These pedagogical lectures present some material, classical or more recent, on (Rational) Conformal Field Theories and their general setting " in the bulk " or in the presence of a boundary. Two well posed problems are the classification of modular invariant partition functions and the determination of boundary conditions consistent with conformal invari-ance. It is shown why the two problems a...

In this paper, we prove upper bounds for the volume spectrum of a Riemannian manifold that depend only on volume, dimension and conformal invariant.

We use the Virasoro master equation to study the space of Lie h-invariant conformal field theories, which includes the standard rational conformal field theories as a small subspace. In a detailed example, we apply the general theory to characterize and study the Lie h-invariant graphs, which classify the Lie h-invariant conformal field theories of the diagonal ansatz on SO(n). The Lie characte...

In this note we investigate a fourth-order geometric flow on closed, compact Riemannian manifolds of dimension greater than four. In the presence of a positive conformal invariant it can be shown that this flow must be positivity preserving.

For polynomials f on the complex plane with a dendrite Julia set we study invariant probability measures, obtained from a reference measure. To do this we follow Keller [K1] in constructing canonical Markov extensions. We discuss ‘liftability’ of measures (both f -invariant and non-invariant) to the Markov extension, showing that invariant measures are liftable if and only if they have a positi...

We demonstrate that any scale-invariant mechanics of one variable exhibits not only 0+1 conformal symmetry, but also the symmetries of a full Virasoro algebra. We discuss the implications for the adS/CFT correspondence.

We demonstrate that any scale-invariant mechanics of one variable exhibits not only 0+1 conformal symmetry, but also the symmetries of a full Virasoro algebra. We discuss the implications for the adS/CFT correspondence.

We discuss Weyl (conformal) transformations in two-dimensional matterless dilaton gravity. We argue that both classical and quantum dilaton gravity theories are invariant under Weyl transformations. PACS number(s): 04.60.Kz, 04.20.Cv

For polynomials f on the complex plane with a dendrite Julia set we study invariant probability measures, obtained from a reference measure. To do this we follow Keller [K1] in constructing canon-ical Markov extensions. We discuss " liftability " of measures (both f – invariant and non–invariant) to the Markov extension, showing that invariant measures are liftable if and only if they have a po...

This article presents a new definition of Branson’s Q-curvature in even dimensional conformal geometry. The Q-curvature is a generalization of the scalar curvature in dimension 2: it satisfies an analogous transformation law under conformal rescalings of the metric and on conformally flat manifolds its integral is a multiple of the Euler characteristic. Our approach is motivated by the recent w...