# Prof. dr. Bernhard Lamel: The world according to Poincaré: A hitchhikers' guide to CR geometry

Source: Mathematics colloquium

**Bernhard Lamel**

Univerza na Dunaju

The Riemann mapping theorem provides a basic tool for complex analysis in one variable: every simply connected domain in the plane is biholomorphically equivalent to the unit disk. Poincare (in a paper from 1907) made the observation that this kind of phenomenon is purely one dimensional. Subsequent research showed that biholomorphic invariants of simply connected pseudoconvex domains in C^{n} are extraordinarily "rich", i.e. there exist (in a sense too) many of them. In our talk, we will survey results about the local equivalence of boundaries of domains, which is a problem intimately linked to the global one. It has been studied intensely in the 20th century, with groundbreaking contributions by E. Cartan, S.S. Chern, J. Moser, and N. Tanaka in the strictly pseudoconvex (or, more generally, Levi-nondegenerate) case; this assumption puts a uniformity restriction on the biholomorphic geometry of the boundaries making them into Cartan geometries. After surveying the history of this "biholomorphic equivalence problem", we will discuss a particular strategy to attack it in the more general setting where the uniformity mentioned before breaks down, leading to a number of recent results.