European Union Pension Directive

[Excerpt] This Directive thus represents a first step on the way to an internal market for occupational retirement provision organised on a European scale. By setting the ‘prudent person’ rule as the underlying principle for capital investment and making it possible for institutions to operate across borders, the redirection of savings into the sector of occupational retirement provision is encouraged, thus contributing to economic and social progress. The prudential rules laid down in this Directive are intended both to guarantee a high degree of security for future pensioners through the imposition of stringent supervisory standards, and to clear the way for the efficient management of occupational pension schemes

Conformal geometry of surfaces in the Lagrangian--Grassmannian and second order PDE

Of all real Lagrangian--Grassmannians $LG(n,2n)$, only $LG(2,4)$ admits a distinguished (Lorentzian) conformal structure and hence is identified with the indefinite M\"obius space $S^{1,2}$. Using Cartan's method of moving frames, we study hyperbolic (timelike) surfaces in $LG(2,4)$ modulo the conformal symplectic group $CSp(4,R)$. This $CSp(4,R)$-invariant classification is also a contact-invariant classification of (in general, highly non-linear) second order scalar hyperbolic PDE in the plane. Via $LG(2,4)$, we give a simple geometric argument for the invariance of the general hyperbolic Monge--Amp\`ere equation and the relative invariants which characterize it. For hyperbolic PDE of non-Monge--Amp\`ere type, we demonstrate the existence of a geometrically associated ``conjugate'' PDE. Finally, we give the first known example of a Dupin cyclide in a Lorentzian space

Invariant Yang-Mills connections over Non-Reductive Pseudo-Riemannian Homogeneous Spaces

We study invariant gauge fields over the 4-dimensional non-reductive pseudo-Riemannian homogeneous spaces G/K recently classified by Fels & Renner (2006). Given H compact semi-simple, classification results are obtained for principal H-bundles over G/K admitting: (1) a G-action (by bundle automorphisms) projecting to left multiplication on the base, and (2) at least one G-invariant connection. There are two cases which admit nontrivial examples of such bundles and all G-invariant connections on these bundles are Yang-Mills. The validity of the principle of symmetric criticality (PSC) is investigated in the context of the bundle of connections and is shown to fail for all but one of the Fels-Renner cases. This failure arises from degeneracy of the scalar product on pseudo-tensorial forms restricted to the space of symmetric variations of an invariant connection. In the exceptional case where PSC is valid, there is a unique G-invariant connection which is moreover universal, i.e. it is the solution of the Euler-Lagrange equations associated to any G-invariant Lagrangian on the bundle of connections. This solution is a canonical connection associated with a weaker notion of reductivity which we introduce.Comment: 34 pages; minor typos corrected; to appear in Transactions of the AM

The New Hampshire, Vol. 107, No. 01 (Aug. 24, 2017)

An independant student produced newspaper from the University of New Hampshire