Graded hypoellipticity of BGG sequences.

This article studies hypoellipticity on general filtered manifolds. We extend the Rockland criterion to a pseudodifferential calculus on filtered manifolds, construct a parametrix and describe its precise analytic structure. We use this result to study Rockland sequences, a notion generalizing elliptic sequences to filtered manifolds. The main application that we present is to the analysis of the Bernstein-Gelfand-Gelfand (BGG) sequences over regular parabolic geometries. We do this by generalizing the BGG machinery to more general filtered manifolds (in a non-canonical way) and show that the generalized BGG sequences are Rockland in a graded sense.

Analytic Torsion of Generic Rank Two Distributions in Dimension Five.

We propose an analytic torsion for the Rumin complex associated with generic rank two distributions on closed 5-manifolds. This torsion behaves as expected with respect to Poincaré duality and finite coverings. We establish anomaly formulas, expressing the dependence on the sub-Riemannian metric and the 2-plane bundle in terms of integrals over local quantities. For certain nilmanifolds, we are able to show that this torsion coincides with the Ray-Singer analytic torsion, up to a constant.

Nonlinear flag manifolds as coadjoint orbits.

A nonlinear flag is a finite sequence of nested closed submanifolds. We study the geometry of Fréchet manifolds of nonlinear flags, in this way generalizing the nonlinear Grassmannians. As an application, we describe a class of coadjoint orbits of the group of Hamiltonian diffeomorphisms that consist of nested symplectic submanifolds, i.e., symplectic nonlinear flags.

Analytic Eigenbranches in the Semi-classical Limit.

We consider a one parameter family of Laplacians on a closed manifold and study the semi-classical limit of its analytically parametrized eigenvalues. Our results establish a vector valued analogue of a theorem for scalar Schrödinger operators on Euclidean space by Luc Hillairet which applies to geometric operators like Witten's Laplacian on differential forms.

The Heat Asymptotics on Filtered Manifolds.

The short-time heat kernel expansion of elliptic operators provides a link between local and global features of classical geometries. For many geometric structures related to (non-)involutive distributions, the natural differential operators tend to be Rockland, hence hypoelliptic. In this paper, we establish a universal heat kernel expansion for formally self-adjoint non-negative Rockland differential operators on general closed filtered manifolds. The main ingredient is the analysis of parametrices in a recently constructed calculus adapted to these geometric structures. The heat expansion implies that the new calculus, a more general version of the Heisenberg calculus, also has a non-commutative residue. Many of the well-known implications of the heat expansion such as, the structure of the complex powers, the heat trace asymptotics, the continuation of the zeta function, as well as Weyl's law for the eigenvalue asymptotics, can be adapted to this calculus. Other consequences include a McKean-Singer type formula for the index of Rockland differential operators. We illustrate some of these results by providing a more explicit description of Weyl's law for Rumin-Seshadri operators associated with curved BGG sequences over 5-manifolds equipped with a rank-two distribution of Cartan type.

Assembling surface mounted components on ink-jet printed double sided paper circuit board.

Printed electronics is a rapidly developing field where many components can already be manufactured on flexible substrates by printing or by other high speed manufacturing methods. However, the functionality of even the most inexpensive microcontroller or other integrated circuit is, at the present time and for the foreseeable future, out of reach by means of fully printed components. Therefore, it is of interest to investigate hybrid printed electronics, where regular electrical components are mounted on flexible substrates to achieve high functionality at a low cost. Moreover, the use of paper as a substrate for printed electronics is of growing interest because it is an environmentally friendly and renewable material and is, additionally, the main material used for many packages in which electronics functionalities could be integrated. One of the challenges for such hybrid printed electronics is the mounting of the components and the interconnection between layers on flexible substrates with printed conductive tracks that should provide as low a resistance as possible while still being able to be used in a high speed manufacturing process. In this article, several conductive adhesives are evaluated as well as soldering for mounting surface mounted components on a paper circuit board with ink-jet printed tracks and, in addition, a double sided Arduino compatible circuit board is manufactured and programmed.