# CR Geometry and Complex Hyperbolic Space Winter 2007

JBE 358 Mondays 3:30-5pm

Organizers: Robert Montgomery and Jie Qing

**January 8, 2007**

*Basic Notions in CR-Geometry, Part I*

*Basic Notions in CR-Geometry, Part I*

**Richard Montgomery **

**January 22, 2007**

*Basic Notions, part II and Left-Invariant CR-Structures on \mathbb(s)^3(the Hitchin Family)*

*Basic Notions, part II and Left-Invariant CR-Structures on \mathbb(s)^3(the Hitchin Family)*

**Richard Montgomery **

**January 29, 2007**

**SEMINAR CANCELLED FOR THIS WEEK.**

**February 5, 2007**

*The Lewy and Nirenberg non-solvability examples in the theory of Linear PDEs and its relation to non-embeddable CR-manifolds*

*The Lewy and Nirenberg non-solvability examples in the theory of Linear PDEs and its relation to non-embeddable CR-manifolds*

**Alex Castro **

In 1956 Hans Lewy provided an example of a linear complex PDE with smooth coefficients which turned out to have no local solutions, contrary to the intuition provided by the Cauchy-Kowalewski from classical analysis. In 1974 (actually, earlier than that) motivated by another problem proposed by Lewy and based on that same old example, Louis Nirenberg constructed explicit examples of homogeneous linear PDE's in \mathbb{R}^3 without solutions either. On the other hand, it's known that the boundary \Sigma of a complex domain in U\subset\mathbb{C}^n one can talk about the concept of "holomorphy" by restricting the Cauchy-Riemann equations to the tangent spaces to the boundary. Moreover, the concept of CR-structure can be abstracted (based on the original geometrical construction) for general manifolds by axiomatizing the essential geometrical features of the initial construction. In fancy language, that's done by considering an involutive distribution on the complexified tangent bundle. In other words, we're considering a maximal subspace of the tangent space which is invariant by multiplication by \imath - a complex structure). Locally, this complex distribution is spanned by complex vector fields(up to scaling) or more precisely complex direction fields. The natural question to ask is when can these abstract CR-structures actually be realized(at least locally) as the boundary of some complex domain. The connection of Nirenberg examples with this geometrical question is that through the right interpretation, one can provide examples of non-realizable 3-dim. CR-manifolds, and these are generic in some sense.

**February 12, 2007**

*Conformal geometry and PDE I*

*Conformal geometry and PDE I*

**Jie Qing **

Conformal geometry in higher dimension in my opinion has grown out from the surface theory in two directions: one is uniformization in terms of finding better Riemannian metrics in a conformal class of metrics; the other is uniformization in terms of Kleinian group theory. We will discuss the development of conformal geometry in both directions and its relation to solving partial differential equations on manifolds. In particular the recent development of scattering operators in conformal geometry and hopefully more clearer connection to general relativity will be introduced. There will be discussions on possible analogous questions when we switch to the context of CR geometry in contrast to the conformal geometry.

**February 21, 2007 **WEDNESDAY MEETING THIS WEEK****

*Conformal geometry and PDE II*

*Conformal geometry and PDE II*

**Jie Qing **

Conformal geometry in higher dimension in my opinion has grown out from the surface theory in two directions: one is uniformization in terms of finding better Riemannian metrics in a conformal class of metrics; the other is uniformization in terms of Kleinian group theory. We will discuss the development of conformal geometry in both directions and its relation to solving partial differential equations on manifolds. In particular the recent development of scattering operators in conformal geometry and hopefully more clearer connection to general relativity will be introduced. There will be discussions on possible analogous questions when we switch to the context of CR geometry in contrast to the conformal geometry.