Speakers

List of speakers and abstracts

Marco Budinich (Università di Trieste)

"On Complex Representations of Clifford Algebra: Majorana and quaternionic spinors"

We show that complex representations of Clifford algebra can always be reduced either to a real or to a quaternionic algebra depending on a duly defined signature of complex space. It follows that spinors are unavoidably either real Majorana spinors or quaternionic spinors and complex spinors fade away. We use this result to support the (1, 3) signature of quaternions for Minkowski space.

Hendrik De Bie (Gent University)

"New progress on Dunkl operators for dihedral groups"

Dunkl operators are differential-difference operators that can be constructed starting from reflection groups. They can be paired with Clifford algebra generators to construct a Dunkl version of the Dirac operator in $\mathbb{R}^n$.

In recent years a lot of progress has been made in the case when the reflection group is a dihedral group, i.e. the symmetry group of a regular polygon in the plane.

In this talk I will give an overview of some of these recent results, potentially including the Dunkl transform, the intertwining operator, Dunkl monogenics as irreducible representations of certain symmetry algebras, and unitary deformations.

H. De Bie, P. Lian, The Dunkl kernel and intertwining operator for dihedral groups. J. Funct. Anal. 280, 108932 (2021).

D. Constales, H. De Bie, P. Lian, Explicit formulas for the Dunkl dihedral kernel and the (k, a)-generalized Fourier kernel. J. Math. Anal. Appl. 460 (2018), 900-926.

H. De Bie, Alexis Langlois-Rémillard, R. Oste and J. Van der Jeugt, Finite-dimensional representations of the symmetry algebra of the dihedral Dunkl--Dirac operator, arXiv:2010.03381, 46 pages.

D. Ciubotaru, H. De Bie, M. De Martino, R. Oste, Deformations of unitary Howe dual pairs. arXiv:2009.05412, 26 pages.

Kamal Diki (Chapman University)

"A new family of QRKHS in the Clifford-Appell case"

We will use the classical Fueter mapping theorem in order to construct a special system of Clifford-Appell polynomials in the quaternionic setting. We will discuss various properties of such polynomials and study their algebraic behaviour with respect to the well-known Cauchy-Kowalevski product. Then, based on this system we can introduce a new family of quaternionic reproducing kernel Hilbert spaces (QRKHS) in this framework. As particular examples we will treat further results on the Hardy and Fock spaces and their related operators such as creation, annihilation, shift and backward shift operators. Finally, if time allows we will discuss some connections and applications of this new approach to the recent theory of quaternionic slice polyanalytic functions and Schur analysis. This talk is based on recent works that are in collaboration with Daniel Alpay, Fabrizio Colombo, Soeren Krausshar and Irene Sabadini.

Rita Fioresi (Università di Bologna)

"Supersymmetry and Supergeometry"

Supersymmetry was discovered by the physicists in the 1970's with the intent to treat on equal ground the two types of elementary particles: bosons and fermions. Since then, many mathematicians including Berezin, Leites, Kostant, Deligne and Manin to cite a few, have worked out the mathematical foundations of supersymmetry introducing the concept of supermanifold and superscheme. In this talk, after a brief introduction of the basic notions of supergeometry, we shall take a tour of the main results available and the current directions of research in mathematics and physical applications.

Francesca Pelosi (Università di Roma 2 "Tor Vergata")

"Spline surfaces with Pythagorean hodograph isoparametric curves in quaternion form."

In this seminar we present the construction of spline surfaces, with spatial Pythagorean hodograph spline curves as isoparametric curves. First a single bi-patch is determined which interpolates two boundary PH curves and two polynomial curves. Using the quaternion representation for the spatial PH curve, it is proved that this construction can be reduced to solving a quadratic equation in a single unknown quaternion. In a second step, all the bi-patches are put together to form a globally regular surface.

Jasna Prezelj (University of Primorska, University of Ljubljana, IMFM)

"Quaternionic Riemann surface and quaternionic logarithm - Part I"

We present the quaternionic and octonionic analogs of the classical Riemann surfaces of the complex logarithm and give a unifying definition of the logarithm in the quaternionic and octonionic settings. The construction of these manifolds has nice peculiarities and the scrutiny of Bernhard Riemann approach to Riemann surfaces, mainly based on conformality, leads to the definition of slice conformal or slice isothermal parameterization of Riemann 4-manifold and 8-manifold. These new classes of manifolds include slice regular quaternionic and octonionic curves, graphs of slice regular functions, the 4 and 8 dimensional sphere, the helicoidal and catenoidal 4 and 8 dimensional manifolds and the deformations from the catenoidal to part of the helicoidal surface.

This is a joint work with G. Gentili and F. Vlacci.

Caterina Stoppato (Università di Firenze)

"Zeros of regular functions on non-symmetric slice domains"

The notion of slice regular functions of one quaternions variable, introduced in 2006, has rapidly grown into a full-fledged theory. It has been developed focusing only on quaternionic domains that are symmetric with respect to the real axis. Recent work in the theory concerns slice regularity over domains that are not axially symmetric. The talk will outline the motivation for these recent developments and focus on the study of zero sets in this new setting.

The results presented in the talk have been obtained jointly with Graziano Gentili.

Luigi Vezzoni (Università di Torino)

"The Calabi-Yau problem in HKT Geometry"

HKT Geometry (HyperKahler with torsion Geometry) is the Geometry of hyperHermitian manifolds equipped with a nondegenerate ∂-closed (2,0)-form Ω. The interest for this kind of manifolds is mainly motivated by theoretical and mathematical physics since they appear on supersymmetric sigma models with Wess-Zumino term and in supergravity theories. From the mathematic point of view HKT manifolds generalize HyperKahler manifolds. The talk mainly focuses on a Calabi-Yau type problem in HKT geometry introduced by Alesker and Verbiskiy in 2010. I will describe a recent study of the problem in some explicit examples and a new approach via a geometric flow.

The talk is based on a work in progress with Lucio Bedulli and Giovanni Gentili.

Fabio Vlacci (Università di Trieste)

"Quaternionic Riemann surface and quaternionic logarithm - Part II"

As it is known, a quaternionic logarithmic function cannot be defined on the entire set H \ {0}. Indeed, for any I in the sphere S of purely imaginary units of H, the quaternionic logarithm coincides with the complex logarithm on the “slice” R + IR = C_I, and hence it cannot be defined on the entire C_I \ {0}. It is probably less known that sometimes a quaternionic logarithm cannot be defined even locally. In particular, at any point x of the negative real axis, there is no way to choose a unique imaginary unit I ∈ S such that

log(x) = ln(−x) + I arg(x).

Any I ∈ S will be fine, and so the quaternionic logarithm is a “sphere-valued” function at such a point, even if we choose a particular determination of arg(x). This is in fact the most apparent difference between the complex and the quaternionic case. In addition to this, except for the case of the principal branch of the quaternionic logarithm, the same phenomenon appears for the case of all points of the positive real axis. For such a point y, we have that

log(y) = ln(y) + I arg(y)

for any I belonging to the entire sphere S. Therefore if we set arg(y) =2kπ with k different from zero, then the quaternionic logarithm is sphere-valued at y as well. After introducing a quaternionic logarithmic function via a suitable quaternionic Riemann surface, we investigate the possibiliry of extending such quaternionic logarithm along paths. Finally, we’ll present some results on the inverse functionial of the ∗− exponential exp_*, introduced in A. ALTAVILLA, C. DE FABRITIIS, “*-exponential of slice regular functions”, Proc. Amer. Math. Soc. 147, 1173-1188, 2019.

This is a joint work with G. Gentili and J. Prezelj.