For a smooth foliated manifold $(M,\mathcal F)$, the basic and the foliated cohomologies are defined by using the de Rham complex of $M$. These cohomologies are related with the cohomology of the manifold by the de Rham spectral sequence of $\mathcal F$. A foliated manifold is an example of a space with two topologies, one coarser than the other. For these spaces one can define a continuous cohomology that, for a foliated manifold, corresponds to the continuous foliated (or leafwise) cohomology. In this paper we introduce a construction for spaces with two topologies based upon the Alexander-Spanier continuous cochains. It allows us to define a spectral sequence, similar to the de Rham spectral sequence for a foliation. In particular, continuous basic and foliated cohomologies are defined and related with the cohomology of the space. For a smooth foliated manifold, we also consider Alexander-Spanier differentiable cochains. We compare the continuous and differentiable cohomologies, and the latter with the de Rham cohomology. We prove that all three spectral sequences are isomorphic from $E_2$ onwards if $\mathcal F\/$ is a Riemannian foliation. As a consequence, we conclude that this spectral sequence is a topological invariant of the Riemannian foliation. We also compute some examples. In particular, we give an isomorphism between the $E_2$ term for a $G$-Lie foliation and the reduced cohomology of $G$ (in the sense of S.-T. Hu) with coefficients in the reduced foliated cohomology of $\mathcal F$.

Which pairs of quotients over ideals on $\mathbb{N}$ can be distinguished without assuming additional set theoretic axioms? Essentially, those that are not isomorphic under the Continuum Hypothesis. A CH-diagonalization method for constructing isomorphisms between certain quotients of countable products of finite structures is developed and used to classify quotients over ideals in a class of generalized density ideals. It is also proved that many analytic ideals give rise to quotients that are countably saturated (and therefore isomorphic under CH).

The goal of this paper is to present, from the $b$-calculus perspective, a program for analyzing generalized resolvent families and heat kernels of pseudodifferential operators on conic manifolds. To carry this out, a new class of parameter-dependent cone pseudodifferential operators is developed and studied.

We show how to extend coarse cohomology for foliations to non-Hausdorff foliations and compute several examples of Reeb type.

We point out an error in the computation of the coarse cohomology of the Sullivan foliation in our paper "Coarse sheaf cohomology for foliations."

We establish the $L^p$ boundedness for a class of singular integral operators under the $H^1$ kernel condition. These operators were introduced by Duoandikoetxea and Rubio de Francia as an extension of the classical Calderón-Zygmund operators.

Let $\Omega$ be a proper subdomain of $\mathbb{R}^n$, $n\ge 2$, and let $\partial{\Omega}$ and $\delta_{\Omega}(x)$ denote, respectively, the boundary of $\Omega$ and the Euclidean distance of the point $x\in \Omega$ to $\mathbb{R}^n \setminus\Omega$. Denote by $K(\Omega)$ the John space of all $C^1$ functions $f:\Omega\rightarrow\mathbb{R}$ with $\sup_{x\in \Omega}\delta_\Omega (x)|\nabla f(x)|<+\infty$. We study $K(\Omega)$-functions via quadratic integral forms and o-minimal structures.

A closed orientable triangulated surface is "nice" if its vertices can be assigned 4 colours in such a way that all 4 colours are used in the closed star of each edge. The 4-colouring can be interpreted as a simplicial map from the surface to the 4-vertex 2-sphere. If the surface has genus $(n-1)^2$, then the degree of this map is at least $n^2$. Conversely we show that, if $n$ is not divisible by 2 and 3, then there are "nice" surfaces of genus $(n-1)^2$ for which the degree of the above map is exactly $n^2$. Complex analytically "nice" surfaces can be viewed as minimally triangulated meromorphic functions of a Riemann surface.

In this paper, we define and study the McShane integral of functions mapping a compact interval $I_0$ in $R^m$ into a Banach space $X$. We compare this integral with the Pettis integral and prove, in particular, that the two integrals are equivalent if $X$ is reflexive and the unit ball of the dual $X^*$ satisfies an additional condition (P). This gives additional information on an implicitly stated open problem of R.A. Gordon and on the work of D.H. Fremlin and J. Mendoza.

The Thom polynomial of a singularity is a universal characteristic class that expresses the cohomology class represented by a singularity locus. In this paper we compute the Thom polynomials of real singularities up to codimension 8 in cohomology with integer coefficients.

We classify the gauge-invariant ideals in the $C^*$-algebras of infinite directed graphs, and describe the quotients as graph algebras. We then use these results to identify the gauge-invariant primitive ideals in terms of the structural properties of the graph, and describe the $K$-theory of the $C^*$-algebras of arbitrary infinite graphs.

We study the geometry and topology of properly embedded minimal surfaces in $M\times\mathbb{R}$, where $M$ is a Riemannian surface. When $M$ is a round sphere, we give examples of all genus and we prove such minimal surfaces have exactly two ends or equal $M\times\{t\}$, for some real $t$. When $M$ has non-negative curvature, we study the conformal type of minimal surfaces in $M\times\mathbb{R}$, and we prove half-space theorems. When $M$ is the hyperbolic plane, we obtain a Jenkins-Serrin type theorem.

Some transportation cost inequalities are established on the path space over a connected complete Riemannian manifold with Ricci curvature bounded from below. The reference distance on the path space is the $L^2$-norm of the Riemannian distance along paths.

The Fourier transform is considered as a Henstock-Kurzweil integral. Sufficient conditions are given for the existence of the Fourier transform and necessary and sufficient conditions are given for it to be continuous. The Riemann-Lebesgue lemma fails: Henstock-Kurzweil Fourier transforms can have arbitrarily large point-wise growth. Convolution and inversion theorems are established. An appendix gives sufficient conditions for interchanging repeated Henstock-Kurzweil integrals and gives an estimate on the integral of a product.

A conjecture of Fuglede states that a bounded measurable set $\Omega\subset\mathbb{R}^d$, of measure $1$, can tile $\mathbb{R}^d$ by translations if and only if the Hilbert space $L^2(\Omega)$ has an orthonormal basis consisting of exponentials $e_\lambda(x) = \exp \{2\pi i\langle{\lambda},{x}\rangle\}$. If $\Omega$ has the latter property it is called {\em spectral}. Let $\Omega$ be a polytope in $\mathbb{R}^d$ with the following property: there is a direction $\xi \in S^{d-1}$ such that, of all the polytope faces perpendicular to $\xi$, the total area of the faces pointing in the positive $\xi$ direction is more than the total area of the faces pointing in the negative $\xi$ direction. It is almost obvious that such a polytope $\Omega$ cannot tile space by translation. We prove in this paper that such a domain is also not spectral, which agrees with Fuglede's conjecture. As a corollary, we obtain a new proof of the fact that a convex body that is spectral is necessarily symmetric, in the case where the body is a polytope.

Starting from a nondecreasing function $K:[0,\infty)\to [0,\infty)$, we introduce a M\"obius-invariant Banach space $Q_K$ of functions analytic in the unit disk in the plane. We develop a general theory of these spaces, which yields new results and also, for special choices of $K$, gives most basic properties of $Q_p$-spaces. We have found a general criterion on the kernels $K_1$ and $K_2$, $K_1\leq K_2$, such that $Q_{K_2}\subsetneqq Q_{K_1}$, as well as necessary and sufficient conditions on $K$ so that $Q_K=\mathcal{B}$ or $Q_K =\mathcal{D}$, where the Bloch space $\mathcal{B}$ and the Dirichlet space $\mathcal{D}$ are the largest, respectively smallest, spaces of $Q_K$-type. We also consider the meromorphic counterpart $Q_K^\#$ of $Q_K$ and discuss the differences between $Q_K$-spaces and $Q_K^\#$-classes.

We study stability properties of amenable locally compact quantum groups under the bicrossed product construction. We obtain as our main result an equivalence between amenability of the bicrossed product and amenability of the matched quantum groups used as building ingredients of the bicrossed product. Finally, we give examples of non-amenable locally compact quantum groups obtained by a bicrossed product construction.

We study the inverse Galois problem with restricted ramification. Let $K$ be an algebraic number field and $G$ be a $2$-group. We consider the question whether there exists an unramified Galois extension $M/K$ with Galois group isomorphic to $G$. We study this question using the theory of embedding problems. Let $L/k$ be a Galois extension and $(\varepsilon): 1\to \mathbf{Z}/2\mathbf{Z}\to E\to \operatorname{Gal} (L/k)\to 1$ a central extension. We first investigate the existence of a Galois extension $M/L/k$ such that the Galois group $\operatorname{Gal} (M/k)$ is isomorphic to $E$ and any finite prime is unramified in $M/L$. As an application, we prove the existence of an unramified extension over cyclic quintic fields with Galois group isomorphic to $32{\Gamma}_5a_2$ under the condition that the class number is even. We also consider the Fontaine-Mazur-Boston Conjecture in the case of abelian $l$-extensions over $\mathbf{Q}$.

Let $\mu$ be a Borel measure on $\mathbb{R}^d$ which may be non-doubling. The only condition that $\mu$ must satisfy is $\mu(Q)\leq c_0l(Q)^n$ for any cube $Q\subset \mathbb{R}^d$ with sides parallel to the coordinate axes, for some fixed $n$ with $0 < n\leq d$. In this note we consider the commutators of fractional integrals with functions of the new BMO introduced by X. Tolsa.

Given a set $E\subset (0,\infty)$, the spherical maximal operator associated to the parameter set $E$ is defined as the supremum of the spherical means of a function when the radii of the spheres are in $E$. The aim of the paper is to study boundedness properties of these operators on the spaces $L^p(|x|^{\alpha})$. It is shown that the range of values of $\alpha$ for which boundedness holds behaves essentially as follows: (i) for $p > n/(n-1)$ and negative $\alpha$ the range does not depend on $E$; (ii) when $\alpha$ is positive it depends only on the Minkowski dimension of $E$; (iii) if $p < n/(n-1)$ and $\alpha$ is negative, sets with the same Minkowski dimension can give different ranges of boundedness.

Let $X$ be a rearrangement-invariant space. An operator $T: X\to X$ is called narrow if for each measurable set $A$ and each $\varepsilon > 0$ there exists $x \in X$ with $x^2= \chi_A,\ \int x d \mu = 0$ and $\| Tx \| < \varepsilon$. In particular, all compact operators are narrow. We prove that if $X$ is a Lorentz function space $L_{w,p}$ on [0,1] with $p > 2$, then there exists a constant $k_X > 1$ such that for every narrow projection $P$ on $L_{w,p}$ $\| \operatorname{Id} - P \| \geq k_X. $ This generalizes earlier results on $L_p$ and partially answers a question of E. M. Semenov. Moreover, we prove that every rearrangement-invariant function space $X$ with an absolutely continuous norm contains a complemented subspace isomorphic to $X$ which is the range of a narrow projection and a non-narrow projection. This gives a negative answer to a question of A. Plichko and M. Popov.

Let $\mu$ be a self-similar measure on $\mathbb{R}$ generated by an equicontractive iterated function system. We prove that the Hausdorff dimension of $\mu^{*n}$ tends to $1$ as $n$ tends to infinity, where $\mu^{*n}$ denotes the $n$-fold convolution of $\mu$. Similar results hold for the $L^q$ dimension and the entropy dimension of $\mu^{*n}$.