We introduce a new method of proving Poisson limit laws in the theory of dynamical systems, which is based on the Chen-Stein method (\cite{Ch}, \cite{St}) combined with the analysis of the homoclinic Laplace operator in \cite{Go} and some other homoclinic considerations. This is accomplished for the hyperbolic toral automorphism $T$ and the normalized Haar measure $P$. Let $(G_n)_{n \ge 0}$ be a sequence of measurable sets with no periodic points among its accumulation points and such that $P(G_n) \to 0$ as $n \to \infty$, and let $(s(n))_{n > 0}$ be a sequence of positive integers such that $\lim_{n\to \infty} s(n)P(G_n)=\lambda$ for some $\lambda>0$. Then, under some additional assumptions about $(G_n)_{n \ge 0}$, we prove that for every integer $k \ge 0$

\[ P\left(\sum_{i=1}^{s(n)} \one _{G_n}\circ T^{i-1} = k\right) \to \lambda^k \exp { (- \lambda)} /k! \]

as $n \to \infty$. Of independent interest is an upper mixing-type estimate, which is one of our main tools.

Given a foliation $\mathcal F$ of a compact manifold $M$ and a vector field $X$ on $M$, we provide some necessary and sufficient conditions for $X$ to become the mean curvature vector of (the leaves of) $\mathcal F$ with respect to some Riemannian metric on $M$.

For $n\ge 2$ we classify all connected $n$-dimensional complex manifolds admitting effective actions of the special unitary group $SU_n$ by biholomorphic transformations.

We consider an ergodic measure-preserving system in which we fix a measurable partition $\mathcal{A}$ and a set $B$ of nontrivial measure. In a version of the Shannon-McMillan-Breiman Theorem, for almost every $x$, we estimate the rate of the exponential decay of the measure of the cell containing $x$ of the partition obtained by observing the process only at the times $n$ when $T^nx\in B$. Next, we estimate the rate of the exponential growth of the first return time of $x$ to this cell. Then we apply these estimates to topological dynamics. We prove that a partition with zero measure boundaries can be modified to an open cover so that the S-M-B theorem still holds (up to $\epsilon$) for this cover, and we derive the \en\ \fu\ on \im s from the rate of the exponential growth of the first return time to the $(n,\epsilon)$-ball around $x$.

We show that if a family of $\mathcal{A}$-harmonic functions that admits a common growth condition is closed in $L^p_{\operatorname{loc}}$, then this family is locally compact on a dense open set under a family of topologies, all generated by norms. This implies that when this family of functions is a vector space, then such a vector space of $\mathcal{A}$-harmonic functions is finite dimensional if and only if it is closed in $L^p_{\operatorname{loc}}$. We then apply our theorem to the family of all $p$-harmonic functions on the plane with polynomial growth at most $d$ to show that this family is essentially small.

Let $M$ be an $n$-dimensional complete connected Riemannian manifold with sectional curvature $K_M\geq 1$ and radius $\operatorname{rad}(M)>\pi /2$. For any $x\in M$, denote by $\operatorname{rad} (x)$ and $\rho (x)$ the radius and conjugate radius of $M$ at $x$, respectively. In this paper we show that if $\operatorname{rad} (x)\leq \rho (x)$ for all $x\in M$, then $M$ is isometric to a Euclidean $n$-sphere. We also show that the radius of any connected nontrivial (i.e., not reduced to a point) closed totally geodesic submanifold of $M$ is greater than or equal to that of $M$.

Let $\a$ be a C$^*$-algebra. In this paper the sets $\ii$ of partial isometries and $\ii_\Delta\subset\ii$ of partial unitaries (i.e., partial isometries which commute with their adjoints) are studied from a differential geometric point of view. These sets are complemented submanifolds of $\a$. Special attention is paid to geodesic curves. The space $\ii$ is a homogeneous reductive space of the group $U_\a \times U_\a$, where $U_\a$ denotes the unitary group of $\a$, and geodesics are computed in a standard fashion. Here we study the problem of the existence and uniqueness of geodesics joining two given endpoints. The space $\ii_\Delta$ is \emph{not} homogeneous, and therefore a completely different treatment is given. A principal bundle with base space $\ii_\Delta$ is introduced, and a natural connection in it defined. Additional data, namely certain translating maps, enable one to produce a \emph{linear} connection in $\ii_\Delta$, whose geodesics are characterized.

A well known result of Privalov shows that if $f$ is a function that is analytic in the unit disc $\Delta =\{z\in \mathbb{C} : \vert z\vert <1\} $, then the condition $f'\in H\sp 1$ implies that $f$ has a continuous extension to the closed unit disc. Consequently, if $B$ is an infinite Blaschke product, then $B'\notin H\sp 1$. This has been proved to be sharp in a very strong sense. Indeed, for any given positive and continuous function $\phi $ defined on $[0, 1)$ with $\phi (r)\to\infty $ as $r\to 1$, one can construct an infinite Blaschke product $B$ having the property that

\[ M_1(r,B')\defeq \frac{1}{2\pi } \int_{-\pi }\sp\pi\vert B'(re\sp{it})\vert\,dt=\og\left (\phi (r)\right ) ,\quad\hbox{as $r\to 1$.} \tag{$*$} \]

All examples of Blaschke products constructed so far to prove this result have their zeros located on a ray. Thus it is natural to ask whether an infinite Blaschke product $B$ such that the integral means $M_1(r,B')$ grow very slowly must satisfy a condition \lq\lq close\rq\rq \, to that of having its zeros located on a ray. More generally, we may formulate the following question: Let $B$ be an infinite Blaschke product and let $\{ a_n\} _{n=1}\sp\infty $ be the sequence of its zeros. Do restrictions on the growth of the integral means $M_1(r,B')$ imply some restrictions on the sequence $\{ \Arg (a_n)\} _{n=1}\sp\infty $?

In this paper we prove that the answer to these questions is negative in a very strong sense. Indeed, for any function $\phi $ as above we shall construct two new and quite different classes of examples of infinite Blaschke products $B$ satisfying ($*$) with the property that every point of $\partial \Delta$ is an accumulation point of the sequence of zeros of $B$.

We give a canonical construction of a balanced big Cohen-Macaulay algebra for a domain of finite type over $\mathbb C$ by taking ultraproducts of absolute integral closures in positive characteristic. This yields a new tight closure characterization of rational singularities in characteristic zero.

A net in the group algebra of a locally compact group which commutes asymptotically with elements from the measure algebra is called quasi-central. In this paper we provide new characterizations of locally compact groups whose group algebras possess quasi-central bounded approximate identities. Reiter-type and structural conditions for such groups are obtained which indicate that these groups behave much like the tractable [SIN]-groups. A general notion of an amenable action on the predual of a von Neumann algebra is developed to prove these theorems. Applications to the cohomology of group and Fourier algebras are discussed.

In this paper, we investigate the commutative algebra of the cohomology ring $H^*(G,k)$ of a finite group $G$ over a field $k$. We relate the concept of quasi-regular sequence, introduced by Benson and Carlson, to the local cohomology of the cohomology ring. We give some slightly strengthened versions of quasi-regularity, and relate one of them to Castelnuovo--Mumford regularity. We prove that the existence of a quasi-regular sequence in either the original sense or the strengthened ones is true if and only if the Dickson invariants form a quasi-regular sequence in the same sense. The proof involves the notion of virtual projectivity, introduced by Carlson, Peng and Wheeler.

As a by-product of this investigation, we give a new proof of the Bourguiba--Zarati theorem on depth and Dickson invariants, in the context of finite group cohomology, without using the machinery of unstable modules over the Steenrod algebra.

Finally, we describe an improvement of Carlson's algorithm for computing the cohomology of a finite group using a finite initial segment of a projective resolution of the trivial module. In contrast to Carlson's algorithm, ours does not depend on verifying any conjectures during the course of the calculation, and is always guaranteed to work.

Assume that $\Gamma$ is a free group on $n$ generators, where $2\le n< +\infty$. Let $\Omega $ be an infinite subset of $\Gamma$ such that $\Gamma \setminus \Omega$ is also infinite, and let $P$ be the projection on the subspace $l^2(\Omega )$ of $l^2(\Gamma )$. We prove that, for some choices of $\Omega$, the C*-algebra $C^*_r(\Gamma ,P)$ generated by the reduced group C*-algebra $C^*_r\Gamma$ and the projection $P$ has exactly two non-trivial, stable, closed ideals of real rank zero. We also give a detailed analysis of the Toeplitz algebra generated by the restrictions of operators in $C^*_r(\Gamma ,P)$ on the subspace $l^2(\Omega )$.

n this paper we derive sharp upper bounds for the first positive eigenvalue of the linearized operator of the higher order mean curvature of a closed hypersurface immersed into a Riemannian space form. Our bounds are extrinsic in the sense that they are given in terms of the higher order mean curvatures and the center(s) of gravity of the hypersurface, and they extend previous bounds recently given by Veeravalli [Ve] for the first positive eigenvalue of the Laplacian operator.

We develop a theory of G-dimension over local homomorphisms which encompasses the classical theory of G-dimension for finitely generated modules over local rings. As an application, we prove that a local ring $R$ of characteristic $p$ is Gorenstein if and only if it possesses a nonzero finitely generated module of finite projective dimension that has finite G-dimension when considered as an $R$-module via some power of the Frobenius endomorphism of $R$. We also prove results that track the behavior of Gorenstein properties of local homomorphisms under composition and decomposition.

In this paper we ask the following question: What is the minimal value of the difference $\ehk(I) - \ehk(I')$ for ideals $I' \supseteq I$ with $l_A(I'/I) =1$? In order to answer to this question, we define the notion of minimal relative Hilbert-Kunz multiplicity for strongly $F$-regular rings. We calculate this invariant for quotient singularities and for the coordinate rings of Segre embeddings: $\bbP^{r-1} \times \bbP^{s-1} \hookrightarrow \bbP^{rs-1}$.

We discuss vanishing of cohomology of finite modules over Cohen-Macaulay local rings $(R, \mathfrak m)$. Special attention is given to the case when the modules are annihilated by $\mathfrak m^2$. (Note that if $\mathfrak m^3=0$, then we can assume the modules satisfy this condition.) In this case we obtain effective versions of conjectures of Auslander-Reiten and Tachikawa.

Let $\mathcal{F}$ be a family of holomorphic functions in a domain $D$; let $k$ be a positive integer; let $h$ be a positive number; and let $a$ be a function holomorphic in $D$ such that $a(z)\not= 0$ for $z\in D$. For $k\not= 2$ we show that if, for every $f\in \mathcal{F}$, all zeros of $f$ have multiplicity at least $ k$, $f(z)=0$ $\Longrightarrow $ $f^{(k)}(z)=a(z)$, and $f^{(k)}(z)=a(z)$ $\Longrightarrow $ $|f^{(k+1)}(z)|\le h$, then $\mathcal{F}$ is normal in $D$. For $k=2$ we prove the following result: Let $s\ge 4$ be an even integer. If, for every $f\in \mathcal{F}$, all zeros of $f$ have multiplicity at least $ 2$, $f(z)=0$ $\Longrightarrow $ $f''(z)=a(z)$, and $f''(z)=a(z)$ $\Longrightarrow $ $|f'''(z)|+|f^{(s)}(z)|\le h$, then $\mathcal{F}$ is normal in $D$. This improves the well-known normality criterion of Miranda.

We study the abstract geometric notion of unitaries in a Banach space characterized in terms of the equivalence of the norm determined by the state space.

Let $p\geq 5$ be prime, $\mathfrak{S}_p$ the set of all characteristic $p$ supersingular j-invariants in $\mathbb{F}_p-\{0,1728\}$, and $\mathfrak{M}_p$ the set of all monic irreducible quadratic polynomials in $\mathbb{F}_p[x]$ whose roots are supersingular j-invariants. A theorem of Dwork and Koike asserts that there are integers $A_p(\alpha),B_p(g),C_p(g)$, and a polynomial $D_p(x)\in\mathbb{F}_p[x]$ of degree $p-1$, for which

\begin{multline*} j(pz) \equiv j(z)^p + pD_p(j(z)) \\ + p\sum_{\alpha\in\mathfrak{S}_p}\frac{A_p(\alpha)}{j(z)-\alpha} + p \sum_{ g(x) \in \mathfrak{M}_p} \frac{B_p(g)j(z)+C_p(g)}{g(j(z))} \pmod{p^2}. \end{multline*}

It is natural to seek a description of the polynomials $D_p(x)$. Here we provide such a description in terms of certain Hecke polynomials.