In 1994, H. Mohebi and M. Radjabalipour proved that every operator in a certain class of operators on reflexive Banach spaces has infinitely many invariant subspaces. In this paper, we prove that the invariant subspace lattice for every operator in the class of operators on (general) Banach spaces is rich, and we give an example of an operator $T$ that has infinitely many invariant subspaces, while the invariant subspace lattice $\operatorname{Lat} (T)$ for $T$ is not rich. Here we call an invariant lattice subspace $\operatorname{Lat} (T)$ for the operator $T$ rich if there exists an infinite dimensional Banach space $E$ such that $\operatorname{Lat} (T)$ contains a sublattice that is order isomorphic to the lattice $\operatorname{Lat}(E)$ of all closed liner subspaces of $E$. Finally we show that the invariant subspace lattice $\operatorname{Lat} (T)$ for a bounded linear operator $T$ on a reflexive Banach space $X$ is reflexive-rich if and only if the invariant subspace lattice $\operatorname{Lat} (T^{*})$ for $T^{*}$ is reflexive-rich.

We relate the integrability of vector fields, and of the vanishing of $p$-torsion, to model-theoretic questions concerning separably closed fields, endowed canonically with a derivation. While each differential field $(F_p(t)^s,D_p)$ is known to be decidable, we show that the asymptotic theory of these fields as a class is undecidable in a strong sense. This precludes a geometric answer to certain generalizations of the Grothendieck-Katz conjecture.

Holomorphic Hermitian vector bundles over the upper half plane that admit a lift of the action of $\text{SL}(2,{\mathbb R})$ are considered. All such vector bundles are described and classified up to isomorphism.

New lower bound inequalities are obtained for generalized upcrossings of ergodic averages. The results and techniques are compared with those of E. Bishop on upper bounds. Moreover, a connection between these results and spatial oscillations is established.

We establish the local well-posedness for a new integrable equation. We prove that the equation has strong solutions that blow up in finite time and obtain the precise blow-up scenario for this equation. Moreover, we provide a framework of weak solutions for the study of soliton interaction.

We compute the flux of Killing fields through ends of constant mean curvature $1$ in hyperbolic space, and we prove a result conjectured by Rossman, Umehara and Yamada: the flux matrix defined by these authors is equivalent to the flux of Killing fields. We next give a geometric description of embedded ends of finite total curvature. In particular, we show that if such an end is asymptotic to a catenoid cousin, then we can associate an axis to it. We also compute the flux of Killing fields through these ends, and we deduce some geometric properties and analogies to minimal surfaces in Euclidean space.

We show that, given a bounded Reinhardt domain $D$ in $\mathbb{C}^n$, there exists a hyperconvex domain $\Omega$ such that $\Omega$ contains $D$ and every holomorphic function on a neighborhood of $\overline{D}$ extends to a neighborhood of $\overline{\Omega}$. As a consequence of this result, we recover an earlier result stating that every bounded fat Reinhardt domain having a Stein neighbourhoods basis must be hyperconvex. We also study the connection between the Caratheodory hyperbolicity of a Reinhardt domain and that of its envelope of holomorphy. We give an example of a Caratheodory hyperbolic Reinhardt domain in $\mathbf{C}^3$, for which the envelope of holomorphy is not Caratheodory hyperbolic, and we show that no such example exists in $\mathbf{C}^2$.

For locally compact abelian groups it is known that if the product of the measures of the support of an $L^1$-function $f$ and its Fourier transform is less than $1$, then $f = 0$ almost everywhere. This is a weak version of the classical qualitative uncertainty principle. In this paper we focus on compact groups. We obtain conditions on the structure of a compact group under which there exists a lower bound for all products of the measures of the support of an integrable function and its Fourier transform, and conditions under which this bound equals $1$. For several types of compact groups, we determine the exact set of values which the product can attain.

We study the compactness of composition operators on the Hilbert space of Dirichlet series with square summable coefficients. In particular, we give some necessary and sufficient conditions for compactness. We also describe the spectrum of such operators, and we extend our work to some weighted spaces.

We correct an error in [1], and provide a new shorter proof of Theorem B$'$.

We extend the regularity criterion of Bayer-Stillman for a graded ideal $\mathfrak {a}$ of a polynomial ring $K[\underline {\bf x}] := K [\underline {\bf x}_0, \dots , {\bf x}_r]$ over an infinite field $K$ to the situation of a graded submodule $M$ of a finitely generated graded module $U$ over a Noetherian homogeneous ring $R = \oplus_{n \geq 0}R_n$, whose base ring $R_0$ has infinite residue fields. If $R_0$ is Artinian, we construct a polynomial $\widetilde{P} \in {\mathbb Q}[{\bf x}]$, depending only on the Hilbert polynomial of $U$, such that $\operatorname{reg}(M) \leq \widetilde{P} ( \max \{ d(M), \operatorname{reg}(U) + 1 \} ) $, where $d(M)$ is the generating degree of $M$. This extends the regularity bound of Bayer-Mumford for a graded ideal $\mathfrak {a} \subseteq K[\underline {\bf x}]$ over a field $K$ to the pair $M \subseteq U$.

The Heine transformation and the theory of Bailey pairs are combined to construct partition functions that are not theta series yet are almost always $0$.

Lacey and Thiele recently obtained a new proof of Carleson's theorem on almost everywhere convergence of Fourier series. This paper is a generalization of their techniques (known broadly as time-frequency analysis) to higher dimensions. In particular, a weak-type (2,2) estimate is derived for a maximal dyadic sum operator on $\mathbb R^{n}$, $n \gt 1$. As an application one obtains a new proof of Sjölin's theorem on weak $L^{2}$ estimates for the maximal conjugated Calderón-Zygmund operator on $\mathbb R^{n}$.

We give new proof of the theorem of K. Zhang [Z] on biting convergence of Jacobian determinants for mappings of Sobolev class ${\mathscr W}^{1,n}(\Omega,\mathbb{R}^n)$. The novelty of our approach is in using ${\mathscr W}^{1,p}$-estimates with the exponents $1\leqslant p \lt n$, as developed in [IS1], [IL], [I1]. These rather strong estimates compensate for the lack of equi-integrability. The remaining arguments are fairly elementary. In particular, we are able to dispense with both the Chacon biting lemma and the Dunford-Pettis criterion for weak convergence in ${\mathscr L}^1(\Omega)$. We extend the result to the so-called Grand Sobolev setting.

Biting convergence of Jacobians for mappings whose cofactor matrices are bounded in ${\mathscr L}^{\frac n{n-1}}(\mathbb{R}^n)$ is also obtained. Possible generalizations to the wedge products of differential forms are discussed.

A space of boundary values is constructed for minimal symmetric discrete Dirac operators in the limit-circle case. A description of all maximal dissipative, maximal accretive and self-adjoint extensions of such a symmetric operator is given in terms of boundary conditions at infinity. We construct a self-adjoint dilation of a maximal dissipative operator and its incoming and outgoing spectral representations, which make it possible to determine the scattering matrix of the dilation. We also construct a functional model of the dissipative operator and its characteristic function. Finally, we prove the completeness of the system of eigenvectors and associated vectors of dissipative operators.

Let $M^n$ be a complete spacelike hypersurface with constant mean curvature $H$ in the de Sitter space $S_1^{n+1}$. We use the operator $\phi =A-HI$, where $A$ is the second fundamental form of $M$, and the roots $B_H^- \le B_H^+$ of a certain second order polynomial, to prove that either $\vert\phi\vert^2\equiv 0$ and $M$ is totally umbilical, or $B_H^-\le\sqrt{\sup \vert\phi\vert^2}\le B_H^+$. For the case $H\geq 2\sqrt{n-1}/n$ we prove the following results: for every number $B$ in the interval $[\max\{0,B_H^-\},B_H^+]$ there is an example of a complete spacelike hypersurface such that $\sqrt{\sup \vert\phi\vert^2}=B$; if $\sqrt{\sup \vert\phi\vert^2}=B_H^-$ is attained at some point, then the corresponding $M$ is a hyperbolic cylinder. We characterize the hyperbolic cylinders as the only complete spacelike hypersurfaces in $S_1^{n+1}$ with constant mean curvature, non-negative Ricci curvature and having at least two ends. We also characterize all complete spacelike hypersurfaces of constant mean curvature with two distinct principal curvatures as rotation hypersurfaces or generalized hyperbolic cylinders.

Let $(X,\,\|\,\cdot\,\|_X)$ be a Banach function space over a nonatomic probability space. We give a necessary and sufficient condition on $X$ for the inequalities $c \|f_{\infty}\|_X \leq \|S(f)\|_X \leq C \|f_{\infty}\|_X$ to hold for all uniformly integrable martingales $f=(f_n)_{n \geq 0}$, where $f_{\infty}=\lim_n f_n$ a.s. and $S(f)=\left\{f_0^{\,2}+\sum_{n=1}^{\infty} (f_n - f_{n-1})^2\right\}^{1/2}$.

Let $(G,S)$ be a finitely generated Coxeter group such that the Coxeter system is indecomposable and the canonical bilinear form is indefinite but non-degenerate. We show that the reduced $C^{\ast}$-algebra of $G$ is simple with unique normalised trace.

For an arbitrary finitely generated Coxeter group we prove the validity of a Haagerup inequality: There exist constants $C \gt 0$ and $\Lambda\in \mathbb{N}$ such that, for any function $f\in l^2(G)$ supported on elements of length $n$ with respect to the generating set $S$ and for all $h\in l^2(G)$, $\|\,f\ast h\,\| \leq C(n+1)^{\frac{3}{2}{\Lambda}}\|\,f\,\|$.

Let $\mathcal{G}$ be a smooth linear algebraic group over the ring of Witt vectors of a finite field $k$. In this paper, we study deformations of representations of a profinite group into the points $\mathcal{G}(k)$ of $\mathcal{G}$ over $k$. We show that the $\mathcal{G}$-deformation functor has a versal deformation ring, and we generalize criteria of Tilouine concerning when this ring is universal. If $\mathcal{G}$ is an algebraic subgroup of $\mathrm{GL}_n$, we study when the $\mathcal{G}$-deformation functor is a subfunctor of the $\mathrm{GL}_n$-deformation functor studied by Mazur. When $\mathcal{G}$ is an orthogonal group, this leads to studying versal versions of results of Serre and Fröhlich about the connection between Stiefel-Whitney classes, spinor norms and Hasse-Witt invariants of orthogonal Galois representations.

It is known that a unital simple $C^*$-algebra $A$ with tracial topological rank zero has real rank zero and topological stable rank one. In this note we construct unital $C^*$-algebras with tracial topological rank zero but topological stable rank two.

For a real number $p$ with $1<p<\infty$ we consider the spectrum of the $p$-Laplacian on graphs, $p$-harmonic morphisms between two graphs, and estimates for the solutions of $p$-Laplace equations on graphs. More precisely, we prove a Cheeger type inequality and a Brooks type inequality for infinite graphs. We also define $p$-harmonic morphisms and horizontally conformal maps between two graphs and prove that these two concepts are equivalent. Finally, we give some estimates for the solutions of $p$-Laplace equations, which coincide with Green kernels in the case $p=2$.