- January 1, 2011 - April 30, 2014
- total support EUR 384.860,13.

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

**People supported by the project**:

- Matthias Hammerl (post doc) January 2011 - September 2012 (junior professorship at the University of Greifswald starting from October 2012).
- Ales Navrat (PhD student) January 2011 - June 2012 (after finishing his PhD, he works at the Masaryk University, Brno).
- Tomas Salac (post doc) January 2013 - December 2013 (position at the Charles University in Prague starting in 2014).
- Chiara de Zanet (PhD student) October 2011 - April 2014
- Christoph Harrach (PhD student) April 2012 - April 2014

**Main international collaboarators**:

- Michael G. Eastwood (Australian National University, Canberra, Australia)
- A. Rod Gover (University of Auckland, New Zealand)
- C. Robin Graham (University of Washington, USA)
- Karin Melnick (University of Maryland, USA)
- Jan Slovak (Masaryk University, Brno, Czech Republic)
- Vladimir Soucek (Charles University, Prague, Czech Republic)

**Scientific Aims**: The project is devoted to exploring the
relation be geometric structures, and in particular parabolic
geometries, and differential equations. This comprises two principal
directions. On the one hand, there is a rich theory of *invariant
differential operators* associated to parabolic geometries, i.e.,
of operators which are intrinsically associated to such a geometry. In
particular, the machinery of BGG-sequences (Bernstein-Gelfand-Gelfand
sequences) provides prowerful tools from algebra and representation
theory to construct and study such operators. Recently, the machinery
has also been applied to study solutions of these differential
operators and in particular the possible zero-loci of such
solutions. Here the project has already made important contributions,
which also led to foundational results on the concept of holonomy of
Cartan geometries.

The second main direction of research is the study of general differential equations by geometric methods. The basic strategy here is that via the canonical differential systems on jet spaces, differential equations can be encoded in distributions (subbundles of the tangent bundle) on manifolds. In several cases (e.g. generic distributions of rank two in dimension five, rank three in dimension six, and rank four in dimension seven) such distributions are equivalent to parabolic geometries. In other cases, ideas from parabolic geometries can be applied to the study of distributions.

**Activities related to the project**:

- Workshop "Cartan Connections, Geometry of Homogeneous Spaces, and Dynamics", July 11-22, 2011, Erwin Schrödinger Institute (ESI), Vienna, organized by A. Cap, C. Frances (Universit\'e Paris-Sud) and K. Melnik (University of Maryland)
- Workshop "The Interaction of Geometry and Representation Theory. Exploring new frontiers", September 3-14, 2012, Erwin Schrödinger Institute (ESI), Vienna, organized by A. Cap, A. Carey (Australian National University), A.R. Gover (Univiersity of Auckland), C.R. Graham (University of Washington) and J. Slovak (Masaryk University).
- Program session "Special Structures and Representation Theory", Conference Differential Geometry and its Applications, August 19-23, 2013, Brno, organized by A. Cap and J. Slovak (Masaryk University).

**Publications related to the project**:

- M. Hammerl, K. Sagerschnig: "The twistor spinors of generic 2- and
3-distributions" Ann. Glob. Anal. Geom.
**39**, 4 (2011) 403-425, available at arXiv:1004.3632 - M. Hammerl: "Coupling solutions of BGG-equations in conformal spin
geometry" J. Geom. Phys.
**62**, 2 (2012) 213-223, available at arXiv:1009.1547 - M. Hammerl, P. Somberg, V. Soucek, J. Silhan: "Invariant
prolongation of overdetermined PDE's in projective, conformal and
Grassmannian geometry" Ann. Glob. Anal. Geom.
**42**, 1 (2012) 121-145, available at arXiv:1103.3141 - A. Cap, V. Soucek: "Subcomplexes in curved BGG-sequences" Math. Ann.
**354**No. 1 (2012) 111-136, available at arXiv:math/0508534 - M. Hammerl, P. Somberg, V. Soucek, J. Silhan: "On a new
normalization for tractor covariant derivatives", J. Europ. Math.
Soc.
**14**no. 6 (2012) 1859-1883, available at arXiv:1003.6090 - A. Cap, A.R. Gover, M. Hammerl: "Projective BGG equations,
algebraic sets, and compactifications of Einstein geometries" to
J. London Math. Soc.
**86**no. 2 (2012) 433-454, available at arXiv:1005.2246 - A. Cap, A.R. Gover, M. Hammerl: "Normal BGG solutions and polynomials",
Internat. J. Math
**23**No. 11 (2012) 1250117 (29 pp.), available at arXiv:1201.0799 - A. Cap, K. Melnick: "Essential Killing fields of parabolic geometries: projective and
conformal structures", Centr. Europ. J. Math.
**11**, 12 (2013) 2053-2061, available at arXiv:1211.5477 - A. Cap, K. Melnick: "Essential Killing fields of parabolic geometries", Indiana Univ. Math. J.
**62**no. 6 (2013) 1917-1953, available at arXiv:1208.5510 - A. Cap, A.R. Gover, M. Hammerl: "Holonomy reductions of Cartan
geometries and curved orbit decompositions", Duke Math. J.
**163**, no. 5 (2014) 1035-1070, available a arXiv:1103.4497 - A. Cap, A.R. Gover, H.R. Macbeth: "Einstein metrics in projective geometry", Geom. Dedicata
**168**No. 1 (2014) 235-244, available at arXiv:1207.0128 - A. Navrat: "Non-standard Operators in Almost Grassmannian Geometry", PhD thesis, University of Vienna, October 2012.
- A. Cap, A.R. Gover: "Projective Compactifications and Einstein Metrics", J. Reine Angew. Math.
**717**(2016) 47-75, available online as preprint arXiv:1304.1869. - A. Cap, T. Salac: "Pushing down the Rumin complex to locally
conformally symplectic quotients ", Differential
Geom. Appl.
**35**Supplement (2014) 255-265, available at arXiv:1312.2712. - M. Hammerl, K. Sagerschnig: "A non-normal Fefferman-type construction of split-signature conformal structures admitting twistor spinors", preprint arXiv:1109.4231.
- T. Salac: "k-Dirac operator and Cartan-Kähler theorem", preprint arXiv:1304.0956.
- A. Cap, A.R. Gover: "Projective Compactness and Conformal Boundaries", Math. Ann.
**366**no. 3-4, 1587-1620, available online at arXiv:1406.4225. - A. Cap, A.R. Gover: "Scalar Curvature and projective compactness", J. Geom. Phys.
**98**(2015) 475-481, available online at arXiv:1409.1698.

**Editorial activities related to the project**:

- Central European
Journal of Mathematics
**10**no. 5, Topical Issue: "Finite dimensional integrable systems, dynamics, and Lie theoretic methods in Geometry and Mathematical Physics"; related to the 2011 activity at the ESI, Editors: A. Cap, V. Matveev, K. Melnick, G. Valent - Differential Geometry and its Application,
**33**Number S1 (March 2014), special issue related to the 2012 workshop at the ESI, Editors: A. Cap, A. Carey, A.R. Gover, C.R. Graham, J. Slovak.

**Talks related to the project**:

- A. Cap: "Normal Weyl structures and special solutions of first BGG operators", 31st Winter School Geometry an Physics, Srni, Czech Republic, January 2011.
- M. Hammerl: "Holonomy reductions of Cartan geometries", 31st Winter School Geometry an Physics, Srni, Czech Republic, January 2011.
- A. Cap: "Introduction to G-structures", University of Auckland, New Zealand, February 2011.
- A. Cap: "The Cartan connection associated to a conformal structure" University of Auckland, New Zealand, February 2011.
- M. Hammerl: "Homogeneous Cartan Geometries and Applications", University of Auckland, New Zealand, March 2011.
- A. Cap: "Essential automorphisms of parabolic geometries", Central European Seminar on differential geometry, March 2011.
- A. Cap: "Parabolic geometries determined by distributions", Workshop "Vector Distributions and sub-Riemannian Systems", Banach Center, Warsaw, Poland, March 2011.
- A. Cap: "Holonomy reductions of Cartan connections and invariant differential equations", keynote lecture, Workshop "Moving Frames in Geometry", CRM, Montreal, Canada, June 2011.
- M. Hammerl: "Holonomeireduktionen von konformen Strukturen", Humboldt University, Berlin, July 2011.
- A. Cap: "Introduction to Cartan geometries", Workshop "Cartan Connections, Geometry of Homogeneous Spaces, and Dynamics", ESI, Vienna, July 2011.
- M. Hammerl: "Holonomy reductions of Cartan connections", Workshop "Cartan Connections, Geometry of Homogeneous Spaces, and Dynamics", ESI, Vienna, July 2011.
- A. Cap: "Normal Weyl structures and solutions of first BGG operators", Workshop "The Geometry of Differential Equations", ANU, Canberra, Australia, September 2011.
- M. Hammerl: "A non-normal Fefferman-type construction", Workshop "The Geometry of Differential Equations", ANU, Canberra, Australia, September 2011.
- A. Cap: "Infinitesimal automorphisms of parabolic geometries and dynamics", 55th Annual Meeting of the Australian Mathematical Society, Wollongong, Australia, September 2011.
- M. Hammerl: "Fefferman-type constructions for parabolic geometries", University of Washington, Seattle, USA, November 2011.
- M. Hammerl: "Comparing ambient and conformal holonomy", Central European Seminar on Differential Geometry, Masaryk University, Brno, November 2011.
- A. Cap: "Conformally invariant overdetermined PDEs and Einstein metrics", General Relativity Seminar, University of Vienna, January 2012.
- A. Cap: "Comparison techniques for infintesimal automorphisms of parabolic geometries", 32nd Winter School Geometry and Physics, Srni, Czech Republic, January 2012.
- C. De Zanet: "Prolongations of nilpotent stratified Lie algebras: Tanaka's theorem", 32nd Winter School Geometry and Physics, Srni, Czech Republic, January 2012.
- M. Hammerl: "Recent progress on a non-normal Fefferman-type construction of split-signature conformal structures", 32nd Winter School Geometry and Physics, Srni, Czech Republic, January 2012.
- A. Cap: "Projective compactifications", Central European Seminar on differential geometry, March 2012.
- C. de Zanet: "Generic rank four distributions in dimension eight", Central European Seminar on differential geometry, March 2012.
- M. Hammerl: "Überbestimmte Systeme in der Geometrie", Ernst-Moritz-Arndt University, Greifswald, May 2012.
- C. de Zanet: "Generic distributions of rank four", Presentation of thesis project, University of Vienna, May 2012.
- A. Cap: "Projective Compactness", Workshop Conformal and CR geometry (12w5072) Banff International Research Station (BIRS) Canada, August 2012.
- M. Hammerl: "Ambient and conformal holonomy", Workshop Conformal and CR geometry (12w5072) Banff International Research Station (BIRS) Canada, August 2012.
- A. Cap: "Relative BGG sequences", Workshop "The Interaction of Geometry and Representation Theory. Exploring new frontiers" Erwin Schrödinger Institute (ESI) Vienna, September 2012.
- A. Cap: "Einstein-metrizability of projective structures", Central European Seminar on differential geometry, November 2012.
- C. de Zanet: "Towards Weyl connections for generic rank four distributions in dimension eight", Central European Seminar on differential geometry, November 2012.
- A. Cap: "Special infinitesimal automorphisms of parabolic geometries" (plenary lecture), 33rd Winter School Geometry and Physics, Srni, Czech Republic, January 2013.
- C. de Zanet: "Generic rank two bracket generating distributions", 33rd Winter School Geometry and Physics, Srni, Czech Republic, January 2013.
- C. Harrach: "Poisson transformation on real hyperbolic space", 33rd Winter School Geometry and Physics, Srni, Czech Republic, January 2013.
- T. Salac: "k-Dirac operator and Cartan Kähler theorem", 33rd Winter School Geometry and Physics, Srni, Czech Republic, January 2013.
- A. Cap: "Essential infinitesimal automorphisms of projective and conformal structures", University of Auckland, February 2013.
- A. Cap: "Projective compactifications of Einstein metrics", Central European Seminar on differential geometry, March 2013.
- A. Cap: "Mechanik, Distributionen und G
_{2}, "Lectures for everybody", Departement of Mathematics, University of Vienna, June 2013 (2 talks). - C. Harrach: "Poission transform for differential forms", Presentation of thesis project, University of Vienna, June 2013.
- A. Cap: "Projective compactness", General Relativity Seminar, Departement of Physics, University of Vienna, June 2013.
- A. Cap: Last two parts of the minicourse "Invariant differential Operators and exterior differential systems", Conference "New Directions in Exterior Differential Systems" (A conference in honor of Robert Bryant's 60th birthday), jointly with J.M. Landsberg (Texas A&M University) and M.G. Eastwood (ANU), July 2013.
- A. Cap: "Holonomy reductions of Cartan geometries", special meeting of the Central European Seminar on differential geometry, Banach Center, Warsaw, Poland, October 2013.
- C. De Zanet: "Generic rank four distributions in dimension nine", ECC-AMEGA seminar, Trest, Czech Republic, October 2013.
- T. Salac: "Pushing down sequences of differential operators from contact to conformally symplectic manifolds", ECC-AMEGA seminar, Trest, Czech Republic, October 2013.
- A. Cap: "A remarkable class of locally conformally symplectic geometries", 34th Winter School Geometry and Physics, Srni, Czech Republic, January 2014.
- C. de Zanet: "Generic distributions of rank 4 in dimension 9", 34th Winter School Geometry and Physics, Srni, Czech Republic, January 2014.
- C. Harrach: "A Poisson transform for differential forms", 34th Winter School Geometry and Physics, Srni, Czech Republic, January 2014.
- T. Salac: "Pushing down the Rumin complex to locally conformally symplectic quotients", 34th Winter School Geometry and Physics, Srni, Czech Republic, January 2014.
- A. Cap: "Relative BGG sequences" (series of 4 lectures), University of Auckland, March and April 2014.
- A. Cap: "C-projective compactness", Workshop: C-projective geometry and related structures, Kioloa, Australia, March 2014.