Applications of parabolic geometries and BGG sequences
Individual research project funded by
the Austrian Science
Fund ("Fonds zur Förderung der wissenschaftlichen Forschung"
- October 1, 2020 - March 31, 2024 (projected)
- total support EUR 405.168,75
Project leader: Andreas Cap, Faculty of Mathematics, University of
People in Vienna related to the project:
Main international collaborators:
- Christoph Harrach (post-doc) started on October 1, 2020
- Zhangwen Guo (pre-doc) started on November 25, 2020
- Michal Wasilewicz (pre-doc) not supported by the project, position financed by the University of Vienna starting from May 1, 2020.
- Boris Doubrov (Belarusian State University, Minsk)
- A. Rod Gover (University of Auckland, New Zealand)
- Pierre Julg (Universitè de Orleans)
- Thomas Mettler (UniDistance Suisse)
- Vladimir Soucek (Charles University, Prague, Czech Republic)
- Dennis The (University of Tromso)
Scientific Aims: Parabolic geometries are a class of geometric structures,
which in their standard descriptions look very diverse. They do admit a uniform
description as Cartan geometries with homogeneous model the quotient of a (real or
complex) semisimple Lie group by a parabolic subgroup. This description already
indicates a strong connection to semi-simple representation theory, which indeed plays
a major role in the study of parabolic geometries. Combined with geometric methods
tailored to the specific situation of second order geometric structures, this leads
to a large number of very efficient tools for the study of parabolic geometries, and
the general theory of these structures is developed very well by now.
One of the core tools for parabolic geometries is provided by
Bernstein-Gelfand-Gelfand sequences (BGG sequences). These are sequences of higher
order invariant differential operators intrinsically associated to any parabolic
geometry. They can be viewed as curved generalizations of the resolution of finite
dimensional irreducible representations by generalized Verma modules which was
obtained by J. Lepowsky as a generalization of the original BGG resolution by Verma
modules. While the theory of BGG sequences dates back to the early 2000's, a
substantial extension of the theory was recently obtained in joint work of A. Cap and
V. Soucek in the form of relative BGG sequences.
The basic aim of the project is to develop the theory of parabolic geometries and BGG
sequences and to apply them to geometric problems beyond the realm of parabolic
geometries. The five main directions of study planned for the project are:
Geometric compactifications and holonomy reductions: This is a
continuation of a long term joint project of A. Cap and A.R. Gover, based on
the theory of holonomy reductions of Cartan geometries that we developed in
joint work with M. Hammerl. These holonomy reductions turn out to provide
model cases for various types of geometric compactifications generalizing
R. Penrose's concept of conformal compactness. This provides interesting
connections to several areas of mathematics (e.g. scattering theory) and
theoretical physics (e.g. general relativity and the
AdS/CFT-correspondence). In particular, we plan to apply tractor methods to
the study of the mass of asymptotically hyperbolic metrics and
generalizations of this. This research also admits applications to the study
of compactifications of symmetric spaces, which gives connections to the
research on Poisson transforms described below.
Poisson transforms: This part of the project builds on joint work with P.
Julg and on C. Harrach's thesis. Generalizing work of P.Y. Gaillard, we construct
Poisson transforms for differential forms that are adapted to flat parabolic
geometries on the boundary (or on boundary components) of certain symmetric spaces.
To a large extent, these questions can be reduced to the construction of invariant
differential forms which can be done using finite dimensional representation
theory. The question of whether the transform produces harmonic forms turns out to
be closely related to the question whether it descends from a (twisted) de-Rham
sequence to the corresponding BGG sequence on the boundary. The main motivation for
these questions is applications to KK-theory for which parts of specific BGG
sequences and Poisson transforms have to be combined to form special complexes that
generate KK-groups. This research has obvious connections to the study of geometric
compactifications. Moreover, there is hope that some of these constructions admit
generalizations to curved settings, for example in the setting of
BGG-machinery and applications: This part of the project builds on the
recent joint work with V. Soucek in which we showed that there is a relative
version of the BGG machinery associated to a pair of nested parabolic
subalgebras. While the machinery used to construct relative BGG sequences is
similar to the original BGG construction, the resulting sequences are of different
nature. In particular, one obtains complexes that are resolutions of certain
sheaves without assumptions (or with only weak assumptions) on local
flatness. While the general theory of these sequences has been established in our
joint articles, this theory has not been worked out in detail in any case. Studying
this for two structures of broader interest is the content of the thesis projects
of Z. Guo and M. Wasilewicz. Related to this is the question of developing a
relative version of tractor calculus, which in turn can be used to study the full
tractor calculus for these structures, which is not completely understood for the
structures in question. A second aim for this part of the project is to generalize
the construction of invariant operators in singular infinitesimal character that is
provided by the relative BGG machinery to the case of maximal parabolics. This will
be studied jointly with V. Soucek.
The bundle of Weyl structures and non-linear invariant PDE: This a joint
project with T. Mettler. Motivated by earlier work by M. Dunajski and T. Mettler we
have obtained a general construction that associated to any $|1|$-graded parabolic
geometry a certain bi-Lagrangean structure on the total space of a natural fiber
bundle. It turns out that sections of that bundle can be identified with Weyl
structures for the corresponding parabolic geometry. This opens the possibility to
study Weyl structures via submanifold geometry in a bi-Lagrangean geometry. The
motivation for this is that special Weyl structure turn out to be related to
non-linear invariant PDE of Mongé-Ampère type. In low dimensions, the latter
turn out to be closely related to convex projective structures, which provides a
very promising relation to topics like character varieties and higher Teichmüller
theory. This opens the possibility for completely new applications of parabolic
Geometry of differential equations: It is well known that path
geometries, which can be equivalently described as a parabolic geometry, provide an
equivalent description of systems of second order ODE. It has been known for quite
some time that there is an analogous description of systems of higher order ODE in
terms of a canonical Cartan geometry (not of parabolic type). In recent joint work
with B. Doubrov and D. The, we have obtained a new construction of this canonical
Cartan geometry, which brings it into very similar form to parabolic geometries. In
particular, this opens the possibility to generalize parts of the (relative) BGG
machinery to the setting of higher order ODE. Studying this and applications to ODE
theory, e.g. on existence of geometric structure on spaces of solutions, will be an
important part of the project.
Activities related to the project:
- Thematic program Geometry for Higher Spin Gravity: Conformal Structures, PDEs, and Q-manifolds, Erwin Schrödinger Institute (ESI), Vienna, August 23 - September 17, 2021, organized jointly with Xavier Bekaert (U Tours), Stefan Fredenhagen (U Vienna), Maxim Grigoriev (Lebedev Inst. and Lomonosov Moscow SU), Alexei Kotov (U Hradec Kralove)
Publications related to the project:
- A. Cap, A.R. Gover, M. Hammerl: "Parabolic Compactification of Homogeneous Spaces", J. Inst. Math. Jussieu 20 no. 4 (2021) 1371-1408, published version availbable via Cambridge Core Share. Also available online as preprint arXiv:1807.04556.
- A. Cap, K. Melnick: "C1 Deformations of almost-Grassmannian structures with strongly essential symmetry", Transform. Groups 26 no. 4 (2021) 1169-1187, published version available via SharedIt here, also available online at arXiv:1902.01801.
- A. Cap, C. Harrach, P. Julg: "A Poisson transform adapted to the Rumin complex", to appear in J. Topol. Anal., DOI: 10.1142/S1793525320500570, available online as preprint arXiv:1904.00635.
- A. Cap, B. Doubrov, D. The: "On C-class equations", to appear in Commun. Anal. Geom., available online as preprint arXiv:1709.01130.
- A. Cap, T. Mettler: "Geometric Theory of Weyl Structures", to appear in Commun. Contemp. Math., DOI: 10.1142/S0219199722500262, available online as preprint arXiv:1908.10325
- A. Cap, A.R. Gover: "A relative mass cocycle and the mass of asymptotically hyperbolic manifolds", preprint arXiv:2108.01373.
- A. Cap, K. Hu: "BGG sequences with weak regularity and applications", preprint arXiv:2203.01300.
Talks related to the project:
- A. Cap: "Asymptotic invariants of metrics", Relativity Seminar, University of Vienna, February 2021
- C. Harrach: "Poisson transforms adapted to BGG complexes", SCREAM opening workshop, Ilawa, Poland, August 2021
- A. Cap: "Cartan Geometries" (series of 2 lectures), Thematic program Geometry for Higher Spin Gravity: Conformal Structures, PDEs, and Q-manifolds, ESI, Vienna, August 2021
- A. Cap: "Asymptotically hyperbolic mass and tractors", Central European Seminar on differential geometry, Brno, Czech Republic, October 2021
- C. Harrach: "Boundary values of Poisson transforms", Central European Seminar on differential geometry, Brno, Czech Republic, October 2021
- A. Cap: "Tractors and AH-mass", QMAP-Seminar, UC Davis, USA (online), January 2022
- A. Cap: "Tractors and the mass of asymptotically hyperbolic manifolds", 42th Winter School "Geometry and Physics", Srni, Czech Republic, January 2022
- Z. Guo: "Weyl structures for path geometries", 42th Winter School "Geometry and Physics", Srni, Czech Republic, January 2022
- C. Harrach: "On boundary values for Poisson transforms of differential forms", 42th Winter School "Geometry and Physics", Srni, Czech Republic, January 2022
- M. Wasilewicz: "Relative BGG sequences for Lagrangean contact structures", 42th Winter School "Geometry and Physics", Srni, Czech Republic, January 2022
- A. Cap: "BGG sequences in applied mathematics", Central European Seminar on differential geometry, Brno, Czech Republic, February 2022.