Let $M$ be a smoothly bounded weakly convex hypersurface in $\mathbb{C}^{n} (n \geq 4)$, $0 \in M$. Let $q \in \mathbb{N}$, $1 \leq q \le n-2$. We derive a homotopy formula for $\bar{\partial}_{b}$ without shrinking on certain submanifolds of $M$ (no finite type condition is assumed for $M$).

This is a study of Arens regularity in the context of quotients of the Fourier algebra on a non-discrete locally compact abelian group (or compact group).

(1) If a compact set $E$ of $G$ is of bounded synthesis and is the support of a pseudofunction, then $A(E)$ is weakly sequentially complete. (This implies that every point of $E$ is a Day point.)

(2) If a compact set $E$ supports a synthesizable pseudofunction, then $A(E)$ has Day points. (The existence of a Day point implies that $A(E)$ is not Arens regular.)

We use be $L^{2}$-methods of proof which do not have obvious extensions to the case of $A_{p}(E)$.

Related results, context (historical and mathematical), and open questions are given.

We study local systems arising from flat line bundles over topologically trivial families $U \rightarrow S$ of hyperplane complements in $\mathbb{P}^{n}$. Imposing some genericity condition on the monodromy, one knows that fiberwise the cohomology of the local system is concentrated in the middle dimension and is computed by the Aomoto complex, a subcomplex of global differential forms on a good compactification $\pi:X \rightarrow S$ with logarithmic poles along $D' = X \backslash U$.

The families $\mathcal{A}'$ considered are obtained by fixing a configuration $\mathcal{A}$ of hyperplanes and moving one additional hyperplane. The line bundle is the structure sheaf, endowed with the connection $d_{\mathrm{rel}} + \omega$, for a logarithmic relative differential form $\omega$. In this situation we construct the Gauß-Manin connection $\nabla$ on $R^{n}\pi_{\ast}(\Omega_{X/S}^{\bullet}(\log \mathcal{D}')$, $d^{\mathrm{rel}}$ $+ \omega)$. We show that these sheaves are free. Using the combinatorics of $\mathcal{A}'$ we give a basis for these sheaves and an algorithm to express the connection $\nabla$ in this basis. The corresponding matrix depends too much on the combinatorics of the arrangement to be given in a closed form, but we illustrate the method with some examples.

These results can be seen as a generalization of the hypergeometric functions.

The paper introduces a new approach to the Cowen-Douglas theory based on the notion of a spanning holomorphic cross-section. This approach is less geometric and enables one to obtain several additional results, including one about the similarity of operators in the Cowen-Douglass classes and another about the representation of these operators as the adjoint of multiplication by $z$ on certain Hilbert spaces of holomorphic functions.

Let $\Phi:\mathbb{S}^{1} \times M \rightarrow M$ be a smooth action of the unit circle $\mathbb{S}^{1}$ on a manifold $M$. In this work, we compute the minimal model of $M$ in terms of the orbit space $B$ and the fixed point set $F \subset B$, as a dg-module over the Sullivan's minimal model of $B$.

We improve the result of Charalambous and Evans [C-E] to show that the Betti number sequence in their example of incomparable minimals among the resolutions for a fixed Hilbert function is indeed minimal. Their example was dependent upon the graded betti numbers. We give an example of a finite length Hilbert function and two cyclic finite length modules attaining the Hilbert function for which the betti number sequences are incomparable, i.e., independent of the grading.

The following conjecture is studied. Let $G$ be a simple nonabelian group. If $H$ is any group which has the same set of character degrees as $G$, then $H \cong G \times A$, where $A$ is abelian. In the present paper this is proved if $G$ is a Suzuki group on some $SL(2,2^{f})$.

We study the pointwise behavior of the smoothed out averages $$P_{n}f(x)= \frac{1}{n} \sum_{k=1}^{n} \frac{1}{\epsilon_{k}} \int_{|t| \lt \epsilon_{k}/2} {f(T_{k+t}x)dt},$$ where $T_{t}$ is a measure preserving flow on a probability space. We show that these are good averages in $L^{P}$, $p \gt 1$, if $\epsilon_{k}$ is a convergent sequence or if they are given by stationary random variables. When $p = 1$ the averages are good if $\lim_{k \rightarrow \infty} \epsilon_{k} = \epsilon \gt 0$

We show that coarse cohomology for foliations can be defined as a sheaf cohomology and use this fact to compute two examples.

We construct nonsingular Chacon transformations with $2$-cuts of type $\mathrm{III}_{\lambda}$, for every $0 \leq \lambda \leq 1$, and type $\mathrm{II}_{\infty}$ and show that their $2$-fold Cartesian product is ergodic.

Using a preparation theorem for subanalytic functions and Lipschitz stratification for compact subanalytic sets we prove that volumes of slices of globally subanalytic sets and density have a log-analytic nature. We also prove that the set of parameters for which the volume of fiber is finite is globally subanalytic.

We give a Lojasiewicz inequality for the $o$-minimal structure generate by Rolle leaves over the globally subanalytic sets. We obtain uniform estimates in the iterated exponentials scale.

We give an alternate proof of one of the inequalities proved recently for martingales (= sums of martingale differences) in a non-commutative $L_{p}$-space, with $1 \lt p \lt \infty$, by Q. Xu and the author. This new approach is restricted to $p$ an even integer, but it yields a constant which is $O(p)$ when $p \rightarrow \infty$ and it applies to a much more general kind of sum which we call $p$-orthogonal. We use mainly combinatorial tools, namely the Möbius inversion formula for the lattice of partitions of a $p$-element set.