Speakers
List of speakers and abstracts (updated August 28th 2017)
Daniele Angella (Università di Firenze) - "Slice-quaternionic Hopf surfaces"
We construct slice-quaternionic Hopf surfaces, namely, manifolds whose universal cover is the quaternionic punctured 2-dimensional plane and whose fundamental group is infinite cyclic. In other words, this gives slice-quaternionic structures on S^1 × S^7: we study their group of automorphisms and their deformations. Joint work with Cinzia Bisi.
Swanhild Bernstein (TU Bergakademie Freiberg) - "Riemann-Hilbert problems in Clifford analysis"
The classical Riemann-Hilbert problem is one of the most famous problems in classical function theory.
Definition (Riemann-Hilbert problem). Let Γ ∈ C be an oriented contour or a set of oriented contours in the complex plane. The contours may be simple or disconnected, or even self intersecting. The orientation de- fines + and − sides of the contour in the usual way. The Riemann-Hilbert problems consists in finding a function u(z) such that
- u(z) is analytic for z ∈ C\Γ,
- u+(z) = u−(z)g(z) for z ∈ Γ with u±(z0) = lim u(z) and u± is on
- z±→z0 the ± side of Γ, g(z) is a given function on Γ.
- u(z)→0, as |z|→∞.
The Riemann-Hilbert problem can be generalized in a lot of different ways and has many applications, for example to orthogonal polynomials and the theory of random matrices.
In this talk we will present different approaches to a Riemann-Hilbert problem of monogenic functions in Clifford analysis, the results and problems. A major obstacles is the non-commutativity of multiplication in Clifford analysis.
Paolo Lella (Università di Torino) - "Approximation of slice functions"
I will report on a joint work with R. Ghiloni in which we address the following natural question: "Which approximations theorems of functions of a complex variable can be extended/generalized to slice functions"?
I will give a quite complete answer to this problem by describing an "approximation turning principle" that permits to extend many important classical results for real- and complex-valued functions to the general case of slice functions.
Alessandro Perotti (Università di Trento) - "Slice-regular functions preserve the orientation of R^4"
The Jacobian determinant of a quaternionic slice-regular function f is always non-negative. From this property and from a detailed study of the fibers of f, we get a topological description of the singular set N_f. We also show that the sign property of the Jacobian implies, in great generality, some of the fundamental results of the theory of slice-regular functions. Some of them are already partially known, others are new ((Quasi-)open mapping theorem, Maximum modulus Principle, Rouché Theorem, Injectivity Theorem).
Paolo Piccinni (Università di Roma "La Sapienza") - "Coomologia degli spazi simmetrici eccezionali di tipo compatto e strutture di Clifford"
Dei dodici spazi simmetrici eccezionali di tipo compatto, due sono varietà complesse kaehleriane, cinque quaternionali kaehleriane, e quattro "ottonionico kaehleriane" (con una sovrapposizione in due casi tra tali strutture). Cercherò di descrivere come le nozioni di struttura di Clifford pari, introdotta qualche anno fa da A. Moroianu e Semmelmann, e quella un po' più antica di sistema di Clifford, consentono di dare una costruzione dei generatori di coomologia associati alle strutture kaehleriane dei tre tipi.
Irene Sabadini (Politecnico di Milano) - "On the on the Segal-Bargmann and Bargmann-Radon transform in the monogenic setting"
The Segal-Bargmann transform can be seen as a unitary map from spaces of square-integrable functions to spaces of square-integrable holomorphic functions. This class of functions is known in the literature under different names: Bargmann or Segal-Bargmann or Fock spaces either in one or several complex variables. The classical Segal-Bargmann transform can be defined starting from the basis of Hermite polynomials and can be extended to Clifford algebra-valued functions. Thus we can consider monogenic functions and prove that the Segal-Bargmann kernel corresponds to the kernel of the Fourier-Borel transform for monogenic functionals. This kernel is also the reproducing kernel for the monogenic Bargmann module.
We also discuss a Bargmann-Radon transform on the real monogenic Bargmann module. This transform is defined as the projection of the real Bargmann module on the closed submodule of monogenic functions spanned by the monogenic plane waves. It is possible to prove a characterization formula for the Bargmann-Radon transform of a function in the real Bargmann module in terms of its complex extension and then its restriction to the nullcone in $\mathbb C^m$. If time permits, we shall also discuss the relation with the Szego-Radon transform.
Giulia Sarfatti (Università di Firenze) - "Quaternionic toric manifolds"
In this seminar we will introduce a possible counterpart to classical symplectic toric manifolds in the quaternionic setting.
We will see a procedure with which, starting from appropriate m-dimensional Delzant polytopes, we obtain 4m-dimensional real manifolds, acted on by m copies of the group of unit quaternions, that can be endowed with a 4-plectic structure and a generalized moment map.
The interest of these manifolds also relies in the fact that they are quaternionic regular (in the sens of slice regularity).
This is a joint work with Graziano Gentili and Anna Gori.
Caterina Stoppato (Università di Firenze) - "Funzioni slice in ambito non associativo"
The theory of slice functions on alternative *-algebras extends the theory of slice regular functions introduced by Gentili and Struppa in 2006. The talk will focus on the properties of slice functions in the non associative case - in particular, in the case of Cayley’s octonions. The results presented in the talk have been obtained jointly with Riccardo Ghiloni and Alessandro Perotti.