Abstract: Cauchy-Riemann manifolds attempt to describe abstractly the geometry of an embedded submanifold in $\mathbb{C}^n$, $n>1$, that is invariant under local biholomorphisms of the ambient space. The most interesting case is the Cauchy-Riemann geometry of boundaries of domains in (higher dimensional) complex space, since this interacts strongly with the analytic function theory on the domain (explaining, for instance, why the Riemann mapping theorem fails in higher dimensions). I'll introduce Cauchy-Riemann geometry, explaining the connection to contact topology, and discuss the problem of determining when an abstract Cauchy-Riemann structure on the standard contact 3-sphere corresponds to that of a hypersurface in $\mathbb{C}^2$.
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