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We correct a mistake in a lemma in the paper cited in the title and show that it did not affect any of the other results of the paper. To this end, we prove results on linearly disjoint field extensions that do not seem to be commonly known. We give an example to show that a separability assumption in one of these results cannot be dropped (doing so had led to the mistake). Further, we discuss recent generalizations of the original classification of defect extensions.

Let H(Dn) be the space of holomorphic functions on the unit polydisk Dn, and let , where p, q> 0, α = (α1,…,αn) with αj > -1, j =1,..., n, be the class of all measurable functions f defined on Dn such thatwhere Mp(f,r) denote the p-integral means of the function f. Denote the weighted Bergman space on . We provide a characterization for a function f being in . Using the characterization we prove the following result: Let p> 1, then the Cesàro operator is bounded on the space .

In this short paper, we apply estimates and ideas from [CM4] to study the ends of a properly embedded complete minimal surface $Σ^{2} ⊂\mathbf{R}^{3}$ with finite topology. The main result is that any complete properly embedded minimal annulus that lies above a sufficiently narrow downward sloping cone must have finite total curvature.

We express the Masur-Veech volume and the area Siegel-Veech constant of the moduli space $\mathcal{Q}_{g,n}$ of genus $g$ meromorphic quadratic differentials with $n$ simple poles as polynomials in the intersection numbers of $\psi$-classes with explicit rational coefficients. The formulae obtained in this article result from lattice point counts involving the Kontsevich volume polynomials that also appear in Mirzakhani's recursion for the Weil-Petersson volumes of the moduli spaces of bordered hyperbolic surfaces with geodesic boundaries. A similar formula for the Masur-Veech volume (though without explicit evaluation) was obtained earlier by Mirzakhani via completely different approach. Furthermore, we prove that the density of the mapping class group orbit of any simple closed multicurve $\gamma$ inside the ambient set of integral measured laminations computed by Mirzakhani coincides with the density of square-tiled surfaces having horizontal cylinder decomposition associated to $\gamma$ among all square-tiled surfaces in $\mathcal{Q}_{g,n}$. We study the resulting densities (or, equivalently, volume contributions) in more detail in the special case $n=0$. In particular, we compute the asymptotic frequencies of separating and non-separating simple closed geodesics on a closed hyperbolic surface of genus $g$ for small $g$ and we show that for large genera the separating closed geodesics are $\sqrt{\frac{2}{3\pi g}}\cdot\frac{1}{4^g}$ times less frequent., Comment: The current paper (as well as the companion paper arXiv:2007.04740) has grown from arxiv:1908.08611. The conjectures stated in arXiv:1908.08611 are proved by A. Aggarwal in arXiv:2004.05042

This is the first of a series of papers where we relate tangent cones of Hermitian-Yang-Mills connections at an isolated singularity to the complex algebraic geometry of the underlying reflexive sheaf, when the sheaf is locally modelled on the pull-back of a holomorphic vector bundle from the projective space. In this paper we shall impose an extra assumption that the graded sheaf determined by the Harder-Narasimhan-Seshadri filtrations of the vector bundle is reflexive. In general we conjecture that the tangent cone is uniquely determined by the double dual of the associated graded object of a Harder-Narasimhan-Seshadri filtration of an algebraic tangent cone, which is a certain torsion-free sheaf on the projective space. In this paper we also prove this conjecture when there is an algebraic tangent cone which is locally free and stable., Final version

Fake projective planes are smooth complex surfaces of general type with Betti numbers equal to those of the usual projective plane. They come in complex conjugate pairs and have been classified as quotients of the two-dimensional ball by explicitly written arithmetic subgroups. In this paper we find equations of a projective model of a conjugate pair of fake projective planes by studying the geometry of the quotient of such surface by an order seven automorphism., Comment: This is a full version of "Research announcement: equations of a fake projective plane", arXiv:1710.04501. Key tables and some M2 and Magma code from the paper are included in separate files for convenience

In a remarkable paper in 2008, Fyodorov and Bouchaud conjectured an exact formula for the density of the total mass of (sub-critical) Gaussian multiplicative chaos (GMC) associated to the Gaussian free field (GFF) on the unit circle. In this paper we will give a proof of this formula. In the mathematical literature this is the first occurrence of an explicit probability density for the total mass of a GMC measure. The key observation of our proof is that the negative moments of the total mass of GMC determine its law and are equal to one-point correlation functions of Liouville conformal field theory in the disk defined by Huang, Rhodes and Vargas. The rest of the proof then consists in implementing rigorously the framework of conformal field theory (BPZ equations for degenerate field insertions) in a probabilistic setting to compute the negative moments. Finally we will discuss applications to random matrix theory, asymptotics of the maximum of the GFF and tail expansions of GMC., 27 pages

In a famous paper, Asai indicated how to develop a theory of Eisenstein series for arbitrary number fields, using hyperbolic 3-space to take care of the complex places. Unfortunately he limited himself to class number 1. The present paper gives a detailed exposition of the general case, to be used for many applications. First, it is shown that the Eisenstein series satisfy the authors’ definition of regularized products satisfying the generalized Lerch formula, and the basic axioms which allow the systematic development of the authors’ theory, including the Cramér theorem. It is indicated how previous results of Efrat and Zograf for the strict Hilbert modular case extend to arbitrary number fields, for instance a spectral decomposition of the heat kernel periodized with respect to SL2 of the integers of the number field. This gives rise to a theta inversion formula, to which the authors’ Gauss transform can be applied. In addition, the Eisenstein series can be twisted with the heat kernel, thus encoding an infinite amount of spectral information in one item coming from heat Eisenstein series. The main expected spectral formula is stated, but a complete exposition would require a substantial amount of space, and is currently under consideration.

In this paper, quantitative bounds in high-frequency central limit theorems are derived for Poisson based $U$-statistics of arbitrary degree built by means of wavelet coefficients over compact Riemannian manifolds. The wavelets considered here are the so-called needlets, characterized by strong concentration properties and by an exact reconstruction formula. Furthermore, we consider Poisson point processes over the manifold such that the density function associated to its control measure lives in a Besov space. The main findings of this paper include new rates of convergence that depend strongly on the degree of regularity of the control measure of the underlying Poisson point process, providing a refined understanding of the connection between regularity and speed of convergence in this framework., Comment: 19 pages

We correct Proposition 3.12 and Lemma 3.13 of the paper published in Vol. 2013, No.13, pp.3006-3052. The corrections do not affect the other statements of the paper. In this note, we correct a flow in the statement of Proposition 3.12 of [1] which also leads to a modification in the statement of Lemma 3.13 of [1]. We recall that in this proposition one considers morphisms of schemes X ?→π Y ?→q S, where q is proper, flat, with equidimensional fibers of dimension n and π is finite, flat and surjective. Imposing certain conditions on the fibers it is claimed that the loci of s € S fulfilling these conditions are open subsets of S. A missing condition should be added and the correct version of Parts (g) and (h) of Proposition 3.12 should be as follows: (g) Ys has no embedded components and the discriminant scheme of π s : Xs → Ys is of pure codimension one and smooth; (h) Ys has no embedded components and the discriminant scheme of π s : Xs → Ys is of codimension one, irreducible and generically reduced.

In previous papers, we examined several properties of an error term in a certain divisor problem related to the derivatives of the Riemann zeta-function. In this paper, we obtain representations of error terms related to the derivatives of some Dirichlet series, which can be regarded as generalized versions of a Dirichlet divisor problem and a Gauss circle problem. We also give the upper bounds of the error terms in terms of exponent pairs.

Consider an elliptic curve defined over an imaginary quadratic field $K$ with good reduction at the primes above $p\geq 5$ and has complex multiplication by the full ring of integers $\mathcal{O}_K$ of $K$. In this paper, we construct $p$-adic analogues of the Eisenstein-Kronecker series for such elliptic curve as Coleman functions on the elliptic curve. We then prove $p$-adic analogues of the first and second Kronecker limit formulas by using the distribution relation of the Kronecker theta function., v2. The current version is the synthesis of {\S}1-{\S}3 of the first version of this article with the content of arXiv:0807.4008 "The Kronecker limit formulas via the distribution relation." {\S}4,{\S}5 of the first version of this paper will be treated in a subsequent article

In this paper we introduce and develop the theory of FI-modules. We apply this theory to obtain new theorems about: - the cohomology of the configuration space of n distinct ordered points on an arbitrary (connected, oriented) manifold - the diagonal coinvariant algebra on r sets of n variables - the cohomology and tautological ring of the moduli space of n-pointed curves - the space of polynomials on rank varieties of n x n matrices - the subalgebra of the cohomology of the genus n Torelli group generated by H^1 and more. The symmetric group S_n acts on each of these vector spaces. In most cases almost nothing is known about the characters of these representations, or even their dimensions. We prove that in each fixed degree the character is given, for n large enough, by a polynomial in the cycle-counting functions that is independent of n. In particular, the dimension is eventually a polynomial in n. In this framework, representation stability (in the sense of Church-Farb) for a sequence of S_n-representations is converted to a finite generation property for a single FI-module., 54 pages. v4: new title, paper completely reorganized; final version, to appear in Duke Math Journal

The object of this paper is to initiate a study of the Fourier coefficients of a weight $1$ harmonic Maass form and relate them to the complex Galois representation associated to a weight $1$ newform, which is the form’s image under a certain differential operator. In this paper, our focus will be on weight $1$ dihedral newforms of prime level $p\equiv3(\operatorname{mod}{4})$ . In this case we give properties of the Fourier coefficients that are similar to (and sometimes reduce to) cases of Stark’s conjectures on derivatives of $L$ -functions. We also give a new modular interpretation of certain products of differences of singular moduli studied by Gross and Zagier. Finally, we provide some numerical evidence that the Fourier coefficients of a mock-modular form whose shadow is exotic are similarly related to the associated complex Galois representation.

Li and Shanmugalingam showed that annularly quasiconvex metric spaces endowed with a doubling measure preserve the property of supporting a $p$-Poincaré inequality under the sphericalization and flattening procedures. Because natural examples such as the real line or a broad class of metric trees are not annularly quasiconvex, our aim in the present paper is to study, under weaker hypotheses on the metric space, the preservation of $p$-Poincaré inequalites under those conformal deformations for sufficiently large $p$. We propose the hypotheses used in a previous paper by the same authors, where the preservation of $\infty$-Poincaré inequality has been studied under the assumption of radially star-like quasiconvexity (for sphericalization) and meridian-like quasiconvexity (for flattening). To finish, using the sphericalization procedure, we exhibit an example of a Cheeger differentiability space whose blow up at a particular point is not a PI space.

In 2007, Shimoda, in connection with a long-standing question of Sally, asked whether a Noetherian local ring, such that all its prime ideals different from the maximal ideal are complete intersections, has Krull dimension at most 2. In this paper, having reduced the conjecture to the case of dimension 3, if the ring is regular and local of dimension 3, we explicitly describe a family of prime ideals of height 2 minimally generated by three elements. Weakening the hypothesis of regularity, we find that, to achieve the same end, we need to add extra hypotheses, such as completeness, infiniteness of the residue field, and the multiplicity of the ring being at most 3. In the second part of the paper, we turn our attention to the category of standard graded algebras. A geometrical approach via a double use of a Bertini theorem, together with a result of Simis, Ulrich, and Vasconcelos, allows us to obtain a definitive answer in this setting. Finally, by adapting work of Miller on prime Bourbaki ideals in local rings, we detail some more technical results concerning the existence in standard graded algebras of homogeneous prime ideals with an (as it were) excessive number of generators.

Let g be a complex reductive Lie algebra with Cartan algebra h. Hotta and Kashiwara defined a holonomic D-module M, on g x h, called Harish-Chandra module. We relate gr(M), an associated graded module with respect to a canonical Hodge filtration on M, to the isospectral commuting variety, a subvariety of g x g x h x h which is a ramified cover of the variety of pairs of commuting elements of g. Our main result establishes an isomorphism of gr(M) with the structure sheaf of X_norm, the normalization of the isospectral commuting variety. It follows, thanks to the theory of Hodge modules, that the normalization of the isospectral commuting variety is Cohen-Macaulay and Gorenstein, confirming a conjecture of M. Haiman. We deduce, using Saito's theory of Hodge D-modules, that the scheme X_norm is Cohen-Macaulay and Gorenstein. This confirms a conjecture of M. Haiman. Associated with any principal nilpotent pair in g, there is a finite subscheme of X_norm. The corresponding coordinate ring is a bigraded finite dimensional Gorenstein algebra that affords the regular representation of the Weyl group. The socle of that algebra is a 1-dimensional vector space generated by a remarkable W-harmonic polynomial on h x h. In the special case where g=gl_n the above algebras are closely related to the n!-theorem of Haiman and our W-harmonic polynomial reduces to the Garsia-Haiman polynomial. Furthermore, in the gl_n case, the sheaf gr(M) gives rise to a vector bundle on the Hilbert scheme of n points in C^2 that turns out to be isomorphic to the Procesi bundle. Our results were used by I. Gordon to obtain a new proof of positivity of the Kostka-Macdonald polynomials established earlier by Haiman., The present paper supersedes an earlier paper arXiv:1002.3311

In this paper we study the Weil-Petersson geometry of $\overline{\mathcal{M}_{g,n}}$, the compactified moduli space of Riemann surfaces with genus g and n marked points. The main goal of this paper is to understand the growth of the diameter of $\overline{\mathcal{M}_{g,n}}$ as a function of $g$ and $n$. We show that this diameter grows as $\sqrt{n}$ in $n$, and is bounded above by $C \sqrt{g}\log g$ in $g$ for some constant $C$. We also give a lower bound on the growth in $g$ of the diameter of $\overline{\mathcal{M}_{g,n}}$ in terms of an auxiliary function that measures the extent to which the thick part of moduli space admits radial coordinates., 26 pages, 7 figures

This paper discusses developments in complex analysis and CR geometry in the last forty years related to the Cauchy–Riemann equations, proper holomorphic mappings between balls, and positivity conditions in complex analysis. The paper includes anecdotes about some of the contributors to these developments.

We deal with finitely generated modules over an artin algebra. In his Philadelphia Notes, M.Auslander showed that any homomorphism is right determined by a module C, but a formula for C which he wrote down has to be modified. The paper includes now complete and direct proofs of the main results concerning right determiners of morphisms. We discuss the role of indecomposable projective direct summands of a minimal right determiner and provide a detailed analysis of those morphisms which are right determined by a module without any non-zero projective direct summand., The paper has been revised and expanded. The terminology has been changed as follows: "essential kernel" is replaced by "intrinsic kernel", "determinator" is replaced by "determiner". Sections 3, 4 and 5 are new

This paper establishes new estimates for linear Schroedinger equations in R^3 with time-dependent potentials. Some of the results are new even in the time-independent case and all are shown to hold for potentials in scaling-critical, translation-invariant spaces. The proof of the time-independent results uses a novel method based on an abstract version of Wiener's Theorem., 49 pages; this is an expanded and improved version of the older paper

The classical principle of least action says that orbits of mechanical systems extremize action; an important subclass are those orbits that minimize action. In this paper we utilize this principle along with Aubry-Mather theory to construct (Birkhoff) regions of instability for a certain three-body problem, given by a Hamiltonian system of $2$ degrees of freedom. We believe that these methods can be applied to construct instability regions for a variety of Hamiltonian systems with $2$ degrees of freedom. The Hamiltonian model we consider describes dynamics of a Sun-Jupiter-comet system, and under some simplifying assumptions, we show the existence of instabilities for the orbit of the comet. In particular, we show that a comet which starts close to an orbit in the shape of an ellipse of eccentricity $e=0.66$ can increase in eccentricity up to $e=0.96$ . In the sequels to this paper, we extend the result to beyond $e=1$ and show the existence of ejection orbits. Such orbits are initially well within the range of our solar system. This might give an indication of why most objects rotating around the Sun in our solar system have relatively low eccentricity.

In the present paper we extend A. Weil's proof of the Siegel-Weil theorem to symplectic, nilpotent $t$ -modules (snt-modules), $t$ being a nilpotent endomorphism of a finite-dimensional, symplectic, vector space satisfying a certain consistency condition with respect to the symplectic structure. This extension is key for our proof of the Siegel-Weil theorem for loop groups in the sequel to this paper.

This paper extends the theory of scalar-valued integrable functions with respect to vector measures defined on $δ$-rings to the case of vector-valued tensor integrable functions with respect to vector measures defined on $δ$-rings. This paper also generalizes some results of G. F. Stefánsson for tensor integration theory of vector-valued functions with respect to vector measures defined on $σ$-algebras.

A natural notion for the signature $C_{\pm}({\mathcal V}(p))$ of the curvature of the zero set ${\mathcal V}(p)$ of a non-commutative polynomial $p$ is introduced. The main result of this paper is the bound \[ \operatorname{deg} p \leq2 C_\pm \bigl({\mathcal V}(p) \bigr) + 2. \] It is obtained under some irreducibility and nonsingularity conditions, and shows that the signature of the curvature of the zero set of $p$ dominates its degree. ¶ The condition $C_+({\mathcal V}(p))=0$ means that the non-commutative variety ${\mathcal V}(p)$ has positive curvature. In this case, the preceding inequality implies that the degree of $p$ is at most two. Non-commutative varieties ${\mathcal V}(p)$ with positive curvature were introduced in Indiana Univ. Math. J. 56 (2007) 1189-1231). There a slightly weaker irreducibility hypothesis plus a number of additional hypotheses yielded a weaker result on $p$. The approach here is quite different; it is cleaner, and allows for the treatment of arbitrary signatures. ¶ In J. Anal. Math. 108 (2009) 19-59), the degree of a non-commutative polynomial $p$ was bounded by twice the signature of its Hessian plus two. In this paper, we introduce a modified version of this non-commutative Hessian of $p$ which turns out to be very appropriate for analyzing the variety ${\mathcal V}(p)$.

In this paper we study the local behavior of a solution to the Lam\'e system with \emph{Lipschitz} coefficients in dimension $n\ge 2$. Our main result is the bound on the vanishing order of a nontrivial solution, which immediately implies the strong unique continuation property. This paper solves the open problem of the strong uniqueness continuation property for the Lam\'e system with Lipschitz coefficients in any dimension.

This paper presents an overview of some of the applications of the martingale inequalities of D. L. Burkholder to $L^p$-bounds for singular integral operators, concentrating on the Hilbert transform, first and second order Riesz transforms, the Beurling–Ahlfors operator and other multipliers obtained by projections (conditional expectations) of transformations of stochastic integrals. While martingale inequalities can be used to prove the boundedness of a wider class of Calderón–Zygmund singular integrals, the aim of this paper is to show results which give optimal or near optimal bounds in the norms, hence our restriction to the above operators. ¶ Connections of Burkholder’s foundational work on sharp martingale inequalities to other areas of mathematics where either the results themselves or techniques to prove them have become of considerable interest in recent years, are discussed. These include the 1952 conjecture of C. B. Morrey on rank-one convex and quasiconvex functions with connections to problems in the calculus of variations and the 1982 conjecture of T. Iwaniec on the $L^p$-norm of the Beurling–Ahlfors operator with connections to problems in the theory of qasiconformal mappings. Open questions, problems and conjectures are listed throughout the paper and copious references are provided.

Let $k$ be a field of characteristic zero and ${\bar k}$ an algebraic closure of $k$. For a geometrically integral variety $X$ over $k$, we write ${\bar k}(X)$ for the function field of ${\bar X}=X\times_k{\bar k}$. If $X$ has a smooth $k$-point, the natural embedding of multiplicative groups ${\bar k}^*\hookrightarrow {\bar k}(X)^* $ admits a Galois-equivariant retraction. In the first part of the paper, over local and then over global fields, equivalent conditions to the existence of such a retraction are given. They are expressed in terms of the Brauer group of $X$. In the second part of the paper, we restrict attention to varieties which are homogeneous spaces of connected but otherwise arbitrary algebraic groups, with connected geometric stabilizers. For $k$ local or global, for such a variety $X$, in many situations but not all, the existence of a Galois-equivariant retraction to ${\bar k}^*\hookrightarrow {\bar k}(X)^* $ ensures the existence of a $k$-rational point on $X$. For homogeneous spaces of linear algebraic groups, the technique also handles the case where $k$ is the function field of a complex surface., To appear in Duke Mathematical Journal. An appendix on the Brauer-Manin obstruction for homogeneous spaces has been added

In this paper we show how to compute the �� norm , �� 0, using the dyadic grid. This result is a consequence of the description of the Hardy spaces H p (R N ) in terms of dyadic and special atoms. Recently, several novel methods for computing the BMO norm of a function f in two dimensions were discussed in (9). Given its importance, it is also of interest to explore the possibility of computing the norm of a BMO function, or more generally a function in the Lipschitz class �α, using the dyadic grid in R N. It turns out that the BMO question is closely related to that of approximating functions in the Hardy space H 1 (R N ) by the Haar system. The approximation in H 1 (R N ) by affine systems was proved in (2), but this result does not apply to the Haar system. Now, if H A (R) denotes the closure of the Haar system in H 1 (R), it is not hard to see that the distance d(f, H A ) of f ∈ H 1 (R) to H A is ∼ � R ∞ 0 f(x) dx �, see (1). Thus, neither dyadic atoms suffice to describe the Hardy spaces, nor the evaluation of thenorm in BMO can be reduced to a straightforward computation using the dyadic intervals. In this paper we address both of these issues. First, we give a characterization of the Hardy spaces H p (R N ) in terms of dyadic and special atoms, and then, by a duality argument, we show how to compute the norm in �α(R N ), α ≥ 0, using the dyadic grid. We begin by introducing some notations. Let J denote a family of cubes Q in R N , and Pd the collection of polynomials in R N of degree less than or equal to d. Given α ≥ 0, Q ∈ J, and a locally integrable function g, let pQ(g) denote the unique polynomial in P(α) such that (g − pQ(g)) χQ has vanishing moments up to order (α). For a locally square-integrable function g, we consider the maximal function M ♯,2 α,J g(x) given by

The class of mixed Tsirelson spaces is an important source of examples in the recent development of the structure theory of Banach spaces. The related class of modified mixed Tsirelson spaces has also been well studied. In the present paper, we investigate the problem of comparing isomorphically the mixed Tsirelson space T((Sn, θn) ∞=1) and its modified version TM((Sn, θn) ∞=1). It is shown that these spaces are not isomorphic for a large class of parameters (θn). 1 ≤ p < ∞. Figiel and Johnson (7) provided an analytic description, based on iteration, of the norm of the dual of Tsirelson's original space. Subse- quently, other examples of spaces were constructed with norms described it- eratively, notable among them were Tzafriri's spaces (20) and Schlumprecht's space(18). Gowers' and Maurey's solution to the unconditional basic se- quence problem (8) is a variation based on the same theme. It has emerged in recent years that, far from being isolated examples, Tsirelson's space and its variants from an important class of Banach spaces. Argyros and Deliyanni (2) were the first to provide a general framework for such spaces by defining the class of mixed Tsirelson spaces. Among the earliest vari- ants of Tsirelson's space was its modified version introduced by Johnson (9). Casazza and Odell (6) showed that Tsirelson's space is isomorphic to its modified version. This isomorphism was exploited to study the struc- ture of the space. The modification can be extended directly to the class of mixed Tsirelson spaces, forming the class of modified mixed Tsirelson spaces. It is thus of natural interest to determine if a mixed Tsirelson space is isomorphic to its modified version. This question has been considered by various authors, e.g., (3, 12), who provided answers in what may be con- sidered "extremal" cases. In the present paper, we show that for a large class of parameters, a mixed Tsirelson space and its modified version are not isomorphic.

The closure under extensions of a class of objects in an abelian category is often an important property of that class. Recently the closure of such classes under transfinite extensions (both direct and inverse) has begun to play an important role in several areas of mathematics, for example, in Quillen’s theory of model categories and in the theory of cotorsion pairs. In this paper we prove that several important classes are closed under transfinite extensions. 1. Definitions and basic results Throughout this paper A will be a Grothendieck category with a fixed projective generator U . We will be concerned with direct and inverse limits of systems of objects of A indexed by the well ordered set of ordinals α, where α ≤ λ (or α < λ) for some ordinal λ. To simplify notation, we will denote such a system (direct or inverse) by (Xα | α ≤ λ) with the associated morphisms understood. Definition 1.1. A direct (inverse) system (Xα | α ≤ λ) is said to be continuous if X0 = 0 and if for each limit ordinal β ≤ λ we have Xβ = lim −→α (or Xβ = lim ←− Xα) with the limit over the α < β. The direct (inverse) system (Xα | α ≤ λ) is said to be a system of monomorphisms (epimorphisms) if all the morphisms in the system are monomorphisms (epimorphisms). In order for a continuous direct system (Xα | α ≤ λ) to be a system of monomorphisms it suffices that Xα → Xα+1 be monomorphism whenever α+1 ≤ λ. This follows from what is called the AB5 axiom of a Grothendieck category. If (Xα | α ≤ λ) is a continuous inverse system such that each Xα+1 → Xα (when α + 1 ≤ λ) is an epimorphism, then (Xα | α ≤ λ) is a system of epimorphisms. This is a consequence of the existence of a projective generator U and the fact that (Hom(U,Xα) | α ≤ λ) is a continuous Received May 13, 2005; received in final form July 10, 2006. 2000 Mathematics Subject Classification. 18E15, 18A30. c ©2007 University of Illinois

In this paper we complete the results of our papers [2], [3] and show how to generate from the hypergeometric function F ⁡ 1 / 6 , 1 / 6 ; 1 ; ζ fundamental solutions for the classical Tricomi operator relative to any point in the elliptic, parabolic, or hyperbolic region of the operator.

In our previous paper we have discussed Poisson properties of cluster algebras of geometric type for the case of a nondegenerate matrix of transition exponents. In this paper we consider the case of a general matrix of transition exponents. Our leading idea that a relevant geometric object in this case is a certain closed 2-form compatible with the cluster algebra structure. The main example is provided by Penner coordinates on the decorated Teichmueller space, in which case the above form coincides with the classic Weil-Petersson symplectic form., 17 pages, 7 figures; Substantial changes: new proof of Theorem 3.3. Corrected formulation and new proof of Theorem 3.4. Some other minor changes as well

To each generic tangle projection from the three-dimensional real vector space onto the plane, we associate a derived endofunctor on a graded parabolic version of the Bernstein-Gel'fand category $\mathcal{O}$. We show that this assignment is (up to shifts) invariant under tangle isotopies and Reidemeister moves and defines therefore invariants of tangles. The occurring functors are defined via so-called projective functors. The first part of the paper deals with the indecomposability of such functors and their connection with generalised Temperley-Lieb algebras which are known to have a realisation via decorated tangles. The second part of the paper describes a categorification of the Temperley-Lieb category and proves the main conjectures of [BFK]. Moreover, we describe a functor from the category of 2-cobordisms into a category of projective functors.

It is known that any locally compact group that is approximable by finite groups must be unimodular. However, this condition is not sufficient. For example, simple Lie groups are not approximable by finite ones. In this paper we consider the approximation of locally compact groups by more general finite algebraic systems. We prove that a locally compact group is approximable by finite semigroups iff it is approximable by finite groups. Thus, there exist some locally compact groups and even some compact groups that are not approximable by finite semigroups. We prove also that whenever a locally compact group is approximable by finite quasigroups (latin squares) it is unimodular. The converse theorem is also true: any unimodular group is approximable by finite quasigroups and even by finite loops. In this paper we prove this theorem only for discrete groups. For the case of non-discrete groups the proof is rather long and complicated and is given in a separate paper.