The Pukánszky invariant associates to each maximal abelian self-adjoint subalgebra (masa) $A$ in a type $\operatorname{II}_1$ factor $M$ a certain subset ot $\mathbb N\cup\{\infty\}$, denoted by $\operatorname{Puk}(A)$. We study this invariant in the context of factors generated by infinite conjugacy class discrete countable groups $G$ with masas arising from abelian subgroups $H$. Our main result is that we are able to describe $\operatorname{Puk}(VN(H))$ in terms of the algebraic structure of $H\subseteq G$, specifically by examining the double cosets of $H$ in $G$. We illustrate our characterization by generating many new values for the invariant, mainly for masas in the hyperfinite type $\operatorname{II}_1$ factor $R$.

Among other results we prove that for a Banach space $X$ and $1 < p < \infty$, all $p$-unconditionally Cauchy sequences in $X$ lie inside the range of a $Y$-valued measure of bounded variation for some Banach space $Y$ containing $X$ if and only if each $\ell_1$-valued $2$-summing map on $X$ induces a nuclear map on $X$ valued in $\ell_q$, $q$ being conjugate to $p$. We also characterise Banach spaces $X$ with the property that all $\ell_2$-valued absolutely summing maps on $X$ are already nuclear as those for which $X^\ast$ has the (GT) and (GL) properties.

Extended Bergman projections from Lebesgue classes onto all Besov spaces on the unit ball are defined and characterized. Right inverses and adjoints of the projections share the property that they are imbeddings of Besov spaces into Lebesgue classes via certain combinations of radial derivatives. Applications to the Gleason problem at arbitrary points in the ball, duality, and complex interpolation in Besov spaces are obtained. The results apply, in particular, to the Hardy space $H^2$, the Arveson space, the Dirichlet space, and the Bloch space.

For holomorphic selfmaps of the open unit disc $\U$ that are not elliptic automorphisms, the Schwarz Lemma and the Denjoy-Wolff Theorem combine to yield a remarkable result: each such map $\phi$ has a (necessarily unique) ``Denjoy-Wolff point'' $\dwp$ in the closed unit disc that attracts every orbit in the sense that the iterate sequence $(\phin)$ converges to $\dwp$ uniformly on compact subsets of $\U$. In this paper we prove that, except for the obvious counterexamples—inner functions having $\dwp\in\U$—the iterate sequence exhibits an even stronger affinity for the Denjoy-Wolff point; $\phin\goesto\dwp$ in the norm of the Hardy space $H^p$ for $1\le p<\infty$. For each such map, some subsequence of iterates converges to $\dwp$ almost everywhere on $\bdu$, and this leads us to investigate the question of almost-everywhere convergence of the entire iterate sequence. Here our work makes natural connections with two important aspects of the study of holomorphic selfmaps of the unit disc: linear-fractional models and ergodic properties of inner functions.

We investigate in this paper some rationality questions related to observable, epimorphic, and Grosshans subgroups of linear algebraic groups over non-algebraically closed fields.

Let $H$ be a separable complex Hilbert space and $B_s(H)$ the Jordan algebra of all Hermitian operators on $H$. Let $\theta:B_s(H)\to B_s(H)$ be a surjective ${\mathbb R}$-linear map which is continuous in the strong operator topology such that $\theta(x)\theta(y)+\theta(y)\theta(x)=0$ for all $x,y\in B_s(H)$ with $xy+yx=0$. We show that $\theta(x)=\lambda uxu^*$ for all $x\in B_s(H)$, where $\lambda$ is a nonzero real number and $u$ is a unitary or anti-unitary operator on $H$.

We consider $O$-sequences that occur for arithmetically Cohen-Macaulay (ACM) schemes $X$ of codimension three in ${\pp}^n$. These are Hilbert functions $\varphi$ of Artinian algebras that are quotients of the coordinate ring of $X$ by a linear system of parameters. Using suitable decompositions of $\varphi$, we determine the minimal number of generators possible in some degree $c$ for the defining ideal of any such ACM scheme having the given $O$-sequence. We apply this result to construct Artinian Gorenstein $O$-sequences $\varphi$ of codimension $3$ such that the poset of all graded Betti sequences of the Artinian Gorenstein algebras with Hilbert function $\varphi$ admits more than one minimal element. Finally, for all $3$-codimensional complete intersection $O$-sequences we obtain conditions under which the corresponding poset of graded Betti sequences has more than one minimal element.

The notion of piecewise definable metric space is introduced. It is shown that the set of all Gromov-Hausdorff limits of piecewise definable spaces belonging to a fixed bounded definable family is again a definable family. The Gromov-Hausdorff limit is taken with respect to the geodesic metric and the word definable means definable in some o-minimal structure over $\mathbb{R}$.

Banach space $X$-valued martingale transforms by a $\mathcal{B}(X)$-valued multiplier sequence are bounded on $L_p(X)$, where $1<p<\infty$ and $X$ is a UMD space, if and only if the multiplier sequence is pointwise R-bounded. This is also true for unconditionally convergent martingales in arbitrary Banach spaces.

The braid index of a surface-knot $F$ is the minimal number among the degrees of all simple surface braids whose closures are ambient isotopic to $F$. We prove that there exists a surface-knot with braid index $k$ for any positive integer $k$. To prove it, we use colorings of surface-knots by quandles and give lower bounds of the braid index of surface-knots.

We investigate complete minimal hypersurfaces in the Euclidean space ${R}^{4}$, with Gauss-Kronecker curvature identically zero. We prove that, if $f:M^{3}\rightarrow {R}^{4}$ is a complete minimal hypersurface with Gauss-Kronecker curvature identically zero, nowhere vanishing second fundamental form and scalar curvature bounded from below, then $f(M^{3})$ splits as a Euclidean product $L^{2}\times {R}$, where $L^{2}$ is a complete minimal surface in $ {R}^{3}$ with Gaussian curvature bounded from below.

We present some results related to the question whether every finite measure $\mu$ defined on a $\sigma$--algebra $\Sigma\sub \Borel[0,1]$ is countably compact. In particular, we show that for every finite measure space $(X,\Sigma,\mu)$, where $X$ is a Polish space and $\Sigma\sub \Borel(X)$, there is a regularly monocompact measure space $(\widehat{X},\widehat{\Sigma},\widehat{\mu})$ and an inverse-measure-preserving function $f:\widehat{X}\to X$.

We show that an analogue of Hilbert's Thirteenth Problem fails in the real subanalytic setting. Namely we prove that, for any integer $n$, the $o$-minimal structure generated by restricted analytic functions in $n$ variables is strictly smaller than the structure of all global subanalytic sets, whereas these two structures define the same subsets in $\mathbb{R}^{n+1}$.

The purpose of this work is to construct a Brownian motion with values in simplicial complexes with piecewise differential structure. In order to state and prove the existence of such a Brownian motion, we define a family of continuous Markov processes with values in an admissible complex. We call a process in this family an isotropic transport process. We first show that the family of isotropic processes contains a subsequence which converges weakly to a measure which we call the Wiener measure. Then, using the finite dimensional distributions of this Wiener measure, we construct a new admissible complex valued continuous Markov process, the Brownian motion. We conclude with a geometric analysis of this Brownian motion and determine the recurrent or transient behavior of such a process.

We consider sequences, of positive density $C$, of positive integers $s$ that are postulated to have the property that %%%%%%%%%%%%%%%%%%%% \begin{align} \notag S(x;a,k) &= \sum_{\substack{s\leq x \\ s\equiv a, \mod k}} 1 = f(a,k)x+O\left(x \log^{-A}x\right) \end{align} %%%%%%%%%%%%%%%%%%%% for any positive constant $A$. Let %%%%%%%%%%%%%%%%%%%% \begin{align} \notag G(x,Q) &= \sum_{k \leq Q} \sum_{0<a\leq k}E^2(x;a,k) \quad (Q\leq x), \end{align} %%%%%%%%%%%%%%%%%%%% where $E(x;a,k) = S(x;a,k) - f(a,k)x.$ Then previously we had shewn that %%%%%%%%%%%%%%%%%%%% \begin{align} \notag G(x,Q) &= D\{1+o(1)\}x^2+O\left(x^2\log^{-A}x\right) \quad (Q/x \rightarrow 0). \end{align} %%%%%%%%%%%%%%%%%%%% When $D=0,$ many particular examples of which are known, this formula supplies little information about $G(x,Q)$ and about how small it can be. The first result obtained in this paper is the lower bound

\begin{align} \notag G(x;Q) &\geq \frac{1}{12}\{3C-2C^2+\min \nolimits^2(C,1-C)+o(1)\}Q^2+o(x^2\log^{-A}x) \end{align}

that is best possible when $C=\frac{1}{2}$ or 1.

The other subject of the paper is the sum

\begin{align} \notag G_\lambda(x,Q) &= \sum_{k\leq Q} \sum_{0<a \leq \lambda k}E^2(x;a,k) \quad (\lambda <1) \end{align}

and its connection with $G(x,Q).$ If $G(x,Q)/Q^2$ be bounded, it is demonstrated that the expected limiting equality of $G_\lambda(x,Q)$ and $\lambda G(x,Q)$ can be false. On the other hand, it is shewn that this equality holds in the appropriate sense for any sequence of $Q$ for which $G(x,Q)/Q^2 \rightarrow \infty.$

We prove that the singular set $S(\mathcal{L})$ of convergence in a Colding-Minicozzi limit minimal lamination $\lc$ is a $C^{1,1}$-curve which is orthogonal to leaves of the limit minimal lamination $\mathcal{L}$ in some neighborhood of $\mathcal{S}(\mathcal{L})$. We also obtain useful information on the related limit lamination metric.

For any stratified pseudomanifold $X$ and any action of the unit circle $\mathbb{S}^1$ on $X$ preserving the stratification and the local structure, the orbit space $X/\mathbb{S}^1$ is also a stratified pseudomanifold. For each perversity $\overline{q}$ in $X$ the orbit map $\pi : X/\mathbb{S}^1$ induces a Gysin sequence relating the $\overline{q}$-intersection cohomologies of $X$ and $X/\mathbb{S}^1$. The third term of this sequence can be given by means of a spectral sequence on $X/\mathbb{S}^1 whose second term is the cohomology of the set of fixed points $X^{S^{1}}$ with values on a constructible sheaf.