# Hypercomplex Analysis and Geometry

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December 1.st  2015 ,  Ferrara University

Department of Mathematics and Computer Science

Machiavelli Street, 30, Ferrara

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### P R O G R A M

Languages : English /  Italian

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MORNING:  ROOM  5  (FIRST  FLOOR)

09:10 - 11:10

Speaker 1 : Caterina Stoppato ( Istituto Nazionale di Alta Matematica ) (40 min.)

Titolo:

L’algebra delle funzioni slice: proprietà degli zeri su *-algebre alternative

Abstract:

Il seminario coprirà alcuni recenti sviluppi nella teoria delle funzioni slice su *-algebre alternative. La classe delle funzioni slice comprende le funzioni slice regolari e, tra queste ultime, i polinomi. In particolare, il seminario ne descriverà le proprietà algebriche ed esporrà alcuni curiosi fenomeni riguardanti gli insiemi degli zeri. Questi saranno illustrati con esempi e, tempo permettendo, con l’enunciato di risultati generali.

Gli argomenti trattati fanno parte di un lavoro in collaborazione con Riccardo Ghiloni e Alessandro Perotti.

Speaker 2:  Alessandro Perotti ( Università Trento ) (40 min.)

Titolo:

L’algebra delle funzioni slice: prodotti e quozienti su *-algebre alternative.

Abstract:

Si presentano alcuni risultati riguardanti le classi di funzioni slice e slice regolari definite su una *-algebra alternativa reale A. Tali insiemi costituiscono a loro volta una *-algebra rispetto al prodotto slice e alla coniugazione slice, ed ereditano da A alcune proprietà algebriche riguardanti, in particolare, i prodotti e gli inversi moltiplicativi.
Gli argomenti trattati fanno parte di un lavoro in collaborazione con Caterina Stoppato e Riccardo Ghiloni.

Speaker 3: Alberto  Saracco ( Università Parma )  (25 min.)

Title:

Carleson measures: from the complex to the hypercomplex case.

Abstract:

Since their introduction in 1962 to solve the corona problem, Carlseon measures became a mathematical object studied per se, as well as an useful tool in many circumstances. In the last 50 years various characterizations of Carleson measures have been given: for Hardy and Bergmann spaces in the disc of C, for the ball of C^n and for strongly pseudoconvex domains. Some of the techniques used in those proof can be adapted to obtain similar characterizations also in the hypercomplex setting. Joint work with Irene Sabadini.

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11:10 – 11:30 : Coffee Break (Sala Studio-Piano Rialzato)

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11:30 - 13:00

Speaker 4:  Fabio Vlacci (Università Firenze)  (40 min.)

Title:

Some Properties of subgroup actions of quaternionic transformations.

Abstract:

In this talk we'll describe some properties of certain discrete subgroups of quaternionic matrices acting in the hyperbolic quaternionic space which are similar to the modular group in the complex case: we highlight similarities and differences. Work in collaboration with J.P Diaz Gonzalez and A. Verjovsky (UNAM Cuernavaca Mexico)

Speaker 5: Nicola  Arcozzi (Università Bologna)  (40 min.)

Title:

Invariant Metrics for the Quaternionic Hardy Space.

Abstract:

We study metrics related with the quaternionic Hardy space. The same role is played by the hyperbolic metric in the complex case: we highlight similarities and differences. Work in collaboration with Giulia Sarfatti.

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13:00 - 14:10   “ light lunch a buffet ’’  (Sala Studio-Piano Rialzato)

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AFTERNOON: ROOM 1 (Ground  Floor)

14:10- 16:00

Speaker 6:  Irene  Sabadini ( Polimi ) (35 min.)

Title:

On the Wiener algebra over the quaternions.

Abstract :

We define the counterpart of the Wiener algebra in the quaternionic setting, both for the discrete and continuous case. We prove a Wiener-Levy type theorem, characterizing the invertible elements in the algebra.

Speaker 7: Fabrizio Colombo (Polimi) (35 min.)

Title :

The $H^\infty$ functional calculus based on the $S$-spectrum.

Abstract :

The $H^\infty$-functional calculus is an extension of the Riesz-Dunford functional calculus for bounded operators to unbounded sectorial operators, and it has been introduced by A. McIntosh. The main motivation was the connection of this calculus with the Kato's square root problem.

A second motivation for the $H^\infty$ functional calculus was the study of evolution equations and in particular the characterization of maximal regularity and the fractional powers of differential operators.

In this talk we extend this calculus to quaternionic operators and to $n$-tuples of noncommuting operators using the theory of slice hyperholomorphic functions and the associated functional calculus, called $S$-functional calculus.

Speaker 8: Jonathan Gantner ( Polimi ) (25 min.)

Title :

Two functional calculi based on slice hyperholmorphicity.

Abstract :

In recent years, the fundamental concepts of operator theory have been extended to the setting of quaternionic linear operators. In particular, the natural generalization of the holomorphic functional calculus has been developed and the associated S-spectrum proved to be the correct notion of spectrum in this setting. The function theory that underlies these results is the theory of slice hyperholomorphic functions.

Using this function theory, we were able to generalize other classic results. This talk presents a functional calculus for infinitesimal generators of strongly continuous groups of operators based on the quaternionic Laplace-Stieltjes-transform. This calculus applies also to functions that are not slice hyperholomorphic at infinity. Moreover, the talk shows how to construct fractional powers of quaternionic linear operators and that the famous Kato formula for the resolvents of fractional powers is also true in the quaternionic setting.

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16:00- 16:20   Coffee Break (Sala Studio-Piano Rialzato)

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16:20- 17:40

Speaker 9:   Amedeo  Altavilla ( Università Politecnica delle Marche ) (30 min.)

Title:

Shadow's measure of the image of a compact set transformed by a quaternionic unitary application.

Abstract:

In this talk I will show the behavior of the measure of the projection on some particular subspaces (i.e. the shadow), of a compact set, under the action of an element of the unitary quaternionic group. The result is the quaternionictranslation of the non-squeezing Theorem by Gromov. The talk is based on a work in collaboration with L.Nicolodi.

Speaker 10:  Vincenzo  Recupero ( Polito ) (40 min.)

Title :

Slice Regular Semigroups.

Abstract :

The notion of (right-linear quaternionic) spherical sectorial operator''
has been recently introduced in [R. Ghiloni, V. Recupero,
\emph{Semigroups over real alternative *-algebras: generation theorems
and spherical sectorial operators}, Trans. AMS, in press]. \newline
In this talk we continue this analysis and we define the notion of
slice regular semigroup'', i.e. a slice regular operatorial function,
defined on a spherical sector in $\mathbb{H}$, which satisfies a suitable
semigroup law induced by the slice-algebraic structure of the space of
right-linear operators. We then prove that an operator is spherical
sectorial if and only if it generates a slice regular semigroup. \newline
This result is also generalized to operators acting on spaces over real
associative *-algebras and has been obtained in collaboration with
Riccardo Ghiloni.

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S P O N S O R S :

• Unife
• Firb2012 Project "Differential Geometry and Geometric Function Theory" RBFR12W1AQ001
• INdAM

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L O C A L    O R G A N I Z E R:

• C. Bisi