"Some locally transitive actions on the Möbius sphere related to conformal holonomy"

Friday, June 11, 11:45-12:45

Recent results giving restrictions on the possibilities for
irreducibly acting conformal holonomy groups H ⊂ O(p+1,q+1)
in various signatures (p,q) identify two candidates for special
conformal holonomy groups which haven't been studied yet: (A) the
irreducibly acting SO(2,1) ⊂ O(3,2) for conformal Lorentzian
3-manifolds; and (B) Spin_{0}(1,8) ⊂ O(8,8) for conformal
manifolds of signature (7,7). The groups have a natural action on the
Moebius spheres S^{2,1} and S^{7,7}, respectively, which is
globally transitive in the latter case, but only locally transitive
in the former. In this talk, we report on recent joint work with A.
J. Di Scala and T. Leistner (on case A), and with F. Leitner (on case
B). In particular, the results of this work show that case A cannot
occur as a conformal holonomy group, while in case B the associated
Fefferman-type construction cannot yield the conformal holonomy
group.

"Einstein metrics and parabolic geometries"

Tuesday, July 12, 9:30 - 10:30

I shall give an overview of a joint work with Rafe Mazzeo, seeing certain parabolic geometries as boundaries at infinity of Einstein metrics. This is a program began several years ago, and I will explain what is known, and what remains to be proved.

"Quasi-Einstein metrics and the tractor calculus""

Wednesday, July 13, 14:00 - 14:30

Quasi-Einstein metrics are a general class of metrics which include conformally Einstein metrics, static metrics, gradient Ricci solitons, and more generally, metrics on the base of a conformally Einstein warped product metric. We will describe how, save for gradient Ricci solitons, these metrics can be naturally formulated in the language of conformal geometry, and moreover, how this leads to a new perspective on many aspects of the Ricci flow. In particular, we will discuss a natural prolongation of the quasi-Einstein condition using the tractor calculus, and some consequences of this perspective.

"Two-jets of conformal fields along their zero sets, in any metric signature"

Monday, July 11, 11:30 - 12:30

The connected components of the zero set of any conformal
vector field *v*, in a pseudo-Riemannian manifold *(M,g)* of
arbitrary signature, are of two types, which may be called
`essential' and `nonessential'. The former components consist
of pointsat which *v* is essential, that is, cannot be turned
into a Killing field by a local conformal change of the metric.
In a component of the latter type, points at which *v* is
nonessential form a relatively-open dense subset that is
at the same time a totally umbilical submanifold of *(M,g)*.
An essential component is always a null totally geodesic
submanifold of *(M,g)*, and so is the set of those points
in a nonessential component at which *v* is essential
(unless this set, consisting precisely of all the singular
points of the component, is empty). Both kinds of null
totally geodesic submanifolds arising here carry a one-form,
defined up to multiplications by functions without zeros,
and satisfying a projective version of the Killing equation.
The conformal-equivalence type of the 2-jet of *v* is
locally constant along the nonessential submanifold of a
nonessential component, and along an essential component
on which the distinguished one-form is nonzero.

Presentation

"Conformal foliations and CR geometry"

Monday, July 11, 10:00 - 11:00

In Euclidean three-space, there are some special foliations called
conformal. They enjoy remarkable properties related to CR geometry in
five dimensions. I shall explain this construction and its geometric
interpretation via twistor theory. The ideas are well-known but this
exposition is joint work with Paul Baird.

Presentation

"Essential singularities for higher dimensional conformal maps"

Tuesday, July 12, 11:00 - 12:00

Conformal immersions between Riemannian manifolds of same dimension at least 3 are a natural higher dimensional analogue of holomorphic maps. Like in the holomorphic setting, there is a notion of removable and essential singularities for conformal immersions. The aim of the talk is to give a classification of all essential singularities having a sufficiently small Hausdorff dimension, and to emphasize their link to Kleinian groups.

"Ambient extension of parallel tractors"

Wednesday, July 13, 9:30 - 10:30

Many interesting holonomy classes of conformal structures are
characterized by the existence of parallel tractors. The question of
whether a parallel tractor has a parallel extension to a tensor on the
ambient space which is parallel for an ambient metric will be
discussed.

An application to Nurowski's conformal structures associated to
generic 2-plane distributions on 5-manifolds produces an
infinite-dimensional family of pseudo-Riemannian metrics whose
holonomy is the split real form of the exceptional Lie group
G_{2}. This is a report on joint work with Travis Willse.

"Holonomy-reductions of Cartan connections"

Tuesday, July 12, 15:30 - 16:15

We discuss the notion of holonomy for Cartan geometries and the
geometric implications of reduced holonomy. At the core of the reduction-theorem
lies a geometric comparison theorem which shows that the main properties
of a holonomy reduction can already be read off
from the corresponding reduction of the homogeneous model. The distinctive
property of a holonomy reduction of a Cartan connection is that it naturally
decomposes the manifold into curved analogs of group orbits, each of which
inherits a reduced Cartan geometry.

As an interesting area of application we show how this reduction-procedure
can be used to study the zero-locus and geometric
properties of normal solutions to the first BGG-equation.
This talk is based on joint work with A. Cap (Univ. Vienna) and A.R.
Gover (Univ. Auckland), [arXiv:1103.4497]

"Explicit formulas for GJMS-operators and Q-curvatures"

Wednesday, July 13, 11:00 - 12:00

It is well-known that in two dimensions the Laplace-Beltrami operator is conformally covariant. The Yamabe operator extends this fact to higher dimensions. It arises by correcting the Laplacian by a multiple of scalar curvature. More generally, one can also correct powers of the Laplacian by lower-order terms as to obtain conformally covariant operators. Although these operators were constructed already in 1990 by Graham, Jenne, Mason and Sparling (using the Fefferman-Graham ambient metric construction), the structure of the lower order correction terms remained notoriously complicated. We describe how one can resolve the structure of GJMS-operators in a surprising and esthetically appealing way. These results also lead to a new formula for Q-curvature.

"First order operators in projective contact geometry - a classification"

Wednesday, July 13, 16:30 - 17:00

In the sense of Cartan, the *contact projective geometry* is a
curved version of the Klein homogeneous space *G/P*, where
*G=Sp(2n, R)* and

Its underlying structure consists of a
manifold *M ^{2n-1}*, a contact subbundle of the tangent
bundle

After recalling the definition of the contact projective geometry, we
define first order invariant differential operators for a general
Cartan geometry, and mention the result of Soucek and Slovak on first
order invariant differential operators acting between the sections of
irreducible **finite** rank bundles over parabolic geometries.

We will prove a similar result for a countable family of bundles over
a projective contact geometry, each of them being associated by an
**infinite** dimensional irreducible representation of the double
cover of the symplectic group. The inducing representations we shall
use are *symplectic* analogues of the spinor-tensor
representations of the *orthogonal* groups. The Segal-Shale-Weil
representation is an example of one of them.

"Predicting / prescribing the location of conformal infinity"

Wednesday, July 13, 17:15 - 17:45

On an Einstein manifold a conformal geodesic has a canonical conformal factor associated to it, which is quadratic in the parameter of the curve and whose roots give the location of conformal infinity. This talk will introduce a set of conformal curves that closely mimics the behaviour of conformal geodesics. In particular the canoncial quadratic conformal factor can be retained. This allows to predict / prescribe the location of conformal infinity. We will discuss some applications.

"Intrinsic Pseudodistances for Parabolic Geometries"

Wednesday, July 13, 15:45 - 16:15

This paper generalizes the Kobayashi intrinsic pseudodistance
construction to arbitrary parabolic geometries. Let *C(G,P)*
denote the category of Cartan geometries modeled on a semisimple
homogenous space *G/P* with *P* parabolic. We define an
intrinsic Kobayashi pseudodistance on the objects of *C(G,P)*
with respect to which connection preserving morphisms are
nonexpansive. In general this construction depends on a suitable
choice of an admissable class *C* of canonical curves through the
origin of *G/P*. In contact projective geometry, for example, we
have two natural choices for *C*: the horizontal contact
projective geodesics (*C g_{-1}*)
and the chains (

"Conformal and isometric transformations of finsler manifolds: Wang theorem, Matsumoto question, Deng-Hu conjecture, conformal invariants of finsler metrics and Lichnerowicz-Obata conjecture"

Tuesday, July 12, 14:00 - 15:00

The talk is based on the joint paper with Marc Troyanov. We introduce
a new construction basing on the convex geometry that associates a
Riemannian metric *g _{F}* (called
the

"Hartogs extension theorem for parabolic geometries"

Thursday, July 14, 15:30 - 16:30

I will explain the proof that holomorphic parabolic geometries exhibit the Hartogs extension phenomenon: any holomorphic parabolic geometry defined on a domain in a Stein manifold extends to a unique holomorphic parabolic geometry on the envelope of holomorphy of that domain.

"Effective Cartan-Tanaka connections on

Wednesday, July 13, 16:30 - 17:15

In a recent expository article (Notices of the AMS, **58** (2011), no.
1, 20-27), Ezhov, McLaughlin and Schmalz showed how to perform in an
effective way Tanaka's prolongation procedure valid generally for filtered
structures of constant type when the distribution is equipped with an
integrable complex structure, so as to derive the principal curvature
invariants and (co)frame(s) associated to strongly pseudoconvex real
hypersurfaces *M ^{3}* in

"The Goldberg-Sachs theorem in higher dimensions"

Wednesday, July 13, 14:45 - 15:15

In four dimensions, the Goldberg-Sachs theorem gives necessary and sufficient conditions on the Weyl tensor and Cotton-York tensor for the existence of a locally integrable distribution of complex null 2-planes on a real or complex (pseudo)-Riemannian manifold. We show how the theorem generalises to higher dimensions in the holomorphic category, and time-permitting, we discuss its real versions.

"Ultrarigid tangents of sub-Riemannian nilpotent groups"

Monday, July 11, 17:00 - 17:45

Margulis and Mostow showed that if two equiregular sub-Riemannian manifolds are quasiconformally equivalent, then almost eve- rywhere they have isomorphic Gromov tangent cones. In other words, the tangent cone is a quasiconformal invariant. Their work extends a result of Pansu which says that two Carnot groups are quasiconformally equivalent if and only if they are isomorphic. In this talk I will present a joint result with E. Le Donne and A. Ottazzi showing that the converse of the theorem of Margulis and Mo- stow fails in a strong sense. In particular, we show that there exist two nilpotent Lie groups equipped with left invariant sub-Riemannian metrics, whose tan- gent cones are isomorphic at every point, but which are not quasiconformally equivalent. This result relies on studying those Carnot groups whose quasiconformal maps can only be translations and dilations. We shall refer to groups with this property as ultrarigid groups. One of the tools in our method is to provide an algebraic characterization of ultrarigidity.

"On split signature conformal metrics induced by half-dimensional projective structures"

Thursday, July 14, 16:45 - 17:30

Constructions of conformal metrics of split signature (2,2) from 2-dimensional projective structures are known due to the works of Nurowski-Sparling and Dunajski-Tod. We plan to revisit this topic from the parabolic point of view, discuss an analogy in general dimension and, hopefully, extend the characterization of the resulting conformal structures.

"Tanaka prolongation and symplectification procedure for filtered structures on manifolds"

Friday, July 15, 11:00 - 12:00

The talk is devoted to local equivalence problem for vector distributions (subbundles of tangent bundles). First I will review the classical approaches to this problem, making special emphasis to the algebraic version of Cartan's method of equivalence developed by N. Tanaka in 1970s. The central object in the Tanaka approach is the notion of a symbol of a distributions at a point, which is a graded nilpotent Lie algebra. The prolongation procedure (i.e. the procedure of getting a canonical frame) can be described in terms of natural algebraic operation in the category of graded Lie algebras. Through this review of Tanaka theory I will motivate the recent approach of B. Doubrov and myself to this problem. Our approach is a combination of a kind of a symplectification of the problem (taking its origin in Pontryagin theory in Optimal Control) and various Tanaka type prolongations. This approach allowed us to make a unified construction of canonical frames for distribution of arbitrary rank independently of their Tanaka symbols, avoiding the problem of classification of graded nilpotent Lie algebras with given number of generators.