arXiv research

Study first BGG operators on homogeneous geometries.

Study parallel tractors and cotractors on almost Grassmannian structures.

Study BGG operators on homogeneous conformal geometries.

For a real or complex semisimple Lie group $G$ and two nested parabolic subgroups $Q\subset P\subset G$, we study parabolic geometries of type $(G,Q)$. Associated to the group $P$, we introduce a class of relative natural bundles and relative tractor bundles and construct some basic invariant differential operators on …

CR Killing operator derived from tractor calculus for CR structures.

We prove that the Casimir operator acting on sections of a homogeneous vector bundle over a generalized flag manifold naturally extends to an invariant differential operator on arbitrary parabolic geometries. We study some properties of the resulting invariant operators and compute their action on various special types…

BGG-operators form sequences of invariant differential operators and the first of these is overdetermined. Interesting equations in conformal geometry described by these operators are those for Einstein scales, conformal Killing forms and conformal Killing tensors. We present a deformation procedure of the tractor conn…

Study of BGG sequences on foliated manifolds with transverse parabolic geometry.

New BGG sequences on manifolds help solve elasticity and relativity problems.

BGG-sequences offer a uniform construction for invariant differential operators for a large class of geometric structures called parabolic geometries. For locally flat geometries, the resulting sequences are complexes, but in general the compositions of the operators in such a sequence are nonzero. In this paper, we sh…

A projective geometry is an equivalence class of torsion free connections sharing the same unparametrised geodesics; this is a basic structure for understanding physical systems. Metric projective geometry is concerned with the interaction of projective and pseudo-Riemannian geometry. We show that the BGG machinery of …

The paper classifies tensors on specific Lorentzian metrics.

New Poisson transforms link differential forms on homogeneous spaces to Riemannian symmetric spaces.

The article constructs differential operators for parabolic geometries.

First BGG operators are a large class of overdetermined linear differential operators intrinsically associated to a parabolic geometry on a manifold. The corresponding equations include those controlling infinitesimal automorphisms, higher symmetries, and many other widely studied PDE of geometric origin. The machinery…

The paper constructs a complex for the Dirac operator in 4 dimensions.

Classifies and constructs intertwining differential operators between vector bundles over real projective space.

The paper proves Strichartz estimates for Schrödinger flows on compact Lie groups.

This is an expanded version of a series of two lectures given at the IMA summer program "Symmetries and Overdetermined Systems of Partial Differential Equations". The main part of the article describes the Riemannian version of the prolongation procedure for certain overdetermined system obtained recently in joint work…

The study extends hypoellipticity to filtered manifolds and applies it to BGG sequences.

A regular normal parabolic geometry of type $G/P$ on a manifold $M$ gives rise to sequences $D_i$ of invariant differential operators, known as the curved version of the BGG resolution. These sequences are constructed from the normal covariant derivative $\na^\om$ on the corresponding tractor bundle $V,$ where $\om$ is…

Constructs BGG resolutions for symplectic case.

Partial AHS-structures extend G-structures and Cartan geometries to manifolds with involutive distributions.

Solves index problem for curved BGG sequences in parabolic geometry.

Study Bernstein-Gelfand-Gelfand complexes on Lipschitz domains, computing cohomology and applying to elasticity models.

BGG-equations are geometric overdetermined systems of PDEs on parabolic geometries. Normal solutions of BGG-equations are particularly interesting and we give a simple formula for the necessary and sufficient additional integrability conditions on a solution. We then discuss a procedure for coupling known solutions of …

Developed calculus for $(2, 3, 5)$ distributions, finding geometric interpretations.

Lecture notes on BGG complexes using Lie groups and algebras.

By studying the development of shock waves out of discontinuity waves, in 1954 P. Lax discovered a class of PDEs, which he called 'completely exceptional', where such a transition does not occur after a finite time. A straightforward integration of the completely exceptionality conditions allowed Boillat to show that s…

We give a geometric construction of the BGG resolutions in singular infinitesimal character in the case of 1-graded complex Lie algebras of type A.

Develops a new calculus for contact structures on manifolds.

For a compact, oriented, hyperbolic $n$-manifold $(M,g)$, realised as $M= Γ\backslash \mathbb{H}^{n}$ where $Γ$ is a torsion-free cocompact subgroup of $SO(n,1)$, we establish and study a relationship between differential geometric cohomology on $M$ and algebraic invariants of the group $Γ$. In particular for $\mathbb{…

This paper constructs Poisson transforms and analyzes their properties on complex hyperbolic spaces.

New construction of Riemannian deformation sequence using differential operators.

New path integrals for elasticity derived from differential complex theory.

We show that infinitesimal automorphisms and infinitesimal deformations of parabolic geometries can be nicely described in terms of the twisted de-Rham sequence associated to a certain linear connection on the adjoint tractor bundle. For regular normal geometries, this description can be related to the underlying geome…

Researchers create exact sequences for isotropic 2-Grassmannian.

We develop a relative version of Kostant's harmonic theory and use this to prove a relative version of Kostant's theorem on Lie algebra (co)homology. These are associated to two nested parabolic subalgebras in a semisimple Lie algebra. We show how relative homology groups can be used to realize representations with low…

Constructs complexes of differential operators on homogeneous spaces.

Embedding theorem for tractor bundles applied to conformal geometry.

We give a complete construction of the Bernstein-Gelfand-Gelfand complex on real or complex projective space using minimal ingredients.

This paper analyses non-regular $|2|$-graded geometries, and show that they share many of the properties of regular geometries -- the existence of a unique normal Cartan connection encoding the structure, the harmonic curvature as obstruction to flatness of the geometry, the existence of the first two BGG splitting ope…

Researchers adapt Poisson transforms for CR structures on complex hyperbolic spaces.

This is the last part of a series of articles on a family of geometric structures (PACS-structures) which all have an underlying almost conformally symplectic structure. While the first part of the series was devoted to the general study of these structures, the second part focused on the case that the underlying struc…

We introduce the notion of a conformally Fedosov structure and construct an associated Cartan connection. When an appropriate curvature vanishes, this allows us to construct a family of natural differential complexes akin to the BGG complexes from parabolic geometry.

The paper develops a heat kernel expansion for Rockland operators on filtered manifolds.

For curved projective manifolds we introduce a notion of a normal tractor frame field, based around any point. This leads to canonical systems of (redundant) coordinates that generalise the usual homogeneous coordinates on projective space. These give preferred local maps to the model projective space that encode geome…

New characterisation of geodesics in various geometries yields conserved quantities.