Manifold interpolation and model reduction

One approach to parametric and adaptive model reduction is via the interpolation of orthogonal bases, subspaces or positive definite system matrices. In all these cases, the sampled inputs stem from matrix sets that feature a geometric structure and thus form so-called matrix manifolds. This work will be featured as a chapter in the upcoming Handbook on Model Order Reduction (P. Benner, S. Grivet-Talocia, A. Quarteroni, G. Rozza, W.H.A. Schilders, L.M. Silveira, eds, to appear on DE GRUYTER) and reviews the numerical treatment of the most important matrix manifolds that arise in the context of model reduction. Moreover, the principal approaches to data interpolation and Taylor-like extrapolation on matrix manifolds are outlined and complemented by algorithms in pseudo-code.Comment: 37 pages, 4 figures, featured chapter of upcoming "Handbook on Model Order Reduction

Adjoint orbits, generalised parallelisable spaces and consistent truncations

The aim of this note is to present some new explicit examples of $O(d,d)$-generalised Leibniz parallelisable spaces arising as the normal bundles of adjoint orbits $\mathcal{O}$ of some semi-simple Lie group $G$. Using this construction, an explicit expression for a generalised frame is given in the case when the orbits are regular, but subtleties arise when they become degenerate. In the case of regular orbits, the resulting space is a globally flat fiber bundle over $\mathcal{O}$ which can be made compact, allowing for a generalised Scherk-Schwartz reduction. This means these spaces should admit consistent supergravity truncations. For degenerate orbits, the procedure hinges on the existence of a suitable metric, allowing for a consistent normalisation of the generalised frame.Comment: 10 pages; typos corrected, references adde

Field reduction and linear sets in finite geometry

Based on the simple and well understood concept of subfields in a finite field, the technique called `field reduction' has proved to be a very useful and powerful tool in finite geometry. In this paper we elaborate on this technique. Field reduction for projective and polar spaces is formalized and the links with Desarguesian spreads and linear sets are explained in detail. Recent results and some fundamental ques- tions about linear sets and scattered spaces are studied. The relevance of field reduction is illustrated by discussing applications to blocking sets and semifields

Geometry of Hyper-K\"ahler Connections with Torsion

The internal space of a N=4 supersymmetric model with Wess-Zumino term has a connection with totally skew-symmetric torsion and holonomy in \SP(n). We study the mathematical background of this type of connections. In particular, we relate it to classical Hermitian geometry construct homogeneous as well as inhomogeneous examples, characterize it in terms of holomorphic data, develop its potential theory and reduction theory.Comment: 21 pages, LaTe