character of Spin representation for the above category and work out the factorization results for an embedding of a finite-dimensional, semi-simple Lie algebra into its untwisted affine Lie algebra. Finally, we discuss the classification of those representations for which Spin is irreducible.

In this paper, we prove an addition formula for the Jacobian theta function using the theory of elliptic functions. It turns out to be a fundamental identity in the theory of theta functions and elliptic function, and unifies many important results about theta functions and elliptic functions. From this identity we can derive the Ramanujan cubic theta function identity, Winquist's identity, a theta function identities with five parameters, and many other interesting theta function identities; and all of which are as striking as Winquist's identity. This identity allows us to give a new proof of the addition formula for the Weierstrass sigma function. A new identity about the Ramanujan cubic elliptic function is given. The proofs are self contained and elementary.

To any algebraic variety X and closed 2-form ω on X, we associate the “symplectic action functional”T(ω) which is a function on the formal loop space LX introduced by the authors earlier. The correspondence ω→T(ω) can be seen as a version of the Radon transform. We give a characterization of the functions of the form T(ω) in terms of factorizability (infinitesimal analog of additivity in holomorphic pairs of pants) as well as in terms of vertex operator algebras.These results will be used in the subsequent paper which will relate the gerbe of chiral differential operators on X (whose lien is the sheaf of closed 2-forms) and the determinantal gerbe of the tangent bundle of LX (whose lien is the sheaf of invertible functions on LX). On the level of liens this relation associates to a closed 2-form ω the invertible function expT(ω).

In any Coxeter group, the set of elements whose principal order ideals are boolean forms a simplicial poset under the Bruhat order. This simplicial poset defines a cell complex, called the boolean complex. In this paper it is shown that, for any Coxeter system of rank n, the boolean complex is homotopy equivalent to a wedge of (n-1)-dimensional spheres. The number of such spheres can be computed recursively from the unlabeled Coxeter graph, and defines a new graph invariant called the boolean number. Specific calculations of the boolean number are given for all finite and affine irreducible Coxeter systems, as well as for systems with graphs that are disconnected, complete, or stars. One implication of these results is that the boolean complex is contractible if and only if a generator of the Coxeter system is in the center of the group. of these results is that the boolean complex is contractible if and only if a generator of the Coxeter system is in the center of the group., Comment: final version, to appear in Advances in Mathematics

Let L be a lattice (that is, a Z-module of finite rank), and let L=P(L) denote the family of convex polytopes with vertices in L; here, convexity refers to the underlying rational vector space V=Q⊗L. In this paper it is shown that any valuation on L satisfies the inclusion–exclusion principle, in the strong sense that appropriate extension properties of the valuation hold. Indeed, the core result is that the class of a lattice polytope in the abstract group L=P(L) for valuations on L can be identified with its characteristic function in V. In fact, the same arguments are shown to apply to P(M), when M is a module of finite rank over an ordered ring, and more generally to appropriate families of (not necessarily bounded) polyhedra.

Consider N = n 1 + n 2 + ⋯ + n p non-intersecting Brownian motions on the real line, starting from the origin at t = 0 , with n i particles forced to reach p distinct target points β i at time t = 1 , with β 1 β 2 ⋯ β p . This can be viewed as a diffusion process in a sector of R N . This work shows that the transition probability, that is the probability for the particles to pass through windows E ˜ k at times t k , satisfies, in a new set of variables, a non-linear PDE which can be expressed as a near-Wronskian; that is a determinant of a matrix of size p + 1 , with each row being a derivative of the previous, except for the last column. It is an interesting open question to understand those equations from a more probabilistic point of view. As an application of these equations, let the number of particles forced to the extreme points β 1 and β p tend to infinity; keep the number of particles forced to intermediate points fixed (inliers), but let the target points themselves go to infinity according to a proper scale. A new critical process appears at the point of bifurcation, where the bulk of the particles forced to − n depart from those going to n . These statistical fluctuations near that point of bifurcation are specified by a kernel, which is a rational perturbation of the Pearcey kernel. This work also shows that such equations are an essential tool in obtaining certain asymptotic results. Finally, the paper contains a conjecture.

We consider a Dirac-type operator D P on a vector bundle V over a compact Riemannian manifold ( M , g ) with a non-empty boundary. The operator D P is specified by a boundary condition P ( u | ∂ M ) = 0 where P is a projector which may be a non-local, i.e., a pseudodifferential operator. We assume the existence of a chirality operator which decomposes L 2 ( M , V ) into two orthogonal subspaces X + ⊕ X − . Under certain conditions, the operator D P restricted to X + and X − defines a pair of Fredholm operators which maps X + → X − and X − → X + correspondingly, giving rise to a superstructure on V. In this paper we consider the questions of determining the index of D P and the reconstruction of ( M , g ) , V and D P from the boundary data on ∂M. The data used is either the Cauchy data, i.e., the restrictions to ∂ M × R + of the solutions to the hyperbolic Dirac equation, or the boundary spectral data, i.e., the set of the eigenvalues and the boundary values of the eigenfunctions of D P . We obtain formulae for the index and prove uniqueness results for the inverse boundary value problems. We apply the obtained results to the classical Dirac-type operator in M × C 4 , M ⊂ R 3 .

We prove that a backward orbit with bounded Kobayashi step for a hyperbolic or strongly elliptic holomorphic self-map of a bounded strongly convex domain in the d-dimensional complex Euclidean space necessarily converges to a boundary fixed point, generalizing previous results obtained by Poggi-Corradini in the unit disk and by Ostapyuk in the unit ball., We have found a gap in the proofs of Lemmas 2.2 and 2.5 of the published version of this paper. Here we add to the previous version a note where we fill these gaps, giving a proof of the main results using different arguments. For the benefit of the reader, we report the complete proof of the main theorem, including the needed background results

In this paper, we establish the Gevrey regularity of solutions for a class of degenerate Monge–Ampère equations in the plane. Under the assumptions that one principal entry of the Hessian is strictly positive and the coefficient of the equation is degenerate with appropriately finite type degeneracy, we prove that the solution of the degenerate Monge–Ampère equation will be smooth in Gevrey classes.

Here we study an amazing phenomenon discovered by Newhouse [S. Newhouse, Non-density of Axiom A(a) on S 2 , in: Proc. Sympos. Pure Math., vol. 14, Amer. Math. Soc., 1970, pp. 191–202; S. Newhouse, Diffeomorphisms with infinitely many sinks, Topology 13 (1974) 9–18; S. Newhouse, The abundance of wild hyperbolic sets and nonsmooth stable sets of diffeomorphisms, Publ. Math. Inst. Hautes Etudes Sci. 50 (1979) 101–151]. It turns out that in the space of C r smooth diffeomorphisms Diff r ( M ) of a compact surface M there is an open set U such that a Baire generic diffeomorphism f ∈ U has infinitely many coexisting sinks. In this paper we make a step towards understanding “how often does a surface diffeomorphism have infinitely many sinks.” Our main result roughly says that with probability one for any positive D a surface diffeomorphism has only finitely many localized sinks either of cyclicity bounded by D or those whose period is relatively large compared to its cyclicity. It verifies a particular case of Palis' Conjecture saying that even though diffeomorphisms with infinitely many coexisting sinks are Baire generic, they have probability zero. One of the key points of the proof is an application of Newton Interpolation Polynomials to study the dynamics initiated in [V. Kaloshin, B. Hunt, A stretched exponential bound on the rate of growth of the number of periodic points for prevalent diffeomorphisms I, Ann. of Math., in press, 92 pp.; V. Kaloshin, A stretched exponential bound on the rate of growth of the number of periodic points for prevalent diffeomorphisms II, preprint, 85 pp.].

A separating algebra is, roughly speaking, a subalgebra of the ring of invariants whose elements distinguish between any two orbits that can be distinguished using invariants. In this paper, we introduce a geometric notion of separating algebra. This allows us to prove that only groups generated by reflections may have polynomial separating algebras, and only groups generated by bireflections may have complete intersection separating algebras.

Given points p and q in the plane, we are interested in separating them by two curves C 1 and C 2 such that every point of C 1 has equal distance to p and to C 2 , and every point of C 2 has equal distance to C 1 and to q. We show by elementary geometric means that such C 1 and C 2 exist and are unique. Moreover, for p = ( 0 , 1 ) and q = ( 0 , − 1 ) , C 1 is the graph of a function f : R → R , C 2 is the graph of −f, and f is convex and analytic (i.e., given by a convergent power series at a neighborhood of every point). We conjecture that f is not expressible by elementary functions and, in particular, not algebraic. We provide an algorithm that, given x ∈ R and e > 0 , computes an approximation to f ( x ) with error at most e in time polynomial in log 1 + | x | e . The separation of two points by two “trisector” curves considered here is a special (two-point) case of a new kind of Voronoi diagram, which we call the zone diagram and which we investigate in a companion paper.

In this paper we study the group $A_0(X)$ of zero dimensional cycles of degree 0 modulo rational equivalence on a projective homogeneous algebraic variety $X$. To do this we translate rational equivalence of 0-cycles on a projective variety into R-equivalence on symmetric powers of the variety. For certain homogeneous varieties, we then relate these symmetric powers to moduli spaces of \'etale subalgebras of central simple algebras which we construct. This allows us to show $A_0(X) = 0$ for certain classes of homogeneous varieties, extending previous results of Swan / Karpenko, of Merkurjev, and of Panin., Comment: Significant revisions made to simplify exposition, also includes results for symplectic involution varieties. Main arguments now rely on Hilbert schemes of points and are valid with only mild characteristic assumptions. 32 pages

The main result of [Kostant, Invent. Math. 158 (1) (2004) 181–226, arXiv:math.GR/0309232] shows that a sequence of complexes associated with the MacDonald identities gives a sequence of universal characters which extends and generalises some representations of the exceptional series given in [Deligne, C. R. Acad. Sci. Paris Ser. Math. 322(4) (1996) 321–326]. This proves the conjectures in [Macfarlane, Pfeiffer, J. Phys. A 36 (2003) 230 5–2317. arxiv.math-ph/0208014]. In this paper, we use a twisted version of this definition to define a sequence of complexes for the first row of the Freudenthal magic square. We also calculate the dimensions of these characters for the special linear groups and for the symplectic groups. This gives two new explicit expressions for the D’Arcais polynomials.

A W-algebra is an associative algebra constructed from a semisimple Lie algebra and its nilpotent element. This paper concentrates on the study of 1-dimensional representations of these algebras. Under some conditions on a nilpotent element (satisfied by all rigid elements) we obtain a criterium for a finite dimensional module to have dimension 1. It is stated in terms of the Brundan-Goodwin-Kleshchev highest weight theory. This criterium allows to compute highest weights for certain completely prime primitive ideals in universal enveloping algebras. We make an explicit computation in a special case in type $E_8$. Our second principal result is a version of a parabolic induction for W-algebras. In this case, the parabolic induction is an exact functor between the categories of finite dimensional modules for two different W-algebras. The most important feature of the functor is that it preserves dimensions. In particular, it preserves one-dimensional representations. A closely related result was obtained previously by Premet. We also establish some other properties of the parabolic induction functor., Comment: 31 pages, v2 36 pages, 4 new subsections added, v3 38 pages few gaps fixed, v4 minor changes, v5 references added

In this paper the representation theory of 2-groups in 2-categories is considered, focusing the attention on the 2-category Rep 2 Mat K ( G ) of representations of a 2-group G in (a version of) Kapranov and Voevodsky's 2-category of 2-vector spaces over a field K. The set of equivalence classes of such representations is computed in terms of the invariants π 0 ( G ) , π 1 ( G ) and [ α ] ∈ H 3 ( π 0 ( G ) , π 1 ( G ) ) classifying G, and the categories of intertwiners are described in terms of categories of vector bundles endowed with a projective action. In particular, it is shown that the monoidal category of finite dimensional linear representations (more generally, the category of [ z ] -projective representations, for any given cohomology class [ z ] ∈ H 2 ( π 0 ( G ) , K ∗ ) ) of the first homotopy group π 0 ( G ) as well as its category of representations on finite sets both live in Rep 2 Mat K ( G ) , the first as the monoidal category of endomorphisms of the trivial representation (more generally, as the category of intertwiners between suitable 1-dimensional representations) and the second as a non-full subcategory of the homotopy category of Rep 2 Mat K ( G ) .

In this paper we present a new entropy penalization problem and we discuss its relations with approximate solutions of Hamilton–Jacobi equations, the convergence of associated discrete schemes, as well as several applications, such as: a generalization of the Hopf–Cole transformation which converts non-linear Hamilton–Jacobi equations into linear evolution equations, the study of fixed point problems, approximation of certain linear evolution equations, and the construction of entropy penalized Mather measures.

As is well known, h-vectors of simple (or simplicial) convex polytopes are characterized. In fact, those h-vectors must satisfy Dehn-Sommerville equations and some other inequalities. Simple convex polytopes determine Gorenstein* simplicial posets and h-vectors are defined for simplicial posets. It is known that h-vectors of Gorenstein* simplicial posets must satisfy Dehn-Sommerville equations and that every component in the h-vectors must be non-negative. In this paper we will show that h-vectors of Gorenstein* simplicial posets must satisfy one more subtle condition conjectured by R. Stanley and complete characterization of those h-vectors. Our proof is purely algebraic but the idea of the proof stems from topology., 12 pages

A special case of Haimanʼs identity [M. Haiman, Vanishing theorems and character formulas for the Hilbert scheme of points in the plane, Invent. Math. 149 (2002) 371–407] for the character of the quotient ring of diagonal coinvariants under the diagonal action of the symmetric group yields a formula for the bigraded Hilbert series as a sum of rational functions in q,t. In this paper we show how a summation identity of Garsia and Zabrocki for Macdonald polynomial Pieri coefficients can be used to transform Haimanʼs formula for the Hilbert series into an explicit polynomial in q,t with integer coefficients. We also provide an equivalent formula for the Hilbert series as the constant term in a multivariate Laurent series.

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 Poincare map contains no 2 × 2 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 d ⩾ 2 , 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.

The operad Lie can be constructed as the operad of primitives Prim As from the operad As of associative algebras. This is reflected by the theorems of Friedrichs, Poincare'-Birkhoff-Witt and Cartier-Milnor-Moore. We replace As by families of free operads P, which include the operad Mag freely generated by a noncommutative non-associative binary operation and the operad of Stasheff polytopes. We obtain Poincare'-Birkhoff-Witt type theorems and collect information about the operads Prim P, e.g. in terms of characteristic functions., Comment: 25pages, uses elsart.cls, paper extracted from habilitation thesis (math.RA/0407074), revised (more details in sections 2,3,4, and 6)

The homotopy limit problem for Karoubiʼs Hermitian K-theory (Karoubi, 1980) [26] was posed by Thomason (1983) [44] . There is a canonical map from algebraic Hermitian K-theory to the Z / 2 -homotopy fixed points of algebraic K-theory. The problem asks, roughly, how close this map is to being an isomorphism, specifically after completion at 2. In this paper, we solve this problem completely for fields of characteristic 0 (Theorems 16, 20). We show that the 2-completed map is an isomorphism for fields F of characteristic 0 which satisfy cd 2 ( F [ i ] ) ∞ , but not in general.

We define and investigate extension groups in the context of Arakelov geometry. The “arithmetic extension groups”ExtˆXi(F,G) we introduce are extensions by groups of analytic types of the usual extension groups ExtXi(F,G) attached to OX-modules F and G over an arithmetic scheme X. In this paper, we focus on the first arithmetic extension group ExtˆX1(F,G) – the elements of which may be described in terms of admissible short exact sequences of hermitian vector bundles over X – and we especially consider the case when X is an “arithmetic curve”, namely the spectrum SpecOK of the ring of integers in some number field K. Then the study of arithmetic extensions over X is related to old and new problems concerning lattices and the geometry of numbers. Namely, for any two hermitian vector bundles F¯ and G¯ over X:=SpecOK, we attach a logarithmic sizesF¯,G¯(α) to any element α of ExtˆX1(F,G), and we give an upper bound on sF¯,G¯(α) in terms of slope invariants of F¯ and G¯. We further illustrate this notion by relating the sizes of restrictions to points in P1(Z) of the universal extension over PZ1 to the geometry of PSL2(Z) acting on Poincare's upper half-plane, and by deducing some quantitative results in reduction theory from our previous upper bound on sizes. Finally, we investigate the behaviour of size by base change (i.e., under extension of the ground field K to a larger number field K′): when the base field K is Q, we establish that the size, which cannot increase under base change, is actually invariant when the field K′ is an abelian extension of K, or when F¯∨⊗G¯ is a direct sum of root lattices and of lattices of Voronoi's first kind. The appendices contain results concerning extensions in categories of sheaves on ringed spaces, and lattices of Voronoi's first kind which might also be of independent interest.

Let R be a finite-dimensional torsion-free special λ-ring. In this paper we generalize the results in Dress and Siebeneicher (Adv. in Math. 70 (1988) 89; 78 (1989) 1) by constructing R-analogue Ω R (G) of the Burnside ring of profinite groups Ω (G) . In particular, we remark that the (Grothendieck) Lie-module denominator identity of free Lie algebras in Oh (Necklace rings and logarithmic functions, preprint, KIAS, 2003) is closely related to the canonical isomorphism between Ω R (G) and Grothendieck's ring of formal power series with coefficients in R and constant term 1.

In this paper we consider deformations of an algebroid stack on an etale groupoid. We construct a differential graded Lie algebra (DGLA) which controls this deformation theory. In the case when the algebroid is a twisted form of functions we show that this DGLA is quasiisomorphic to the twist of the DGLA of Hochschild cochains on the algebra of functions on the groupoid by the characteristic class of the corresponding gerbe.

We introduce the_Brauer loop scheme_ E := {M in M_N(C) : M\cp M = 0}, where \cp is a certain degeneration of the ordinary matrix product. Its components of top dimension, floor(N^2/2), correspond to involutions \pi in S_N having one or no fixed points. In the case N even, this scheme contains the upper-upper scheme from [Knutson '04] as a union of (N/2)! of its components. One of those is a degeneration of the_commuting variety_ of pairs of commuting matrices. The_Brauer loop model_ is a quantum integrable stochastic process introduced in [de Gier--Nienhuis '04], and some of the entries of its Perron-Frobenius eigenvector were observed (conjecturally) to match the degrees of the components of the upper-upper scheme. We extend this, with proof, to_all_ the entries: they are the degrees of the components of the Brauer loop scheme. Our proof of this follows the program outlined in [Di Francesco--Zinn-Justin '04]. In that paper, the entries of the Perron-Frobenius eigenvector were generalized from numbers to polynomials, which allowed them to be calculated inductively using divided difference operators. We relate these polynomials to the multidegrees of the components of the Brauer loop scheme, defined using an evident torus action on E. In particular, we obtain a formula for the degree of the commuting variety, previously calculated up to 4x4 matrices., Comment: 31 pages, 4 figures; v2 has tiny edits and an extra circle action

International audience; In this article, we generalize a localization theorem of Lovasz and Simonovits [Random walks in a convex body and an improved volume algorithm, Random Struct. Algorithms 4-4 (1993) 359-412] which is an important tool to prove dimension-free functional inequalities for log-concave measures. In a previous paper [Fradelizi and Guedon, The extreme points of subsets of s-concave probabilities and a geometric localization theorem, Discrete Comput. Geom. 31 (2004) 327-335], we proved that the localization may be deduced from a suitable application of Krein-Milman's theorem to a subset of log-concave probabilities satisfying one linear constraint and from the determination of the extreme points of its convex hull. Here, we generalize this result to more constraints, give some necessary conditions satisfied by such extreme points and explain how it may be understood as a generalized localization theorem. Finally, using this new localization theorem, we solve an open question on the comparison of the volume of sections of non-symmetric convex bodies in R-n by hyperplanes. A surprising feature of the result is that the extremal case in this geometric inequality is reached by an unusual convex set that we manage to identify. (C) 2005 Elsevier Inc. All rights reserved.

In this paper, we prove that in small parameter regions, arbitrary unitary matrix integrals converge in the large $N$ limit and match their formal expansion. Secondly we give a combinatorial model for our matrix integral asymptotics and investigate examples related to free probability and the HCIZ integral. Our convergence result also leads us to new results of smoothness of microstates. We finally generalize our approach to integrals over the othogonal group., 41 pages, important modifications, new section about orthogonal integrals

We study universality problems in Banach space theory. We show that if A is an analytic class, in the Effros–Borel structure of subspaces of C ( [ 0 , 1 ] ) , of non-universal separable Banach spaces, then there exists a non-universal separable Banach space Y, with a Schauder basis, that contains isomorphs of each member of A with the bounded approximation property. The proof is based on the amalgamation technique of a class C of separable Banach spaces, introduced in the paper. We show, among others, that there exists a separable Banach space R not containing L 1 ( 0 , 1 ) such that the indices β and r N D are unbounded on the set of Baire-1 elements of the ball of the double dual R ∗ ∗ of R. This answers two questions of H.P. Rosenthal. We also introduce the concept of a strongly bounded class of separable Banach spaces. A class C of separable Banach spaces is strongly bounded if for every analytic subset A of C there exists Y ∈ C that contains all members of A up to isomorphism. We show that several natural classes of separable Banach spaces are strongly bounded, among them the class of non-universal spaces with a Schauder basis, the class of reflexive spaces with a Schauder basis, the class of spaces with a shrinking Schauder basis and the class of spaces with Schauder basis not containing a minimal Banach space X.

Two new mock theta functions of the sixth order are defined. The main theorem in this paper (Theorem 1.1) provides four transformation formulas relating the new mock theta functions with Ramanujan's mock theta functions of the sixth order. Two further representations of the new mock theta functions are established. Lastly, a hitherto unproved entry from Ramanujan's lost notebook related to sixth order mock theta functions is proved.

Let G be a profinite group and q an indeterminate. In this paper, we introduce and study a q-analog of the Mobius function and the cyclotomic identity arising from the lattice of open subgroups of G. When q is any integer, we show that they have close connections with the functors W G q , Nr G q , and Nr ˆ G q introduced in [Y.-T. Oh, q-Deformation of Witt–Burnside rings, Math. Z. 257 (2007) 151–191]. In particular, we interpret the multiplicative property of the inverse of the table of marks and the Mobius function of G as a composition property of certain functors. Classification of W G q , Nr G q , and Nr ˆ G q up to strict natural isomorphism as q varies over the set of integers and its application will be dealt with, too.