- October 1, 2020 - March 31, 2024 (projected)
- total support EUR 405.168,75

**Project leader**: Andreas Cap, Faculty of Mathematics, University of
Vienna.

**People in Vienna related to the project**:

- 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:

**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: "C
^{1}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.