A CR manifolds are smooth manifolds endowed with a CR structure, namely a complex subbundle of the complexified tangent bundle whose intersection with its conjugate consists only of the zero section, and which satisfies a formal Frobenius integrability condition. A remarkable example is given by real hypersurfaces in the n-dimensional complex space, all of which inherit a natural CR structure from the ambient complex structure.

Within the theory of Partial Differential Equations (PDEs), CR structures can be regarded as a bundle-theoretic reformulation of a first-order system of PDEs known as the tangential Cauchy–Riemann equations. Their solutions are the so-called CR functions. As first observed by H. Lewy, for a solution of the Dirichlet problem for the Cauchy–Riemann equations on a domain of n-dimensional complex space to be smooth up to the boundary, its restriction to the boundary must be a CR function.

It is therefore clear that Cauchy–Riemann geometry (i.e., the study of CR manifolds) lies at the intersection of three major areas of mathematics: Differential Geometry, Complex Analysis, and the theory of Partial Differential Equations.

My research activity in Cauchy–Riemann and pseudohermitian geometry has focused on

• ramifications of the harmonic maps theory (such as subelliptic harmonic map from a strictly pseudoconvex CR manifold and related topics)

• CR embeddability and CR rigidity problems

• sub-Riemannian geometry of the Levi distribution

• Lorentzian geometry of the canonical circle bundle over a strictly pseudoconvex CR manifold equipped with the Fefferman metric

The set of square-integrable complex-valued functions on a bounded domain in the n-dimensional complex space that admit a holomorphic representative in their Lebesgue class can be organized into a Reproducing Kernel Hilbert Space, whose kernel is known as the Bergman kernel (named after S. Bergman).

Surprisingly, (weighted) Bergman kernels also arise naturally in the quantization theory of mechanical systems whose classical phase space carries a Kähler metric (following the works A. Odzijewicz) .

My research activity in this direction has focused on

• geometric aspects of the Bergman metric on a bounded domain and the associated foliation by CR hypersurfaces 

•  boundary behaviour of geometric and analytical objects defined in terms of the Bergman metric

• Kostant–Souriau–Odzijewicz quantization theory

• applications of complex analysis methods developed in quantization theory, involving weighted complex kernels, to problems in signal theory, learning theory, and robotics