Workshop on Cartan Geometry




May 29 - 31, 2019                    KIAS 8101

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Tohru Morimoto (Nara)

- Title: Infinite dimensional extrinsic geometries

- Abstract: I will talk about extrinsic geometry of a morphism $ varphi : (M, mathfrak f ) to  L/ L^0 $ from a filtered manifold to a homogeneous filtered  manifold in which $L$ is an infinite dimensional Lie groups or rather an infinite dimensional Lie pseudo-group of transformations.


Boris Doubrov (Minsk)

- Title: Symplectic deformations of rational homogeneous varieties and contact flag structures

- Abstract: We consider local deformations of rational homogeneous varieties preserving its symbol. In particular, we prove the existence of the canonical frame bundle and describe the cohomology spaces that define the set of fundamental invariants of such deformations. In case, the symmetry group of the homogeneous model preserves a symplectic form, we define the notion of a symplectic deformation and show how these deformations are related to so-called flag structures on contact manifolds. As an example, we discuss such structures on the space of abnormal curves for (2,3,5) distributions.


Thomas Leistner (Adelaide)

- Title: The ambient obstruction tensor and conformal holonomy

- Abstract. The obstruction tensor is a conformally covariant tensor that obstructs the existence of an analytic Ricci flat ambient metric in the sense of Fefferman and Graham. In the talk, I will describe a new relation between the obstruction tensor and the holonomy of the normal conformal Cartan connection. This relation implies several results on the vanishing and the rank of the obstruction tensor, for example for conformal structures with twistor spinors. As the main tool we introduce the notion of a conformal holonomy distribution whose integrability is closely related to the exceptional conformal structures in dimensions five and six that were found by Nurowski and Bryant. This is joint work with Andree Lis- chewski from the Humboldt-University Berlin.


Katharina Neusser (Brno)

- Title: Projective geometry of Sasaki-Einstein structures and their compactification

- Abstract: This talk will be concerned with Sasaki manifolds, which can be characterized as (pseudo-) Riemannian manifolds whose metric cone is Kähler. We will show that Sasaki manifolds admit a natural description in terms of projective differential geometry. In particular, we will see that Sasaki-Einstein manifolds may be characterized as projective manifolds equipped with certain unitary holonomy reductions of their canonical Cartan connections.  This characterization will also allow us to describe a natural geometric compactification of complete non-compact (indefinite) Sasaki-Einstein manifolds. This talk is based on joint work with Rod Gover and Travis Willse.


Andreas Cap (Vienna)

- Title: Holonomy reductions of Cartan geometries and geometric compactifications

- Abstract: In my talk, I will discuss the idea of a holonomy reduction of a Cartan geometry. Surprisingly, such reductions in many cases give rise to a decomposition of the underlying manifold into strata of different dimensions that inherit geometric structures of different types. In the simplest case, one deals with a manifold with boundary and obtains a concept of structures in the interior and on the boundary that are tied together by the Cartan geometry in the background. This provides examples of and motivation for various notions of geometric compactifications, some of which will be discussed in the second part of the talk.


Michail Zhitomirskii (Haifa)

- Title: Normal forms, first invariants, and symmetries in local classification problems with functional moduli

- Abstract: I will explain a general framework which gives known and new results for all local classification problems with functional moduli, including Riemannian metrics and conformal structures,  (2,3,5), (3,6) and (3,5) distributions, and all G-structures on filtered 3-manifold.


Abraham Smith (Stout)

- Title: The Characteristic Variety as an organizing principle

- Abstract: The characteristic variety arises as a specialty topic in the theory of EDS/PDEs, but its geometry undergirds the structure of PDEs. The characteristic variety (really the characteristic scheme) lets one apply the classical methods of matrix decomposition to PDEs, effectively producing what some have called “non-linear representation theory”.  By this, we mean that geometric structures can be realized as parametrized families of Jordan forms, and properties like hyperbolicity and integrability are reflected in special conditions on those forms.  This talk summarizes these relationships, with particular focus on the Eikonal system.


Qifeng Li (KIAS)

- Title: Construct Cartan connections modeled after non-compact homogeneous spaces

- Abstract: In this talk we will give an existence theorem for Cartan connections on manifolds equipped with geometric system modeled after G/H, where G is a complex (possibly non-reductive) linear algebraic group, and H is a (possibly non-parabolic) subgroup.  As an application, we give a characterization of symplectic grassmannians by the family of lines passing through the base point. This is a joint work with Professor Jun-Muk Hwang.


Wojciech Krynski (Warsaw)

- Title: Cayley's cubics and differential equations

- Abstract: In this talk I'll consider Cayley structures understood as fields of Cayley’s cubic surfaces over a 4-dimensional manifold. I shall motivate their study by showing their similarity to indefinite conformal structures and their link to differential equations. The flat Cayley structure is the most symmetric non-metric example within the class of 4-dimiensional causal (or cone) structures.

For the Cayley structures an extension of certain notions defined for indefinite conformal structures in dimension four are introduced, e.g., half-flatness, existence of a null foliation, ultra-half-flatness, an associated pair of second order ODEs, and a dispersionless Lax pair. I'll present a solution to the equivalence problem for the Cayley structures and find the local generality of several classes of Cayley structures. The talk is based on a joint work with Omid Makhmali.


Colleen Robles (Durham)

- Title: What representation theory can tell us about the cohomology of a hyperkahler manifold

- Abstract: The cohomology (with complex coefficients) of a compact kahler manifold M admits an action of the algebra sl(2,C), and this action plays an essential role in the analysis of the cohomology.  In the case that M is a hyperkahler manifold Verbitsky and Looijenga—Lunts showed there is a family of such sl(2,C)’s generating an algebra isomorphic to so(4,b_2-2), and this algebra similarly can tell us quite a bit about the cohomology of the hyperkahler.  I will describe some results of this nature for both the Hodge numbers and Nagai’s conjecture on the nilpotent logarithm of monodromy arising from a degeneration.  This is joint work with Mark Green, Radu Laza and Yoonjoo Kim.