The researcher of the Uppsala University (Sweden) Marcus Vaktnäs recently completed a stay at the University of La Laguna, invited by Ruymán Cruz Barroso, Professor Vaktnäs, from the Department of Mathematical Analysis, visited the university. During his visit, they explored their shared research interests and made progress on a scientific publication aimed at advancing the theory of multiple orthogonality on the unit circle. Vaktnäs also gave a lecture on his latest research at the Seminar on Mathematical Analysis and Applied Mathematics, organized by the University Institute of Mathematics and Applications.
The research stay is part of the University of La Laguna's (ULL) grant program for short research stays for researchers from foreign institutions. This initiative is funded by the Vice-Rectorate for Research and Transfer of the University of La Laguna and by the Canary Islands Agency for Research, Innovation and the Information Society (ACIISI) of the Government of the Canary Islands.
The visit stems from a collaboration between the two researchers, born from their shared interest in the study of multiple orthogonal polynomials on the unit circle. It is worth noting that researchers in the Applied Mathematics department at the University of La Laguna (ULL) have been working for decades in the field of orthogonality and quadrature, a set of numerical integration rules that allow for the calculation of approximate integral values and are fundamental for solving a wide variety of problems. Although it is a theoretical and abstract area of research, the families of orthogonal polynomials studied have numerous applications in disciplines such as Mathematics, Physics, Engineering, Medicine, and Economics. “When we go to the ophthalmologist, for example, to have possible optical aberrations in our eyes diagnosed, a very particular family of orthogonal polynomials, extensively studied in the literature, comes into play,” explains Cruz Barroso.
According to both researchers, this line of study had received scant attention until 2024, with only two publications specifically focused on the subject. One of these was authored by Ruymán Cruz Barroso and professors Ramón Orive and Carlos Díaz Mendoza, also from the Department of Applied Mathematics at the University of La Laguna (ULL). In this context, the contributions of Marcus Vaktnäs and Rostyslav Kozhan, the latter also a researcher at Uppsala University, have given new impetus to this theory. Their work has allowed for a reformulation of the conditions for multiple orthogonality on the unit circle and a revision of the approaches developed in previous publications.
Based on this shared scientific interest, the collaboration between the two researchers began to take shape during the first half of 2026, with a short research stay by Ruymán Cruz Barroso at Uppsala University. The purpose of this cooperation is to explore the possibilities of applying the theoretical advances developed at the Swedish university to the field of numerical integration, one of Cruz Barroso's specialties.
“The main objective of this stay is to work together on mathematics and tackle problems together, sharing a work table, seeking solutions, and exploring alternatives on a daily basis. This dynamic is much more productive than doing it by email or through platforms like Zoom,” says Vaktnäs. The Swedish researcher also highlights the interest shown by the ULL department hosting him in his line of work. “I think this line of research is currently generating more interest at the University of La Laguna than it did in Uppsala,” he admits. This involvement fosters the exchange of ideas and allows for constructive debates about his research.
During his stay, Vaktnäs also gave a lecture at the Seminar on Mathematical Analysis and Applied Mathematics, entitled “What is a multiple orthogonal polynomial on the unit circle?”. In it, he presented his research on multiple orthogonality with respect to measures defined on the unit circle. To address this problem, the Uppsala University researcher has developed an alternative approach that introduces a new notion of a multiple orthogonal polynomial. This approach has yielded significant results that had not been established until now, especially regarding the uniqueness of polynomials and the location of their zeros. Throughout the lecture, Vaktnäs explained these ideas and related them to simpler concepts from the classical theory of orthogonal polynomials, with the aim of contextualizing the topic and making it accessible to a non-specialist audience.
Both Vaktnäs and Cruz Barroso believe that the initial scientific exchanges held this year lay the groundwork for future, highly significant scientific contributions. They also emphasize the value of short research stays, which not only foster a closer and more collaborative working dynamic but also strengthen ties with other specialists and expand collaborative networks with fellow researchers who share their scientific interests.
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