Preface

The roots of the project to write this book originated in the early nineteen–

nineties, when several streams of mathematical ideas met after being developed

more or less separately for several decades. This resulted in an amazing interac-

tion between active groups of mathematicians working in several directions. One

of these directions was the study of conformally invariant differential operators re-

lated, on the one hand, to Penrose’s twistor program (M.G. Eastwood, T.N. Bailey,

R.J. Baston, C.R. Graham, A.R. Gover, and others) and, on the other hand, to

hypercomplex analysis (V. Souˇ cek, F. Sommen, and others). It turned out that,

via the canonical Cartan connection or equivalent data, conformal geometry is just

one instance of a much more general picture. Over the years we noticed that the

foundations for this general picture had already been developed in the pioneering

work of N. Tanaka. His work was set in the language of the equivalence problem

and of differential systems, but independent of the much better disseminated de-

velopments of the theory of differential systems linked to names like S.S. Chern,

R. Bryant, and M. Kuranishi. While Tanaka’s work did not become widely known,

it was further developed, in particular, by K. Yamaguchi and T. Morimoto, who

put it in the setting of filtered manifolds and applied it to the geometric study

of systems of PDE’s. A lot of input and stimulus also came from Ch. Fefferman’s

work in complex analysis and geometric function theory, in particular, his parabolic

invariant theory program and the relation between CR–structures and conformal

structures. Finally, via the homogeneous models, all of these studies have close

relations to various parts of representation theory of semisimple Lie groups and Lie

algebras, developed for example by T. Branson and B. Ørsted.

Enjoying the opportunities offered by the newly emerging International Erwin

Schr¨ odinger Institute for Mathematical Physics (ESI) in Vienna as well as the long

lasting tradition of the international Winter Schools “Geometry and Physics” held

every year in Srn´ ı, Czech Republic, the authors of this book started a long and

fruitful collaboration with most of the above mentioned people. Step by step, all of

the general concepts and problems were traced back to old masters like Schouten,

Veblen, Thomas, and Cartan, and a broad research program led to a conceptual

understanding of the common background of the various approaches developed

more recently. In the late nineties, the general version of the invariant calculus

for a vast class of geometrical structures extended the tools for geometric analy-

sis on homogeneous vector bundles and the direct applications of representation

theory expanded in this way to situations involving curvatures. In particular, the

celebrated Bernstein–Gelfand–Gelfand resolutions were recovered in the realm of

general parabolic geometries by V. Souˇ cek and the authors and the cohomological

substance of all these constructions was clarified.

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