Speakers

List of speakers and abstracts

Nicola Arcozzi (Università di Bologna)

"Carleson measures for the Dirichlet space on the bi-disc"

In 1979 Alice Chang characterized the Carleson measures for the Hardy space on the bi-disc, some years after Lennart Carleson had exhibited a counterexample showing that his celebrated disc result could not be extended to the two parameter case. In 1980 David Stegenga characterized the Carleson measures for the Dirichlet space on the disc, using ideas originating from Vladimir Maz'ya's "isocapacitary inequality".

Stegenga's characterization extends to the Dirichlet space on the bi-disc, but the proof is much more involved, the main reason being that in bi-parameter potential theory the maximum principle for potentials of measures fails. We sketch the flow of ideas underlying the proof.

Most of the seminar is based on research work with Pavel Mozolyako, Karl-Mikael Perfekt, and Giulia Sarfatti.

Luca Baracco (Università di Padova)

"Propagation of regular extendability"

For functions of two quaternionic variables that are regular in the sense of Fueter, we establish a result similar in spirit to the Hanges and Treves theorem. Namely, we show that a ball contained in the boundary of a domain is a propagator of regular extendability across the boundary. This is a joint work with M. Fassina and S. Pinton.

Cinzia Bisi (Università di Ferrara)

"On Runge Pairs and Topology of Axially Symmetric Domains."

Subsequently to the very interesting and seminal paper of Colombo, Sabadini and Struppa in 2011, we will present a new theorem on Runge Pairs of Axially Symmetric Domains in the quaternionic setting, that can be easily generalized to the Clifford environment. A global approach (in which we strongly believe) and a complete study of the homology groups of these domains, instead of a slicewise analysis, have permitted us to improve the results in Colombo, Sabadini and Struppa’s original paper and a more deep understanding of the subject. This is part of a joint project with J. Winkelmann.

Hennie De Schepper (Ghent University)

"Integral representation formulae for and generation of osp(4|2) monogenic functions"

Similarly as hermitian Clifford analysis emerges in Euclidean space of even dimension as a refinement of euclidean Clifford analysis by the introduction of a complex structure on the vector space, quaternionic Clifford analysis arises as a further refinement by the introduction of a so-called hypercomplex structure, i.e. three complex structures which submit to the quaternionic multiplication rules, the dimension now being a fourfold. In the hermitian framework, the fundamental symmetry group is U(n), which is isomorphic to the subgroup of SO(2n) of matrices which are commuting with the complex structure; in the quaternionic framework, this role is taken by the symplectic group Sp(p), which is isomorphic with the subgroup of SO(4p) of matrices which are commuting with all three complex structures. Two, respectively four, differential operators are constructed, which are invariant under the action of the corresponding symmetry group. Their simultaneous null solutions are respectively called hermitian and quaternionic monogenic functions.

The Cauchy integral formula is an important result for holomorphic functions in the complex plane, which allows to reconstruct a holomorphic function in the interior of a bounded domain from its boundary values on the smooth boundary of that domain. This formula was generalized to the framework of monogenic functions and also used to generate monogenic from given boundary values. In this contribution, we compare the ways in which Cauchy integral representation formulae can be established in the hermitian and quaternionic Clifford settings and we investigate under which conditions given boundary values may extend to hermitian or quaternionic monogenic functions.

Riccardo Ghiloni (Università di Trento)

"Slice regular functions in several variables"

In this talk we present basic concepts and results of a new function theory in several quaternionic variables, obtained jointly with Alessandro Perotti. The mentioned theory works with variables ranging in arbitrary real alternative *-algebras, including octonions and all Clifford algebras.

Carlotta Giannelli (Università di Firenze)

"Rational moving frames on Pythagorean-hodograph curves: theory and applications"

In computer aided geometric design, the discipline that provides both the theoretical backgrounds and the numerical tools for the automated manipulation of geometric data, curves and surfaces are generally specified in parametric rational form. When moving frames along spatial trajectories are considered, rational forms can be achieved by considering the class of Pythagorean–hodograph (PH) space curves, which admit a compact description using the algebra of quaternions. Among all moving frames on a given space curve, two examples are of special importance in view of different geometric features: the Frenet frame, which defines a local reference system in terms of the local intrinsic geometry of the curve; and the rotation–minimizing frame, which exhibits the least possible frame rotation along the curve. The talk will address the characterization of different remarkable classes of PH curves with associated rational moving frames, together with their application to the the design of rational rigid body motions.

Anna Gori (Università di Milano)

"On compact affine quaternionic curves and surfaces"

This talk is devoted to the study of affine quaternionic manifolds and to a possible classification of all compact affine quaternionic curves and surfaces. It is established that on an affine quaternionic manifold there is one and only one affine quaternionic structure. A direct result, based on the celebrated Kodaira Theorem states that the only compact affine quaternionic curves are the quaternionic tori. As for compact affine quaternionic surfaces, the study of their fundamental groups, together with the inspection of all nilpotent hypercomplex simply connected 8-dimensional Lie Groups, identifies a path towards their classification.

This talk is based on a work in collaboration with Graziano Gentili and Giulia Sarfatti.

Samuele Mongodi (Politecnico di Milano)

"Complex structures on real associative algebras and slice-regular functions"

Abstract: I will present a study of the linear complex structures which are induced on a real associative algebra by multiplication by one of its elements. This (elementary) geometric object dictates the behavior of slice-regular functions; in particular its natural complex structure endows slice-regular functions of the many properties that they share with holomorphic functions of one complex variable.

As an example, this space of complex structure is embedded in the grassmannian and the total space of the pullback of the tautological bundle controls the behavior of the zeroes of slice-regular functions.

Restricting the attention to those complex structures which are also orthogonal with respect to a given scalar product, we recover the "spheres of imaginary units" studied by Ghiloni and Perotti and the constructions attached to them.

Moreover, the twistor transform, defined by Gentili, Salamon, Stoppato, can be generalized to an associative algebra, given a never vanishing section of a certain bundle over this set of complex structures.

Andrea Spiro (Università di Camerino)

"Quaternionic Kaehler manifolds and harmonic spaces"

In a sequence of theoretical physics papers of the '90s, V. Ogievetsky and coll. introduced the notion of "harmonic space" of anhyperkaehler (hk) manifold, an appropriate holomorphic bundle over the twistor bundle of the manifold. The interest for this object comes from the fact that it allows a complete local descriptions of the instantons and the hyperkaehler metrics in terms of holomorphic functions, the prepotentials.

On the other hand, up to coverings, the hk manifolds coincide with the Ricci flat manifolds that are in the larger family of the quaternionic kaehler (qk) manifolds. In this talk, after a review of all needed definitions (what is a qk manifold, a twistor bundle, etc.), we present the results of a recent joint work with Chandrashekar Devchand and Massimiliano Pontecorvo, where we extend the theory of harmonic spaces and prepotentials to arbitrary non Ricci flat qk manifolds.