Dept Math & Stat UiT, Forskningsparken B459
  10-17-24 Apr 2024 Wed 12:30-14:00

David Sykes (Masaryk University Brno & TU Wien)

TFS Guest Professorship program, 3-part lecture series:

"Connecting intrinsic and extrinsic studies of 2-nondegenerate CR geometries"

Real hypersurfaces in complex spaces inherit a geometric structure induced by restricting Cauchy-Riemann equations to the hypersurface. The basic problem of finding biholomorphisms mapping one hypersurface onto another is only well understood for a limited class of hypersurfaces, and has a fundamental relationship to their induced CR geometries. These geometries can alternatively be defined intrinsically, and today's prevailing techniques for their analysis generally fit within one of two regimes, an extrinsic viewpoint focused on a CR manifold's embedding and an intrinsic viewpoint studying only CR manifolds' intrinsically defined structures. This lecture series is focused on connecting extrinsic and intrinsic studies of 2-nondegenerate CR geometries, and will present fundamentals along with several recent results on the local geometry of 2-nondegenerate hypersurfaces.

The first talk will present an introduction to 2-nondegenerate CR hypersurface geometry, covering the area's key definitions and foundational concepts. Central in this introduction is a construction through which a 2-nondegenerate structure gives rise to a canonical atlas of Lie algebra valued 1-forms. The atlas is similar although not equivalent to a more standard object, the families of gauges that define Cartan connections. We will investigate several CR invariants encoded in this atlas. The second talk will present results on the extrinsic descriptions of 2-nondegenerate CR hypersurfaces. The extrinsic viewpoint allows us to uniquely associate a model 2-nondegenerate structure with every point in the considered hypersurfaces. Each model hypersurface can then be viewed as a local invariant of the structure they are associated with, and we will cover how the local invariants studied in the first talk appear within a structure's associated models. Contrary to the better understood Levi-nondegenerate setting, the moduli space of 2-nondegenerate models is not discrete, and we will cover several results about these models' local geometry. The final talk will present results obtained from intrinsic descriptions of the CR structures. Major theorems we will cover are solutions to local equivalence problems obtained through constructions of canonical absolute parallelisms. We will discuss applications of these constructions and their relationship to the previous talk's extrinsically defined models.