We will expound on the theme that the spectrum and arithmetic of compact, locally symmetric spaces arising from congruent arithmetic lattices should mutually determine each other.

One side of these expectations can be considered as an archimedean analogue of Tate's conjecture. We verify that the arithmetic is determined in the examples constructed by Sunada.

On the other hand for the spaces arising from quaternion division algebras we give examples of isospectral but non-isometric spaces generalizing the examples constructed by Vigneras. Such examples can be interpreted as saying that `Galois conjugation' preserves the spectrum of such spaces.

The design and processing of materials with novel physical and mechanical properties requires a fundamental understanding of the connections between processing, microstructure, and properties. For example, mechanical properties in pure metals and alloys can be varied by manipulating the polycrystalline grain size or the size of the compositional domains through heat treatment, while elastic strain provides a way to tune the optical properties of self-assembled quantum dots during growth. In an analogous manner, the biological function of cell membranes is strongly affected by the details of the local "microstructure".

Typically, microstructural evolution takes place across multiple length and time scales, ranging from atomistic to mesoscopic ones. In this talk I will describe our recent efforts in developing physically-based, coarse-grained continuum models, which bridge the atomistic and mesoscopic scales, to elucidate lateral organization and non-equilibrium dynamics of heterogeneous lipid bilayers. In particular, I will focus on spatially organized, dynamic heterogeneities in the local lipid composition ("lipid rafts") which have been implicated in many important cellular processes including signal transduction, membrane trafficking, cytoskeleton organization, and pathogen entry.

Duality for extended real-valued convex functions is a well-studied, even classical subject based on works of Fenchel, Moreau, Rockafellar, among many others. A corresponding satisfying theory for functions mapping into the power set of a partially ordered locally convex space is still missing. Such a theory seems to be very desirable since it has already been observed e.g. by Luc in 1989 that the dual of a convex vector optimization problem ’is set-valued in nature’. Moreover, the concept of convex set-valued risk measures has been defined recently in financial mathematics which asks for a corresponding dual representation theory.

We shall present a duality concept that is based on a new notion of affine minorants for set-valued functions and show that almost every concept (e.g. properness, sublinearity, conjugates, inf-convolution) and result (e.g. biconjugation and Fenchel-Rockafellar duality theorems) known in the scalar case can be formulated within this new framework. A special feature of the methodology is that proofs do not rely on the corresponding scalar theory – as in almost every duality theory for vector optimization problems. On the other hand, every main result can equivalently be expressed as a result for a family of scalar problems. Finally, we shall show the theory at work when applied to linear vector optimization problems and to set-valued risk measures.