By Zoran Grujic: Exploiting the localization of the vortex-stretching mechanism to small spatio-temporal scales obtained in [Zoran Gruji\'c, \textit{Localization and geometric depletion of vortex-stretching in the 3D NSE}, Comm. Math. Phys \textbf{290} (2009), 861--870] and geometric structure of the leading-order vortex-stretching term, a family of regularity classes approximating a critical, NSE-scaling invariant class is obtained.

By Gui-Qiang Chen, Zhongmin Qian: We study the initial-boundary value problem of the Navier-Stokes equations for incompressible fluids in a domain in $\mathbb{R}^{3}$ with compact and smooth boundary, subject to the kinematic and Navier boundary conditions. We first reformulate the Navier boundary condition in terms of the vorticity, which is motivated by the Hodge theory on manifolds with boundary from the viewpoint of differential geometry, and establish basic elliptic estimates for vector fields subject to the kinematic and Navier boundary conditions. Then we develop a spectral theory of the Stokes operator acting on divergence-free vector fields on a domain with the kinematic and Navier boundary conditions. Finally, we employ the spectral theory and the necessary estimates to construct the Galerkin approximate solutions and establish their convergence to global weak solutions, as well as local strong solutions, of the initial-boundary value problem. Furthermore, we show as a corollary that, when the slip length tends to zero, the weak solutions constructed converge to a solution to the incompressible Navier-Stokes equations subject to the no-slip boundary condition for almost all time. The inviscid limit of the strong solutions to the unique solution of the initial-boundary value problem with the slip boundary condition for the Euler equations is also established.

By Debraj Chakrabarti, Rasul Shafikov: A general class of singular real hypersurfaces, called \textit{subanalytic}, is defined. For a subanalytic hypersurface $M$ in $\mathbb{C}^{n}$, Cauchy-Riemann (or simply CR) functions on $M$ are defined, and certain properties of CR functions discussed. In particular, sufficient geometric conditions are given for a point $p$ on a subanalytic hypersurface $M$ to admit a germ at $p$ of a smooth CR function $f$ that cannot be holomorphically extended to either side of $M$. As a consequence it is shown that a well-known condition of the absence of complex hypersurfaces contained in a smooth real hypersurface $M$, which guarantees one-sided holomorphic extension of CR functions on $M$, is neither a necessary nor a sufficient condition for one-sided holomorphic extension in the singular case.

By Jeffrey Diller, Romain Dujardin, Vincent Guedj: We consider the dynamics of a meromorphic map on a compact Kaehler surface whose topological degree is smaller than its first dynamical degree. The latter quantity is the exponential rate at which iterates of the map expand the cohomology class of a Kaehler form. Our goal in this article and its sequels is to carry out a program for constructing and analyzing a natural measure of maximal entropy for each such map. Here we take the first step, using the linear action of the map on cohomology to construct and analyze invariant currents with special geometric structure. We also give some examples and consider in more detail the special cases where the surface is irrational or the self-intersections of the invariant currents vanish.

By Dilian Yang: In this paper, we initiate the study of endomorphisms and modular theory of the graph C*-algebras $\mathcal{O}_{\theta}$ of a 2-graph $\mathbb{F}_{\theta}^{+}$ on a single vertex. We prove that there is a semigroup isomorphism between unital endomorphisms of $\mathcal{O}_{\theta}$ and its unitary pairs with a \textit{twisted property}. We study when endomorphisms preserve the fixed point algebra $\mathfrak{F}$ of the gauge automorphisms and its canonical masa $\mathfrak{D}$. Some other properties of endomorphisms are also investigated.\par As far as the modular theory of $\mathcal{O}_{\theta}$ is concerned, we show that the algebraic *-algebra generated by the generators of $\mathcal{O}_{\theta}$ with the inner product induced from a distinguished state $\omega$ is a modular Hilbert algebra. Consequently, we obtain that the von Neumann algebra $\pi(\mathcal{O}_{\theta})''$ generated by the GNS representation of $\omega$ is an AFD factor of type III$_1$, provided $\ln m/\ln n \not\in \mathbb{Q}$. Here $m$, $n$ are the numbers of generators of $\mathbb{F}_{\theta}^{+}$ of degree $(1,0)$ and $(0,1)$, respectively.\par This work is a continuation of [Davidson, K.R., Power, S.C., Yang, D., \textit{Atomic representations of rank 2 graph algebras}, J. Funct. Anal. \textbf{255} (2008), 819--853; Davidson, K.R., Power, S.C., Yang, D., \textit{Dilation theory for rank 2 graph algebras}, J. Operator Theory (to appear); Davidson, K.R., Yang, D., \textit{Periodicity in rank 2 graph algebras}, Canad. J. Math. \textbf{61} (2009), 1239--1261].

By L. Olsen: Let $K$ be a compact subset of $\mathbb{R}^{d}$ and write $\mathcal{P}(K)$ for the family of Borel probability measures on $K$. In this paper we study different fractal and multifractal dimensions of measures $\mu$ in $\mathcal{P}(K)$ that are generic in the sense of prevalence. We first prove a general result, namely, for an arbitrary ``dimension'' function $\Delta: \mathcal{P}(K) \to \mathbb{R}$ satisfying various natural scaling and monotonicity conditions, we obtain a formula for the ``dimension'' $\Delta(\mu)$ of a prevalent measure $\mu$ in $\mathcal{P}(K)$; this is the content of Theorem 1.1. By applying Theorem 1.1 to appropriate choices of $\Delta$ we obtain the following results: \begin{itemize}\item By letting $\Delta(\mu)$ equal the (lower or upper) local dimension of $\mu$ at a point $x \in K$ and applying Theorem 1.1 to this particular choice of $\Delta$, we compute the (lower and upper) local dimension of a prevalent measure $\mu$ in $\mathcal{P}(K)$. \item By letting $\Delta(\mu)$ equal the multifractal spectrum of $\mu$ and applying Theorem 1.1 to this particular choice of $\Delta$, we compute the multifractal spectrum of a prevalent measure $\mu$ in $\mathcal{P}(K)$. \item Finally, by letting $\Delta(\mu)$ equal the Hausdorff or packing dimension of $\mu$ and applying Theorem 1.1 to this particular choice of $\Delta$, we compute the Hausdorff and packing dimension of a prevalent measure $\mu$ in $\mathcal{P}(K)$.\end{itemize} Perhaps surprisingly, in all cases our results are very different from the corresponding results for measures that are generic in the sense of Baire category.

By Anna Skripka: We construct higher order spectral shift functions, which represent the remainders of Taylor-type approximations for the value of a function at a perturbed self-adjoint operator by derivatives of the function at an initial unbounded operator. In the particular cases of the zero and the first order approximations, the corresponding spectral shift functions have been constructed by M.G. Krein [M.G. Krein, \textit{On the trace formula in perturbation theory}, Mat. Sbornik N.S. \textbf{33(75)} (1953), 597--626] and L. S. Koplienko [L.S. Koplienko, \textit{The trace formula for perturbations of nonnuclear type}, Sibirsk. Mat. Zh. \textbf{25} (1984), 62--71], respectively. The higher order spectral shift functions obtained in this paper can be expressed recursively via the lower order ones, in particular, Krein's and Koplienko's spectral shift functions. This extends the recent results of \cite{ds} for bounded operators.

By Stefano Francaviglia, Jean-Francois Lafont: Let $M$ be a complete locally compact $\mbox{CAT}(0)$-space, and $X$ an asymptotic cone of $M$. For $\gamma \subset M$ a $k$-dimensional flat, let $\gamma_{\omega}$ be the $k$-dimensional flat in $X$ obtained as the ultralimit of $\gamma$. In this paper, we identify various conditions on $\gamma_{\omega}$ that are sufficient to ensure that $\gamma$ bounds a $(k+1)$-dimensional half-flat. As applications we obtain: (1) constraints on the behavior of quasi-isometries between locally compact $\mbox{CAT}(0)$-spaces; (2) constraints on the possible non-positively curved Riemannian metrics supported by certain manifolds; (3) a correspondence between metric splittings of a complete, simply connected non-positively curved Riemannian manifolds, and metric splittings of its asymptotic cones; and (4) an elementary derivation of Gromov's rigidity theorem from the combination of the Ballmann, Burns-Spatzier rank rigidity theorem and the classic Mostow rigidity theorem.

By T. J. Christiansen, P. D. Hislop: We prove that the resonance counting functions for Schroedinger operators $H_{V} = - \Delta + V$ on $L^{2}(\mathbb{R}^{d})$, for $d \geq 2$ \textit{even}, with generic, compactly-supported, real- or complex-valued potentials $V$, have the maximal order of growth $d$ on each sheet $\Lambda_{m}$, $m \in \mathbb{Z} \setminus \{0\}$, of the logarithmic Riemann surface. We obtain this result by constructing, for each $m \in \mathbb{Z} \setminus \{0\}$, a plurisubharmonic function from a scattering determinant whose zeros on the physical sheet $\Lambda_{0}$ determine the poles on $\Lambda_m$. We prove that the order of growth of the counting function is related to a suitable estimate on this function that we establish for generic potentials. We also show that for a potential that is the characteristic function of a ball, the resonance counting function is bounded below by $C_{m} r^{d}$ on each sheet $\Lambda_{m}$, $m \in \mathbb{Z} \setminus \{0\}$.

By Irene Fonseca, Stefan Kroemer: Two-scale techniques are developed for sequences of maps $\{u_k\} \subset L^{p}(\Omega;\mathbb{R}^{M})$ satisfying a linear differential constraint $\mathcal{A}u_k = 0$. These, together with $\Gamma$-convergence arguments and using the unfolding operator, provide a homogenization result for energies of the type \begin{align*} \MoveEqLeft[5] F_{epsilon}(u) \coloneqq \int_{\Omega}f \left(x,\frac{x}{epsilon},u(x)\right)\, \mathrm{d}x\\ &\mbox{with }u \in L^{p}(\Omega;\mathbb{R}^M),\ \mathcal{A}u = 0, \end{align*} that generalizes current results in the case where $\mathcal{A} = \mbox{curl}$.

By D. Danielli, N. Garofalo, D.M. Nhieu, S. D. Pauls: In the paper [D. Danielli, N. Garofalo, D.M. Nhieu, S.D. Paulsen, \textit{Instability of graphical strips and a positive answer to the Bernstein problem in the Heisenberg group $\mathbb{H}^{1}$}, J. Differential Geom. \textbf{81} (2009), 251--295}, we proved that the only stable $C^{2}$ minimal surfaces in the first Heisenberg group $\mathbb{H}^{1}$ which are graphs over some plane and have empty characteristic locus must be vertical planes. This result represents a sub-Riemannian version of the celebrated theorem of Bernstein.\par In this paper we extend the result in [ibid.] to $C^{2}$ complete embedded minimal surfaces in $\mathbb{H}^{1}$ with empty characteristic locus. We prove that every such a surface without boundary must be a vertical plane. This result represents a sub-Riemannian counterpart of the classical theorems of Fischer-Colbrie and Schoen, [D. Fischer-Colbrie and R. Schoen, \textit{The structure of complete stable minimal surfaces in $3$-manifolds of nonnegative scalar curvature}, Comm. Pure Appl. Math. \textbf{33} (1980), 199--211], and do Carmo and Peng, [M. do Carmo and C.K. Peng, \textit{Stable complete minimal surfaces in $\mathbb{R}^{3}$ are planes}, Bull. Amer. Math. Soc. (N.S.) \textbf{1} (1979), 903--906], and answers a question posed by Lei Ni.

By Joseph A. Cima, Stephan Ramon Garcia, William T. Ross, Warren R. Wogen: A \textit{truncated Toeplitz operator} $A_{\phi}: \mathcal{K}_{\Theta} \to \mathcal{K}_{\Theta}$ is the compression of a Toeplitz operator $T_{\phi}: H^{2} \to H^{2}$ to a model space $\mathcal{K}_{\Theta} \coloneqq H^{2} \ominus \Theta H^{2}$. For $\Theta$ inner, let $\mathcal{T}_{\Theta}$ denote the set of all bounded truncated Toeplitz operators on $\mathcal{K}_{\Theta}$. Our main result is a necessary and sufficient condition on inner functions $\Theta_{1}$ and $\Theta_{2}$ which guarantees that $\mathcal{T}_{\Theta_{1}}$ and $\mathcal{T}_{\Theta_{2}}$ are spatially isomorphic (i.e., $U\mathcal{T}_{\Theta_{1}} = \mathcal{T}_{\Theta_{2}}U$ for some unitary $U: \mathcal{K}_{\Theta_{1}} \to \mathcal{K}_{\Theta_{2}}$). We also study operators which are unitarily equivalent to truncated Toeplitz operators and we prove that every operator on a finite dimensional Hilbert space is similar to a truncated Toeplitz operator.

By Alexander J. Izzo: It is shown that $C(X)$ is the only uniform algebra on a compact Hausdorff space $X$ that is invariant under a transitive action on its maximal ideal space by a locally compact group that can be approximated by Lie groups.