arXiv research

Lecture notes on BGG complexes using Lie groups and algebras.

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

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

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.

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{…

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

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.

Using the Penrose transform, we construct analogues of the BGG (Bernstein-Gelfand-Gelfand) resolutions in certain singular infinitesimal characters, in the holomorphic geometric setting, over the Lagrangian Grassmannian. We prove the exactness of the constructed complex over the big affine cell.

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

We construct exact sequences of invariant differential operators acting on sections of certain homogeneous vector bundles in singular infinitesimal character, over the isotropic $2$-Grassmannian. This space is equal to $G/P$, where $G$ is $\operatorname{Sp}(2n,\mathbb{C})$, and $P$ its standard parabolic subgroup havin…

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

Solves index problem for curved BGG sequences in parabolic geometry.

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 paper constructs discrete Hessian and divdiv complexes on triangulations and proves their cohomology isomorphic to continuous versions.

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