Abstract:
We relate generalized Lebesgue decompositions of measures along curve fragments ("Alberti representations") and Weaver derivations. This correspondence leads to a geometric characterization of the local norm on the Weaver cotangent bundle of a metric measure space $(X,\mu)$: the local norm of a form $df$ "sees" how fast $f$ grows on curve fragments "seen" by $\mu$. This implies a new characterization of differentiability spaces in terms of the $\mu$-a.e. equality of the local norm of $df$ and the local Lipschitz constant of $f$. As a consequence, the "Lip-lip" inequality of Keith must be an equality. We also provide dimensional bounds for the module of derivations in terms of the Assouad dimension of $X$.

Abstract:
We investigate the relationship between measurable differentiable structures on doubling metric measure spaces and derivations. We prove: [1] a decomposition theorem for the module of derivations into free modules; [2] the existence of a measurable differentiable structure assuming that one can control the pointwise upper Lipschitz constant of a function through derivations; [3] an extension of a result of Keith about the choice of chart functions.

Abstract:
Using an inverse system of metric graphs as in: J. Cheeger and B. Kleiner, "Inverse limit spaces satisfying a Poincar\'e inequality", we provide a simple example of a metric space $X$ that admits Poincar\'e inequalities for a continuum of mutually singular measures.

Abstract:
We relate Ambrosio-Kirchheim metric currents to Alberti representations and Weaver derivations. In particular, given a metric current $T$, we show that if the module $\mathscr{X}(\|T\|)$ of Weaver derivations is finitely generated, then $T$ can be represented in terms of derivations; this extends previous results of Williams. Applications of this theory include an approximation of $1$-dimensional metric currents in terms of normal currents and the construction of Alberti representations in the directions of vector fields.

Abstract:
We show that at generic points blow-ups/tangents of differentiability spaces are still differentiability spaces; this implies that an analytic condition introduced by Keith as an inequality (and later proved to actually be an equality) passes to tangents. As an application, we characterize the $p$-weak gradient on iterated blow-ups of differentiability spaces.

Abstract:
For each $\beta>1$ we construct a family $F_\beta$ of metric measure spaces which is closed under the operation of taking weak-tangents (i.e.~blow-ups), and such that each element of $F_\beta$ admits a $(1,P)$-Poincar\'e inequality if and only if $P>\beta$.

Abstract:
We construct new examples of normal (metric) currents using inverse systems of cube complexes. For any $N\ge 2$ we provide examples of $N$-dimensional normal currents whose associated vector fields are simple, and whose supports are purely $2$-unrectifiable and have Nagata dimension $N$. We show that in $l^\infty$ normal currents can be realized as limits in the flat distance of currents associated to cube complexes.

Abstract:
The paper examines the relationship between banking and securities activities in the light of financial market developments (securitisation, institutionalization of investment, emergence of complex financial instruments, conglomeration and consolidation), with particular reference to Europe. The enhanced links between banking and securitiesbusinesses have generated increased and new risks to financial institutions. However, banks' stability remains crucial for the stability of the financial system as a whole, because of their unique role as provider of liquidity. The paper also addresses the implications of the banking-securities combination for regulatory and supervisory arrangements. The exporting of prudential requirements traditional in banking (such as capital ratios) into the securities field, and the importing of securities regulation (such as transparency requirements) into the banking sector, can be deemed mutually beneficial. As regards supervision, there is a need to monitor the continued effectiveness of the current framework. This entails strengthening co-operation both at the national level and on a cross-border basis among sectoral supervisors in the micro-prudential field, and between them and central banks in the macro-prudential field.

Abstract:
The paper examines the relationship between banking and securities activities in the light of financial market developments (securitisation, institutionalization of investment, emergence of complex financial instruments, conglomeration and consolidation), with particular reference to Europe. The enhanced links between banking and securities businesses have generated increased and new risks to financial institutions. However, banks' stability remains crucial for the stability of the financial system as a whole, because of their unique role as provider of liquidity. The paper also addresses the implications of the banking-securities combination for regulatory and supervisory arrangements. The exporting of prudential requirements traditional in banking (such as capital ratios) into the securities field, and the importing of securities regulation (such as transparency requirements) into the banking sector, can be deemed mutually beneficial. As regards supervision, there is a need to monitor the continued effectiveness of the current framework. This entails strengthening co-operation both at the national level and on a cross-border basis among sectoral supervisors in the micro-prudential field, and between them and central banks in the macro-prudential field.

Abstract:
For every n, we construct a metric measure space that is doubling, satisfies a Poincare inequality in the sense of Heinonen-Koskela, has topological dimension n, and has a measurable tangent bundle of dimension 1.