The celebrated Riemann mapping theorem of one complex variable theory says that any proper, simply connected region in the plane is conformally equivalent to the unit disc. This astonishing result has powerfully affected the entire subject, both in the pure and in the applied aspect.

Several complex variables is quite different. It follows from results of Henri Poincaré in 1906, and of many others in modern times, that two domains can be topologically equivalent and even geometrically close need but not biholomorphic.

One standard device for studying a holomorphic mapping on a domain Ω is to restrict the mapping to the boundary. In one complex variable this is an uninteresting procedure because the boundary has no complex structure. But in several complex variables it leads to a system of partial differential equations on the boundary (by restriction of the Cauchy-Riemann equations). These are the tangential Cauchy-Riemann equations. A function that is annihilated by the tangential Cauchy-Riemann equations is called a CR function. An odd-dimensional manifold that supports a complex structure analogous to that for the boundary of a domain in space is called a CR manifold. A mapping of CR manifolds that preserves the CR structure is called a CR mapping.

The purpose of this workshop on CR complexity was threefold: (i) to define and determine the fundamental notion of CR complexity, (ii) to organize CR complexity theory into a broad framework that will be useful in CR geometry and also apply to other parts of mathematics, and (iii) to bring active senior researchers and young mathematicians together to work in a focused manner that will forge interactions and guide future research.

Research of the workshop organizers, and of many of the workshop participants, has suggested new phenomena that can arise in the study of CR mappings. Let us discuss for example the unit sphere, which is in some sense the simplest CR manifold. Consider a proper holomorphic mapping between balls in complex Euclidean spaces of possibly different dimensions. It turns out that, when the dimension of the domain ball is at least 2, and the mapping is smooth up to the boundary, then the mapping must be rational. The restriction of this mapping to the boundary sphere clearly defines a nonconstant CR mapping between spheres. This gives rise to the following question which is fundamental to the philosophy of this workshop:

How complicated can a CR mapping between spheres be, given the domain and target dimensions? What are the appropriate measures of complexity?

First let us discuss the one-dimensional situation. In C we are considering finite Blaschke products. The complexity of such a function (or mapping) is measured by the number of factors or, equivalently, by the degree of the divisor. In higher dimensions it is well known (Pinchuk and Alexander) that a proper holomorphic mapping between equidimensional balls is an automorphism (i.e., one-to-one, onto, and invertible). Thus the mapping must be a rational function of degree 1. In addition, the mapping is spherically equivalent to the identity, so it is not complicated. When the target ball has dimension less than that of the domain ball, the mapping must be constant---so the situation is trivial. On the other hand, when the dimension of the target ball is much greater than that of the domain ball, then the collection of inequivalent CR mappings becomes large without bound. Proper CR mappings of arbitrarily high degree are possible, provided only that the dimension of the target ball is large enough.

In low dimensions there are definite restrictions on what can happen. Some of the pioneering results in this subject area are due to Webster and to Faran. Webster show that a CR mapping of the sphere S^2n-1 to the sphere S²ⁿ⁺¹ for n ≥ 3 must, up to conjugation by a self-mapping of the ball, be of the form z → (z, 0). In short, up to an obvious normalization, the mapping must be trivial. The proof uses the deep differential invariants of Chern and Moser mentioned above. Faran examined the situation for n = 2 and determined that, in that context, there are in fact four CR mappings (up to conjugation by self-mappings of the ball).

One means of attacking the issue of complexity of CR mappings is by way of finite jet determination. An instance of finite jet determination is this: If φ : D → D is a conformal self-map of the unit disc, if φ(0) = 0, and if φ'(0) = 1, then φ is the identity. This result in fact follows from the classical Schwarz lemma. The basic point is that just knowing 0-order and 1-order data at a point uniquely determines the mapping. There are also results like this at the boundary, and so finite jets can play a role in classifying CR mappings.

Another interesting connection is with the theory of finite maps. The germ of a mapping F: C ⁿ → C ⁿ is finite if the inverse image of any point is a finite set. There are fascinating questions about whether the inverse image of a singular complex variety under a finite map must also be singular. One can understand CR mappings by the way they behave on varieties; so the geometric connection between the two circles of ideas is clear.

A major theme of this workshop has been to relate the dimensions of the domain and target balls with the complexity of a CR mapping between these balls. All of the lines of thought described above play a role in attacking this fundamental problem. Almost all the pioneers in this active subject area were present at the workshop.

Participants in this workshop have discovered remarkable connections of complexity of CR mappings with Schur's algorithm (an idea of Putinar), with the higher-dimensional Cauchy-Riemann equations, with formal logic, and with Hilbert's Seventeenth Problem. The complexity theory of CR mappings reaches out in many different directions.

Workers in this subject have found interesting number-theoretic interpretations of results in the complexity theory of CR mappings. For example, there was recently found a new method of primality testing using ideas from the complexity of CR mappings. Studying groups of CR mappings gives rise to interesting algebraic interpretations of results. Connections such as these make modern mathematics rich and exciting. The interface of fundamentally different areas of mathematics adds depth and texture to the subject. The workshop's developments will lay the underpinnings for the future development of this new field of complexity of CR mappings.