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In the authors' first paper, Beurling-Rudin-Korenbljum type characterization of the closed ideals in a certain algebra of holomorphic functions was used to describe the lattice of invariant subspaces of the shift plus a complex Volterra operator. Current work is an extension of the previous work and it describes the lattice of invariant subspaces of the shift plus a positive integer multiple of the complex Volterra operator on the Hardy space. Our work was motivated by a paper by Ong who studied the real version of the same operator., We deleted a proposition and a corollary from section 4, and simplified the proof of the main theorem. **The article has been published in Archiv der Mathematik**

In this paper we classify, up to analytic isomorphism, the family of almost stretched Artinian complete intersection A=R/I with a given Hilbert function, in the case R is a power series ring with an arbitrary number of variables., 20 pages. This paper generalizes a previous version where the result was proven for a power series ring in two variables

International audience; In the context of Mather's theory of Lagrangian systems, we study the decomposition in chain-transitive classes of the Mather invariant sets. As an application, we prove, under appropriate hypotheses, the semi-continuity of the so-called Aubry set as a function of the Lagrangian. In the study of Lagrangian systems, John Mather introduced several invariant sets composed of globally minimizing extremals. He developed methods to construct several orbits undergoing interesting behaviors in phase space under some assumptions on these invariant sets, see [14]. In order to pursue this theory and to apply it on examples, it is necessary to have tools to describe precisely the invariant sets. At least two points of view can be adopted. One can study the invariant set from a purely topological point of view in the style of Conley as compact metric spaces with flows, and study their transitive components. One can also study these set from the point of view of action minimization, and decompose them in invariant subsets that have been called static classes. These points of view are very closely related, but each of them has specific features. For example, understanding the decomposition in static classes is necessary for the variational construction of interesting orbits, while the topological decomposition behaves well under perturbations. Our goal in the present paper is to explicit the links between these two decompositions. We explain that the topological decomposition is finer than the variational one, and that they coincide for most (but not all) systems. As an application, we prove a result of semi-continuity of the so-called Aubry set as a function of the Lagrangian, under certain non-degeneracy hypotheses. The semi-continuity of the Aubry set is a subtle problem, which has remained open for several years, until John Mather gave a counter example, see §18 in [16]. In the same paper, he also states without proof that semi-continuity holds under appropriate hypotheses. Our result extends the one of Mather. The methods we use are inspired from the recent work of Fathi, Figalli and Rifford, [9].

In this paper we present a p-adic algorithm to compute the zeta function of a nondegenerate curve over a finite field using Monsky-Washnitzer cohomology. The paper vastly generalizes previous work since all known cases, e.g. hyperelliptic, superelliptic and C_{ab} curves, can be transformed to fit the nondegenerate case. For curves with a fixed Newton polytope, the property of being nondegenerate is generic, so that the algorithm works for almost all curves with given Newton polytope. For a genus g curve over F_{p^n}, the expected running time is O(n^3g^6 + n^2g^{6.5}), whereas the space complexity amounts to O(n^3g^4), assuming p is fixed., 41 pages

Fomin and Kirillov initiated a line of research into the realization of the cohomology and K-theory of generalized flag varieties G/B as commutative subalgebras of certain noncommutative algebras. This approach has several advantages, which we discuss. This paper contains the most comprehensive result in a series of papers related to the mentioned line of research. More precisely, we give a model for the T -equivariant K-theory of a generalized flag variety KT (G/B) in terms of a certain braided Hopf algebra called the Nichols-Woronowicz algebra. Our model is based on the Chevalley-type multiplication formula for KT (G/B) due to the first author and Postnikov; this formula is stated using certain operators defined in terms of so-called alcove paths (and the corresponding affine Weyl group). Our model is derived using a type-independent and concise approach. Dedicated to Professor Kenji Ueno on the occasion of his sixtieth birthday

In foundational papers, Gross, Zagier, and Kohnen established two formulas for arithmetic intersection numbers of certain Heegner divisors on integral models of modular curves. In [GZ1], only one imaginary quadratic discriminant plays a role. In [GZ2] and [GKZ], two quadratic discriminants play a role. In this paper we generalize the two-discriminant formula from the modular curves X0(N) to certain Shimura curves defined over Q. Our intersection formula was stated in [Ro], but the proof was only outlined there. Independently, the general formula was given, in a weaker and less explicit form, in [Ke2]; there it was proved completely. This paper is thus a synthesis of parts of [Ro] and [Ke2]. The intersection multiplicities computed here were used in [Ku] to derive a relation between height pairings and special values of the derivatives of certain Eisenstein series. We note also that Zhang [Zh] has generalized all of [GZ1] from ground field Q to general totally real ground fields F , working with general Shimura curves. So we certainly expect that all of [GKZ] should generalize similarly. Our work here can be viewed as a step in this direction. We are happy to thank B. Gross and S. Kudla for their encouragement during the long period in which this work was done.

The main goal of the present paper is two-fold. First we extend the theory of toroidal embeddings introduced by Kempf, Knudsen, Mumford and Saint-Donat to the class of toroidal varieties with stratifications (which is the main body of the paper). Second we give a proof of the following weak factorization theorem as an application and illustration of the theory: A birational map between complete nonsingular varieties over an algebraically closed field K of characteristic zero is a composite of blow ups and blow downs with smooth centers. Another proof of the weak factorization theorem appeared in a joint paper with Abramovich, Karu and Matsuki (math.AG/9904135) In that paper the theorem is stated and proven in general for proper algebraic and analytic spaces., Comment: 69 pages, a revised version with conceptual simplifications. The present version is self-contained and it does not rely on weak factorization theorem for toric varieties

It is known that every semigroup of normal completely positive maps $P = {P_t: t\geq 0}$ of $B(H)$, satisfying $P_t(1) = 1$ for every $t\geq 0$, has a minimal dilation to an E_0-semigroup acting on $B(K)$ for some Hilbert space K containing H. The minimal dilation of P is unique up to conjugacy. In a previous paper a numerical index was introduced for semigroups of completely positive maps and it was shown that the index of P agrees with the index of its minimal dilation to an E_0-semigroup. However, no examples were discussed, and no computations were made. In this paper we calculate the index of a unital completely positive semigroup whose generator is a bounded operator $ L: B(H)\to B(H) $ in terms of natrual structures associated with the generator. This includes all unital CP semigroups acting on matrix algebras. We also show that the minimal dilation of the semigroup $P={\exp{tL}: t\geq 0}$ to an \esg\ is is cocycle conjugate to a CAR/CCR flow., Comment: 31 pp. AMS-TeX 2.0

Let g be a symmetrisable Kac-Moody algebra, and U_h(g) the corresponding quantum group. We showed in arXiv:1610.09744 and arXiv:1610.09741 that the braided quasi-Coxeter structure on integrable, category O representations of U_h(g) which underlies the R-matrix actions arising from the Levi subalgebras of U_h(g) and the quantum Weyl group action of the generalised braid group B_g can be transferred to integrable, category O representations of g. We prove in this paper that, up to unique equivalence, there is a unique such structure on the latter category with prescribed restriction functors, R--matrices, and local monodromies. This extends, simplifies and strengthens a similar result of the second author valid when g is semisimple, and is used in arXiv:1512.03041 to describe the monodromy of the rational Casimir connection of g in terms of the quantum Weyl group operators of U_h(g). Our main tool is a refinement of Enriquez's universal algebras, which is adapted to the PROP describing a Lie bialgebra graded by the non-negative roots of g., Expanded Introduction and Sec. 5 to discuss convolution product (5.11), cosimplicial structure on basis elements (5.13) and module structure on coinvariants (5.15). Minor revisions in Sec. 7.1 (gradings), 7.4 (deformation DY modules), 9.7 (exposition), 15.7 (Drinfeld double) and 15.15 (rigidity for diagrammatic KM algebras). Final version, to appear in Adv. Math. 81 pages

Macaulay's Inverse System gives an effective method to construct Artinian Gorenstein k-algebras. To date a general structure for Gorenstein k-algebras of any dimension (and codimension) is not understood. In this paper we extend Macaulay's correspondence characterizing the submodules of the divided power ring in one-to-one correspondence with Gorenstein d-dimensional k-algebras. We discuss effective methods for constructing Gorenstein graded rings. Several examples illustrating our results are given., 19 pages, to appear in Advances in Mathematics

We generalize the nonabelian Poincare duality theorems of Salvatore in [Sal01] and Lurie in [Lur09] to the case of not necessarily grouplike E_n-algebras (in the category of spaces). We define a stabilization procedure based on McDuff's "brining points in from infinity" maps from [McD75]. For open connected parallelizable n-manifolds, we prove that, after stabilizing, the topological chiral homology of M with coefficients in an E_n-algebra A, is homology equivalent to Map^c(M,B^n A), the space of compactly supported maps to the n-fold classifying space of A. The two models of topological chiral homology used in this paper are Andrade's model from [And10] and Salvatore's from [Sal01]., 33 pages, 5 figures. arXiv admin note: text overlap with arXiv:1210.7377

If a knot K bounds a genus one Seifert surface F in the 3-sphere and F contains an essential simple closed curve alpha that has induced framing 0 and is smoothly slice, then K is smoothly slice. Conjecturally, the converse holds. It is known that if K is slice, then there are strong constraints on the algebraic concordance class of such alpha, and it was thought that these constraints might imply that alpha is at least algebraically slice. We present a counterexample; in the process we answer negatively a question of Cooper and relate the result to a problem of Kauffman. Results of this paper depend on the interplay between the Casson-Gordon invariants of K and algebraic invariants of alpha., 17 pages, 1 figure. Typographical corrections

By two well-known results, one of Ax, one of Lubotzky and van den Dries, a profinite group is projective iff it is isomorphic to the absolute Galois group of a pseudo-algebraically closed field. This paper gives an analogous characterization of relatively projective profinite groups as absolute Galois groups of regularly closed fields.

In a recent paper Ben-Zvi and Nadler proved that the induction map from $B$-bundles of degree 0 to semistable $G$-bundles of degree 0 over an elliptic curve is a small map with Galois group isomorphic to the Weyl group of $G$. We generalize their result to all connected components of $\Bun_G$ for an arbitrary reductive group $G$. We prove that for every degree (i.e. topological type) there exists a unique parabolic subgroup such that any semistable $G$\nobreakdash-bundle of this degree has a reduction to it and moreover the induction map is small with Galois group the relative Weyl group of the Levi. This provides new examples of simple automorphic sheaves which are constituents of Eisenstein sheaves for the trivial local system., Comment: 19 pages; contains one table; more typos corrected; there was a gap in the proof (a subsection was added to repair that); added a few references and some remarks about the slope map; comments are welcomed!

In this paper, we study formal deformations of Poisson structures, especially for three families of Poisson varieties in dimensions two and three. For these families of Poisson structures, using an explicit basis of the second Poisson cohomology space, we solve the deformation equations at each step and obtain a large family of formal deformations for each Poisson structure which we consider. With the help of an explicit formula, we show that this family contains, modulo equivalence, all possible formal deformations. We show moreover that, when the Poisson structure is generic, all members of the family are non-equivalent., Comment: 27 pages

Let $X$ be a compact real analytic manifold, and let $T^*X$ be its cotangent bundle. Let $Sh(X)$ be the triangulated dg category of bounded, constructible complexes of sheaves on $X$. In this paper, we develop a Fukaya $A_\infty$-category $Fuk(T^*X)$ whose objects are exact, not necessarily compact Lagrangian branes in the cotangent bundle. We write $Tw Fuk(T^*X)$ for the $A_\infty$-triangulated envelope of $Fuk(T^*X)$ consisting of twisted complexes of Lagrangian branes. Our main result is that $Sh(X)$ quasi-embeds into $Tw Fuk(T^*X)$ as an $A_\infty$-category. Taking cohomology gives an embedding of the corresponding derived categories., 56 pages; to appear in JAMS

Let G be a compact Lie group. Let E be a principal G-bundle over a closed manifold M, and Ad(E) its adjoint bundle. In this paper we describe a new Frobenius algebra structure on h_*(Ad(E)), where h_* is an appropriate generalized homology theory. Recall that a Frobenius algebra has both a product and a coproduct. The product in this new Frobenius algebra is induced by the string topology product. In particular, the product can be defined when G is any topological group and in the case that E is contractible it is precisely the Chas-Sullivan string product on H_*(LM). We will show that the coproduct is induced by the Freed-Hopkins-Teleman fusion product. Indeed, when M is replaced by BG and h_* is K-theory the coproduct is the completion of the Freed-Hopkins-Teleman fusion structure. We will then show that this duality between the string and fusion products is realized by a Spanier-Whitehead duality between certain Thom spectra of virtual bundles over Ad(E)., Comment: 10 pages

Let R be a left Artinian ring, and M a faithful left R-module which is minimal, in the sense that no proper submodule or proper homomorphic image of M is faithful. If R is local, and socle(R) is central in R, we show that length(M/J(R)M)+length(socle(M))$\leq$ length(socle(R))+1, strengthening a result of T. Gulliksen. Most of the rest of the paper is devoted to the situation where the Artinian ring R is not necessarily local, and does not necessarily have central socle. In the case where R is a finite-dimensional algebra over an algebraically closed field, we get an inequality similar to the above, with the length of socle(R) interpreted as its length as a bimodule, and with the final summand +1 replaced by the Euler characteristic of a bipartite graph determined by the structure of socle(R). More generally, that inequality holds if, rather than assuming k algebraically closed, we assume that R/J(R) is a direct product of full matrix algebras over k, and exclude the case where k has small finite cardinality. Examples show that the restriction on the cardinality of k is needed; we do not know whether versions of our result hold with other hypotheses significantly weakened. The situation for faithful modules with only one minimality property, i.e., having no faithful proper submodules or having no faithful proper homomorphic images, is more straightforward: The length of M/J(R)M in the former case, and of socle(M) in the latter, is $\leq$ length(socle(R)) (again meaning length as a bimodule). The final section, essentially independent of the rest of the note, obtains these bounds, and shows that every faithful module over a left Artinian ring has a faithful submodule with the former minimality condition and a faithful factor module with the latter. The proofs involve some nice general results on decompositions of modules., Comment: 16 pages. Copy at http://math.berkeley.edu/~gbergman/papers may be updated more frequently than arXiv copy

This paper is an invitation to Fourier analysis in the context of reduced twisted C*-crossed products associated with discrete unital twisted C*-dynamical systems. We discuss norm-convergence of Fourier series, multipliers and summation processes. Our study relies in an essential way on the (covariant and equivariant) representation theory of C*-dynamical systems on Hilbert C*-modules. It also yields some information on the ideal structure of reduced twisted C*-crossed products., Comment: Some misprints corrected, Section 5 has been slightly reorganized

Cotangent sums are associated to the zeros of the Estermann zeta function. They have also proven to be of importance in the Nyman-Beurling criterion for the Riemann Hypothesis. The main result of the paper is the proof of the existence of a unique positive measure {\mu} on $\mathbb{R}$, with respect to which certain normalized cotangent sums are equidistributed. Improvements as well as further generalizations of asymptotic formulas regarding the relevant cotangent sums are obtained. We also prove an asymptotic formula for a more general cotangent sum as well as asymptotic results for the moments of the cotangent sums under consideration. We also give an estimate for the rate of growth of the moments of order 2k, as a function of k., Comment: 78 pages, 4 figures

An extension B\subset A of algebras over a commutative ring k is an H-extension for an L-bialgebroid H if A is an H-comodule algebra and B is the subalgebra of its coinvariants. It is H-Galois if the canonical map A\otimes_B A\to A\otimes_L H is an isomorphism or, equivalently, if the canonical coring (A\otimes_L H:A) is a Galois coring. In the case of a Hopf algebroid H=(H_L,H_R,S) any H_R-extension is shown to be also an H_L-extension. If the antipode is bijective then also the notions of H_R-Galois extensions and of H_L-Galois extensions are proven to coincide. Results about bijective entwining structures are extended to entwining structures over non-commutative algebras in order to prove a Kreimer-Takeuchi type theorem for a finitely generated projective Hopf algebroid H with bijective antipode. It states that any H-Galois extension B\subset A is projective, and if A is k-flat then already the surjectivity of the canonical map implies the Galois property. The Morita theory, developed for corings by Caenepeel, Vercruysse and Wang, is applied to obtain equivalent criteria for the Galois property of Hopf algebroid extensions. This leads to Hopf algebroid analogues of results for Hopf algebra extensions by Doi and, in the case of Frobenius Hopf algebroids, by Cohen, Fishman and Montgomery., 19 pages, no figure v2: Prop 3.1 added. v3: Prop 3.1 is withdrawn, without affecting any other result in the paper

We compute equivariant Euler characteristics of locally free sheaves on curves, thereby generalizing several results of Kani and Nakajima. For instance, we extend Kani's computation of the Galois module structure of the space of global meromorphic differentials which are logarithmic along the ramification locus from the tamely ramified to the weakly ramified case., Comment: 18 pages; v2: 22 pages, further result added, paper restructured and revised

In this paper we consider extremal and almost extremal bounds on the normal Hilbert coefficients of ${\mathfrak m}$-primary ideals of an analytically unramified Cohen-Macaulay ring $R$ of dimension $d>0$ and infinite residue field. In these circumstances we show that the associated graded ring of the normal filtration of the ideal is either Cohen-Macaulay or almost Cohen-Macaulay.

The purpose of this paper is to provide a new proof of Bando-Mabuchi's uniqueness theorem of K\"ahler Einstein metrics on Fano manifolds, based on Chen's weak C^{1,1} geodesic without using any further regularities. Unlike the smooth case, the lack of regularities on the geodesic forbids us to use spectral formula of the weighed Laplacian operator directly. However, we can use smooth geodesics to approximate the weak one, then prove that a sequence of eigenfunctions will converge into the first eigenspace of the weighted Laplacian operator., Comment: we correct one mistake in the previous version: the eigenfunctions of weighted Laplacian operator is not real valued in general. But this condition has not been used essentially in our proof

The purpose of this paper is to give two applications of Fourier transforms and generic vanishing theorems: - we give a cohomological characterization of principal polarizations - we prove that if $X$ an abelian variety and $\Theta $ a polarization of type $(1,...,1,2)$, then a general pair $(X,\Theta )$ is log canonical, Comment: 10 pages

Extending the Levy-Steinitz rearrangement theorem in ℝn, which in turn extended Riemann’s theorem, Banaszczyk proved in 1990/93 that a metrizable, locally convex space is nuclear if and only if the domain of sums of every convergent series (i.e. the set of all elements in the space which are sums of a convergent rearrangement of the series) is a translate of a closed subspace of a special form. In this paper we present an apparently complete analysis of the domains of sums of convergent series in duals of metrizable spaces or, more generally, in (DF)-spaces in the sense of Grothendieck.

The notion of topological entropy is originally defined for a single action. Later it was extended by Kieffer for arbitrary discrete amenable groups. Recently Friedland defined topological entropy for any discrete group actions amenable or not. Lind, Schmidt and Ward extended Yuzvinskii's addition formula from single action to abelian group actions. A result of Ward and Zhang suggests that such extension might be possible in the amenable case. In this paper we prove that the addition formula can not be extended for groups of finite type with nonzero Euler-characteristcs e.g. for free groups or surface groups., Comment: 7 pages LaTex

In this paper we study the theory of operators on complex Hilbert spaces, which attain their minimum in the unit sphere. We prove some important results concerning the characterization of the N*, and also AN* operators, see respectively Definition 1.1 and Definition 1.3. The injective property plays an important role in these operators, and shall be established by these classes., 18 pages. arXiv admin note: substantial text overlap with arXiv:1007.1477

In the paper, the authors discover an integral representation, some inequalities, and complete monotonicity of Bernoulli numbers of the second kind., 10 pages

It is known that all complete intersection Artinian standard graded algebras of codimension 3 have the Weak Lefschetz Property. Unfortunately, this property does not continue to be true when you increase the number of minimal generators for the ideal defining the algebra. For instance, it is not more valid for almost complete intersection Artinian standard graded algebras of codimension 3. On the other hand, the Hilbert functions of all Weak Lefschetz Artinian graded algebras are unimodal and with the positive part of their first difference forming an $O$-sequence (i.e. are Weak Lefschetz sequences). In this paper we show that all the Hilbert functions of the almost complete intersection Artinian standard graded algebras of codimension 3 are Weak Lefschetz sequences.

Schinzel's Hypothesis H is a general conjecture in number theory on prime values of polynomials that generalizes, e.g., the twin prime conjecture and Dirichlet's theorem on primes in arithmetic progression. We prove an arithmetic analog of this conjecture for polynomial rings over pseudo algebraically closed fields. This implies results over large finite fields. A main tool in the proof is an irreducibility theorems \`a la Hilbert., Comment: The paper was shorten, some errors were corrected

The motivation for this paper comes from the following question on comparison of norms of conformal martingales $X$, $Y$ in $\R^d$, $d\geq 2$. Suppose that $Y$ is differentially subordinate to $X$. For $01$ is not an integer, which has further interesting applications to stopped Bessel processes and to the behavior of smooth functions on Euclidean domains. The inequality for conformal martingales, which has its roots on the study of the $L^p$ norms of the Beurling-Ahlfors singular integral operator \cite{BW}, extends a recent result of Borichev, Janakiraman and Volberg \cite{BJV2}.

In this paper we consider the space $BMO_o(\mathbb{R},X)$ of bounded mean oscillations and odd functions on $\mathbb{R}$ taking values in a UMD Banach space $X$. The functions in $BMO_o(\mathbb{R},X)$ are characterized by Carleson type conditions involving Bessel convolutions and $\gamma$-radonifying norms. Also we prove that the UMD Banach spaces are the unique Banach spaces for which certain $\gamma$-radonifying Carleson inequalities for Bessel-Poisson integrals of $BMO_o(\mathbb{R},X)$ functions hold., Comment: 29 pages

We study the question on whether the famous Golod-Shafarevich estimate, which gives a lower bound for the Hilbert series of a (noncommutative) algebra, is attained. This question was considered by Anick in his 1983 paper 'Generic algebras and CW-complexes', Princeton Univ. Press., where he proved that the estimate is attained for the number of quadratic relations $d \leq \frac{n^2}{4}$ and $d \geq \frac{n^2}{2}$, and conjectured that this is the case for any number of quadratic relations. The particular point where the number of relations is equal to $ \frac{n(n-1)}{2}$ was addressed by Vershik. He conjectured that a generic algebra with this number of relations is finite dimensional. We prove that over any infinite field, the Anick conjecture holds for $d \geq \frac{4(n^2+n)}{9}$ and arbitrary number of generators $n$, and confirm the Vershik conjecture over any field of characteristic 0. We give also a series of related asymptotic results., Comment: 17 pages, to appear in the Proceedings of the Royal Society Edinburgh A

The radial (or Laplacian) masa in a free group factor is the abelian von Neumann algebra generated by the sum of the generators (of the free group) and their inverses. The main result of this paper is that the radial masa is a maximal injective von Neumann subalgebra of a free group factor. We also investigate tensor products of maximal injective algebras. Given two inclusions $B_i\subset M_i$ of type $\mathrm{I}$ von Neumann algebras in finite von Neumann algebras such that each $B_i$ is maximal injective in $M_i$, we show that the tensor product $B_1\ \bar{\otimes}\ B_2$ is maximal injective in $M_1\ \bar{\otimes}\ M_2$ provided at least one of the inclusions satisfies the asymptotic orthogonality property we establish for the radial masa. In particular it follows that finite tensor products of generator and radial masas will be maximal injective in the corresponding tensor product of free group factors., 25 Pages, Typos corrected and exposition improved

In this paper we present a complete description of a stochastic semigroup of finite-dimensional projections in Hilbert space. The geometry of such semigroups is characterized by the asymptotic behavior of the widths of compact subsets with respect to the subspaces generated by the semigroup operators., 13 pages

We give an explicit combinatorial Chevalley-type formula for the equivariant K-theory of generalized flag varieties G/P which is a direct generalization of the classical Chevalley formula. Our formula implies a simple combinatorial model for the characters of the irreducible representations of G and, more generally, for the Demazure characters. This model can be viewed as a discrete counterpart of the Littelmann path model, and has several advantages. Our construction is given in terms of a certain R-matrix, that is, a collection of operators satisfying the Yang-Baxter equation. It reduces to combinatorics of decompositions in the affine Weyl group and enumeration of saturated chains in the Bruhat order on the (nonaffine) Weyl group. Our model easily implies several symmetries of the coefficients in the Chevalley-type formula. We also derive a simple formula for multiplying an arbitrary Schubert class by a divisor class, as well as a dual Chevalley-type formula. The paper contains other applications and examples., Comment: v2: Major revision: several new sections and an appendix added, references added, exposition improved. New material includes: generalization to G/P, two symmetries of coefficients, Pieri-type formula, dual Chevalley-type formula, a conjecture for quantum K-theory. v3: Minor updates and corrections

In this paper we develop a theory of free holomorphic functions on noncommutative Reinhardt domains generated by positive regular free holomorphic functions in n noncommuting variables. We show that the free biholomorphic classification of these domains is the same as the classification, up to unital completely isometric isomorphisms, of the corresponding noncommutative domain algebras., Comment: 40 pages, corrected typos

For the deterministic dyadic model of turbulence, there are examples of initial conditions in $l^2$ which have more than one solution. The aim of this paper is to prove that uniqueness, for all $l^2$-initial conditions, is restored when a suitable multiplicative noise is introduced. The noise is formally energy preserving. Uniqueness is understood in the weak probabilistic sense., 13 pages, no figures. Submitted to the Proceedings of the American Mathematical Society

In this paper we study the L p boundary value problems for $${\mathcal{L}(u)=0}$$ in $${\mathbb{R}^{d+1}_+}$$ , where $${\mathcal{L}=-{\rm div} (A\nabla )}$$ is a second order elliptic operator with real and symmetric coefficients. Assume that A is periodic in x d+1 and satisfies some minimal smoothness condition in the x d+1 variable, we show that the L p Neumann and regularity problems are uniquely solvable for 1

In this paper we determine the support of the irreducible spherical representation (i.e., the irreducible quotient of the polynomial representation) of the rational Cherednik algebra of a finite Coxeter group for any value of the parameter c. In particular, we determine for which values of c this representation is finite dimensional. This generalizes a result of Varagnolo and Vasserot (2009) [20], who classified finite dimensional spherical representations in the case of Weyl groups and equal parameters (i.e., when c is a constant function). Our proof is based on the Macdonald–Mehta integral and the elementary theory of distributions., National Science Foundation (U.S.) (Grant DMS-0504847), National Science Foundation (U.S.) (Grant DMS-0854764)

We establish a link between two geometric approaches to the representation theory of rational Cherednik algebras of type A: one based on a noncommutative Proj construction, used in [GS]; the other involving quantum hamiltonian reduction of an algebra of differential operators, used in [GG]. In the present paper, we combine these two points of view by showing that the process of hamiltonian reduction intertwines a naturally defined geometric twist functor on D-modules with the shift functor for the Cherednik algebra. That enables us to give a direct and relatively short proof of the key result, [GS, Theorem 1.4] without recourse to Haiman's deep results on the n! theorem. We also show that the characteristic cycles defined independently in these two approaches are equal, thereby confirming a conjecture from [GG]., 37 pp

The purpose of this paper is to describe a general and simple setting for defining $(g,p+q)$-string operations on a Poincar\'e duality space and more generally on a Gorenstein space. Gorenstein spaces include Poincar\'e duality spaces as well as classifying spaces or homotopy quotients of connected Lie groups. Our presentation implies directly the homotopy invariance of each $(g,p+q)$-string operation as well as it leads to explicit computations., Comment: 30 pages and 2 figures

The theory of exact C*-algebras was introduced by Kirchberg and has been influential in recent development of C*-algebras. A fundamental result on exact C*-algebras is a local characterization of exactness. The notion of weakly exact von Neumann algebras was also introduced by Kirchberg. In this paper, we give a local characterization of weak exactness. As a corollary, we prove that a discrete group is exact if and only if its group von Neumann algebra is weakly exact. The proof naturally involves the operator space duality., 8 pages

Let $\CP^1_{k,m}$ be the orbifold structure on $\CP^1$ obtained via uniformizing the neighborhoods of 0 and $\infty$ respectively by $z\mapsto z^k$ and $w\mapsto w^m.$ The diagonal action of the torus on the projective line induces naturally an orbifold action on $\CP^1_{k,m}.$ In this paper we prove that if k and m are co-prime then Givental's prediction of the equivariant total descendent Gromov-Witten potential of $\CP^1_{k,m}$ satisfies certain Hirota Quadratic Equations (HQE for short). We also show that after an appropriate change of the variables, similar to Getzler's change in the equivariant Gromov-Witten theory of $\CP^1$, the HQE turn into the HQE of the 2-Toda hierarchy, i.e., the Gromov-Witten potential of $\CP^1_{k,m}$ is a tau-function of the 2-Toda hierarchy. More precisely, we obtain a sequence of tau-functions of the 2-Toda hierarchy from the descendent potential via some translations. The later condition, that all tau-functions in the sequence are obtained from a single one via translations, imposes a serious constraint on the solution of the 2-Toda hierarchy. Our theorem leads to the discovery of a new integrable hierarchy (we suggest to be called the Equivariant Bi-graded Toda Hierarchy). We conjecture that this new hierarchy governs, i.e., uniquely determines, the equivariant Gromov-Witten invariants of $\CP^1_{k,m}.$, 40 pages, 1 figure

In the present paper we study tensor C*-categories with non-simple unit realised as C*-dynamical systems (F,G,\beta) with a compact (non-Abelian) group G and fixed point algebra A := F^G. We consider C*-dynamical systems with minimal relative commutant of A in F, i.e. A' \cap F = Z, where Z is the center of A which we assume to be nontrivial. We give first several constructions of minimal C*-dynamical systems in terms of a single Cuntz-Pimsner algebra associated to a suitable Z-bimodule. These examples are labelled by the action of a discrete Abelian group (which we call the chain group) on Z and by the choice of a suitable class of finite dimensional representations of G. Second, we present a construction of a minimal C*-dynamical system with nontrivial Z that also encodes the representation category of G. In this case the C*-algebra F is generated by a family of Cuntz-Pimsner algebras, where the product of the elements in different algebras is twisted by the chain group action. We apply these constructions to the group G = SU(N)., Comment: 34 pages; References updated and typos corrected. To appear in International Journal of Mathematics

In the present paper, we shall show that for any prime number p, every finite p-group occurs as the Galois Group of the maximal unramified p-extension over a certain number field of finite degree. We shall also show that for any given pro-p-group G with countably many generators, there exists a number field (not necessary of finite degree) whose maximal unramified p-extension has Galois group isomorphic to G. This means that the set of the isomorphism classes of the Galois groups of the maximal unramified p-extensions over the number fields (including of infinite degree) is precisely equal to that of all the pro-p-groups with countably many generators., Comment: In this new version, a considerable improvement has been done, that is, Theorems 1 and 2 have been established for all the prime number p (without restrictive assumption C(p) on the prime number p)

In this paper we classify all configurations of singular fibers of elliptic fibrations on the double cover of $\mathbf{P}^2$ ramified along six lines in general position.

We consider two operator space versions of type and cotype, namely $S_p$-type, $S_q$-cotype and type $(p,H)$, cotype $(q,H)$ for a homogeneous Hilbertian operator space $H$ and $1\leq p \leq 2 \leq q\leq \infty$, generalizing "$OH$-cotype 2" of G. Pisier. We compute type and cotype of some Hilbertian operator spaces and $L_p$ spaces, and we investigate the relationship between a homogeneous Hilbertian space $H$ and operator spaces with cotype $(2,H)$. As applications we consider operator space versions of generalized little Grothendieck's theorem and Maurey's extension theorem in terms of these new notions., Comment: 23 pages, The whole paper is rewritten, and the title has been changed

International audience; This paper is a following to math.RT/0410454. For a finite group of Lie type we study the endomorphisms, commuting with the group action, of a Deligne-Lusztig variety associated to a regular element of the Weyl group. We state some general conjectures, in particular that the endomorphism algebra induced on the cohomology is a cyclotomic Hecke algebra, and prove them in some cases. On the way we also prove results on centralizers in braid groups.