We present Shelah's famous theorem in a version for modules, together with a self-contained proof and some examples. This exposition is based on lectures given at CRM in October 2006.

We denote by $\mathbb{T}$ the unit circle and by $\mathbb{D}$ the unit disc. Let $\mathcal{B}$ be a semi-simple unital commutative Banach algebra of functions holomorphic in $\mathbb{D}$ and continuous on $\overline{\mathbb{D}}$, endowed with the pointwise product. We assume that $\mathcal{B}$ is continously imbedded in the disc algebra and satisfies the following conditions:

The space of polynomials is a dense subset of $\mathcal{B}$.

$\lim_{n\to +\infty}\|z^n\|_{\mathcal{B}}^{1/ n}=1$.

There exist $k \geq 0$ and $C > 0$ such that $$ \bigl| 1- |\lambda| \bigr|^{k} \bigl\| f \bigr\|_{\mathcal{B}} \leq C \bigl\| (z-\lambda) f \bigr\|_{\mathcal{B}}, \quad (f \in \mathcal{B},\, |\lambda| < 2).$$

When $\mathcal{B}$ satisfies in addition the analytic Ditkin condition, we give a complete characterisation of closed ideals $I$ of $\mathcal{B}$ with countable hull $h(I)$, where $$h(I) = \bigl\{ z \in \overline{\mathbb{D}} : f(z) = 0, \quad (f \in I) \bigr\}.$$ Then, we apply this result to many algebras for which the structure of all closed ideals is unknown. We consider, in particular, the weighted algebras $\ell^1(\omega$) and $L^1(\mathbb{R}^{+},\omega)$.

We describe some properties of braided generalizations of Thompson's groups, introduced by Brin and Dehornoy. We give slightly different characterizations of the braided Thompson's groups $BV$ and $\widehat{BV}$ which lead to natural presentations which emphasize one of their subgroup-containment properties. We consider pure braided versions of Thompson's group $F$. These groups, $BF$ and $\widehat{BF}$, are subgroups of the braided versions of Thompson's group $V$. Unlike $V$, elements of $F$ are order-preserving self-maps of the interval and we use pure braids together with elements of $F$ thus again preserving order. We define these pure braided groups, give normal forms for elements, and construct infinite and finite presentations of these groups.

Let $\Omega\subset\mathbb{R}^n$ be open and suppose that $f\colon \Omega\to\mathbb{R}^n$ is a bilipschitz mapping such that $Df\in BV_{\operatorname{loc}}(\Omega,\mathbb{R}^{n^2})$. We show that under these assumptions the inverse satisfies $Df^{-1}\in BV_{\operatorname{loc}}(f(\Omega),\mathbb{R}^{n^2})$.

In this paper we obtain convergence results for the series of differences of Cesàro averages along lacunary sequences in the setting of weighted $L^p$-spaces. These results give some information about how the Cesàro averages converge. The paper extends results of an earlier work by R. L. Jones and J. Rosenblatt. The operators considered are essentially convolution operators given by kernels more singular than the ones in the article by Jones and Rosenblatt.

In this paper we examine how the notion of algebra of quotients for Lie algebras ties up with the corresponding well-known concept in the associative case. Specifically, we completely characterize when a Lie algebra $Q$ is an algebra of quotients of a Lie algebra $L$ in terms of the associative algebras generated by the adjoint operators of $L$ and $Q$ respectively. In a converse direction, we also provide with new examples of algebras of quotients of Lie algebras and these come from associative algebras of quotients. In the course of our analysis, we make use of the notions of density and multiplicative semiprimeness to link our results with the maximal symmetric ring of quotients.

Let $V$ be a vector space over a field $F$. If $G\!\leq\! GL(V,F)$, the central dimension of $G$ is the $F$-dimension of the vector space $V/C_V(G)$. In [DEK] and [KS], soluble linear groups in which the set $\mathcal{L}_{\operatorname{icd}}(G)$ of all proper infinite central dimensional subgroups of $G$ satisfies the minimal condition and the maximal condition, respectively, have been described. On the other hand, in [MOS], periodic locally radical linear groups in which $\mathcal{L}_{\operatorname{icd}}(G)$ satisfies one of the weak chain conditions (the weak minimal condition or the weak maximal condition) have been characterized. In this paper, we begin the study of the non-periodic case by describing locally nilpotent linear groups in which $\mathcal{L}_{\operatorname{icd}}(G)$ satisfies one of the two weak chain conditions.

Let $\Delta $ be the open unit disc in $\mathbb{C}$. Given a continuous function $\varphi \colon b\Delta \rightarrow \mathbb{C}\setminus \{ 0\}$ denote by $\mathcal{W} (\varphi )$ the winding number of $\varphi$ around the origin. We prove that a continuous function $f\colon b\Delta\rightarrow \mathbb{C}$ extends meromorphically through $\Delta $ if and only if there is a number $N\in \mathbb{N}\cup\{ 0\}$ such that $\mathcal{W} (Pf+Q)\geq -N$ for every pair $P$, $Q$ of polynomials such that $Pf+Q\not= 0$ on $b\Delta$. If this is the case then the meromorphic extension has at most $N$ poles in $\Delta$.

We give a simple proof of the existence of parabolic curves for diffeomorphisms in $(\mathbb{C}^2,{0})$ tangent to the identity with isolated fixed point.

Energy inequalities are derived for an elliptic-hyperbolic operator arising in plasma physics. These inequalities imply the existence of distribution and weak solutions to various closed boundary-value problems. An existence theorem is proven for a related class of Keldysh equations, and the failure of expected methods for obtaining uniqueness is discussed. The proofs use ideas recently introduced by Lupo, Morawetz, and Payne for a generalized Tricomi operator. The existence of strong solutions under open boundary conditions is also proven.