Keynote Speakers

Title of the talk: Some fundamental motivations for the development of hypercomplex function theories 

Abstract: As is well known, there are many ways to extend classical complex function theory to higher dimensions, but there is no unique canonical way. One of the founders of hypercomplex function theory, R. Fueter, introduced quaternionic holomorphic functions to provide an approach to solving Hilbert's class field problem (12th problem formulated in the  Zahlbericht from 1896). It became clear to R. Fueter that Abelian functions, which are four-fold periodic functions in two complex variables, do not provide all Galois field extensions of a biquadratic number field, because in the several complex variables setting the periods have to satisfy the so-called Riemann condition. This condition did not allow him to construct all field extensions, but only certain subfields. To get rid of this condition, Fueter discovered that one can consider quaternionic functions that satisfy a generalized Cauchy-Riemann equation instead. But it turned out that this function class is not endowed with the ordinary concept of differentiability in the sense of the usual differential quotient. Nearly 50 years later, H. Malonek found a new differentiability notion that exactly describes this function theory. His permutational product provided a new interpretation and allowed for a direct link between classical aspects of function theory with hypercomplex function theories. Finally, we come back to Fueter's original problem. Over the last 25 years, we discovered that hypercomplex function theories also offer a richer theory of modular forms, including Eisenstein- and Poincaré series. The octonionic setting provides a further canonical setting to construct Galois field extensions, this time of triquadratic number fields. Recently, we were able to fill up this gap. Over nearly 100 years, it became clear that (different) hypercomplex function theories provide new powerful tools to study deep fundamental problems of mathematics.

Short Bio, including current research interests

Sören Krausshar currently holds the position of Full Professor and Chair of Mathematics at the University of Erfurt (since 2014). He received his degree in Mathematics from RWTH Aachen University of Technology in 1998. Subsequently, he earned his PhD from the University of Aveiro/RWTH Aachen in 2000, with a dissertation entitled "Eisenstein Series in Clifford Analysis," and completed his Habilitation at Ghent University/RWTH Aachen in 2004, presenting a thesis on "Automorphic Forms in Clifford Analysis." Before his current appointment, Dr. Krausshar held positions at the University of Aveiro (1998-2000), Ghent University (2000-2006), Fortis Bank AG Brussels (2006), Katholieke Universiteit Leuven (2007-2009), Universität Paderborn (2009-2010), and Technische Universität Darmstadt (2010-2013). His principal research interests encompass Clifford and octonionic analysis, partial differential equations and mathematical physics, fractional integro-differential operators, harmonic analysis on manifolds, and higher-dimensional modular forms.  

Title of the talk: Numerical range in the realm of quaternions

Abstract: The spectrum of an operator in a Hilbert space is an important topic in functional analysis, with many applications in both theoretical and applied contexts. Alongside the spectrum, the numerical range—which is the image of the unit sphere under a certain quadratic form—is a useful but less widely known tool. 

The geometric structure of the numerical range depend strongly on the ground field being the complex numbers or the skew-field of Hamilton's quaternions. In the complex case, the numerical range of both bounded and unbounded operators has been deeply studied and it can be used to help to locate the spectrum and give important information about operator's behavior. However, when extending this concept to quaternionic Hilbert spaces, the non-commutative nature of quaternions introduces substantial challenges. Classical tools from complex operator theory no longer apply directly and must be carefully reformulated. While recent work has begun to address the numerical range for bounded operators in quaternionic Hilbert spaces, the unbounded case remains mostly unexplored.

In this talk, I explore the properties and shape of the numerical range of bounded and unbounded operators on quaternionic Hilbert spaces. I discuss its relation with the spectrum, how it reflects important features of the operator, and what new ideas are needed to work in this non-commutative setting.

This is a joint work with Luís Carvalho and Sérgio Mendes.

Short Bio, including current research interests

Cristina Diogo is an Associate Professor in the Department of Mathematics at ISCTE – University Institute of Lisbon and member of the Center for Mathematical Analysis, Geometry, and Dynamical Systems at Instituto Superior Técnico, University of Lisbon.

Her research interests lie in the fields of operator theory and functional analysis, with a recent focus on the spectrum and numerical range of linear operators on complex and quaternionic Hilbert spaces. She is particularly interested in the connection between these sets and their geometric properties, which raise interesting and challenging questions that bring together different areas of mathematics, including algebra, analysis, and geometry.