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ORTHONET winter 2024

V School and VI Meeting

 

 

The ORTHONET Network brings together researchers focused on orthogonal polynomials and special functions, exploring their deep connections with fields such as approximation theory, operator theory, number theory, information theory, Fourier series, numerical analysis, integrable systems and probability. This cross-disciplinary approach extends beyond pure mathematics, finding applications in mathematical physics, science, and technology.

A core goal of ORTHONET is to foster collaboration within the Spanish scientific community, uniting those who specialize in orthogonality and its diverse applications. By doing so, the network seeks to raise the quality and impact of research in this area. Another priority is facilitating the transfer of expertise to other scientific and technological fields where orthogonality offers untapped potential.

The ORTHONET network receives funding through project RED2022-134580-T, as part of the National Program aimed at advancing scientific and technical research and promoting its transfer. This initiative is supported by the 2021-2023 State Plan for Scientific, Technical, and Innovation Research, under the Ministry of Science, Innovation, and Universities.

logo ministerio

The Organizing Committee gratefully acknowledges the support of the Faculty of Physical Sciences for providing the infrastructure.

                                                                      


School Programme

  • Dates: December 16-19
  • Venue: Faculty of Physical Sciences, UCM, Theoretical Physics Department Seminar Room, second floor (capacity: 25 people)
  • Professors:
    • María Ángeles García Ferrero (Mini-Course: Exceptional orthogonal polynomials and Darboux transformations)
    • Andrei Martínez-Finkelshtein (Mini-Course: Logarithmic Potential Theory)
    • Walter Van Assche (Mini-Course: Multiple Orthogonal Polynomials—Theory and Applications)
  • Schedule:
    • First morning session: 9:00 AM - 11:00 AM. Walter Van Assche: Multiple Orthogonal Polynomials—Theory and Applications
    • Cofee break: 11:00 AM- 11:30 AM
    • Second morning session: 11:30 AM - 1:30 PM. Andrei Martínez-Finkelshtein: Logarithmic Potential Theory
    • Lunch: 1:30 PM - 3:00 PM
    • Afternoon session: 3:00 PM - 5:00 PM. María Ángeles García Ferrero: Exceptional orthogonal polynomials and Darboux transformations
    • Discusion: 5:00 PM- 6:00 PM
  • Each professor will teach four two-hour classes. From Monday to Wednesday, between 5:00 PM and 6:00 PM, there will be discussions on open problems or ongoing research with the students.

The ORTHONET V School

The school is aimed at young researchers in training in fields related to those covered by the ORTHONET network. It consists of three mini-courses taught by internationally renowned experts. The goal of these mini-courses is to train our students in developing research areas, as well as in advanced and fundamental techniques in the theory of orthogonal polynomials, special functions, and their applications.

The school will feature the following professors:

Andrei Martínez-Finkelshtein: 

Andrei began teaching as Professor of Mathematics at Baylor University in January 2018. He is a full professor of applied mathematics at the University of Almeria, Spain, where he served since 1994. Earlier in his career, he taught at Havana University, Cuba (1991-1994), and held visiting positions at various institutions in Spain (Universidad Autónoma de Madrid and Universidad Carlos III de Madrid) and the United States (University of South Florida and Vanderbilt University). He is an active member of several professional societies and currently serves as the Program Director of the SIAM Activity Group on "Orthogonal Polynomials and Special Functions," having previously been its vice-chair (2017-2019). Additionally, he sits on the editorial board of six academic journals.

Mini-Course: Logarithmic Potential Theory

This is a short introduction to the logarithmic potential theory in the complex plane. The central ideas are the concepts of energy and equilibrium. I discuss some classical results characterizing the equilibrium distribution and the extension of these notions to more general settings when an external field or constraints on the distribution are present. The tools provided by potential theory have a profound impact on different branches of analysis and probability. I will illustrate these applications with examples from approximation theory, ordinary differential equations, and complex dynamics.

 

María Ángeles García Ferrero:

María Ángeles is a CSIC researcher at the Instituto de Ciencias Matemáticas (ICMAT) in Madrid. Previously, she held positions at the University of Barcelona, the Basque Center for Applied Mathematics in Bilbao, the Heidelberg University and the Max-Planck Institute for Mathematics in the Sciences in Leipzig. She pursued her thesis at the ICMAT. She received her PhD diploma from the Universidad Complutense de Madrid in 2018 and she graduated in Physics from the Universidad de Valladolid in 2014. In 2020, she was awarded the José Luis Rubio de Francia prize by the Royal Spanish Mathematical Society and in 2019, she received the Vicent Caselles prize by RSME and BBVA Foundation.

Mini-Course: Exceptional orthogonal polynomials and Darboux transformations

Exceptional orthogonal polynomials (XOPs) arise as eigenfunctions of Sturm-Liouville problems and form complete bases in of square integrable functions with weights. Nevertheless, contrary to the classical polynomials of Hermite, Laguerre and Jacobi, their sets of degrees miss finitely many natural numbers. 

This course aims to introduce XOPs, whose existence, despite the “exceptionality” in their name, is not so exceptional. We will learn how rational Darboux transformations may lead to the construction of families of XOPs, and even more, to their full classification. 

Walter Van Assche:

Walter has been a professor emeritus in the Department of Mathematics at KU Leuven since October 1, 2023. He previously served as the department head from 2006 to 2015 and remains active in the Analysis section. Until September 30, 2000, he held the position of Research Director at the Belgian National Fund for Scientific Research, now known as the Research Foundation – Flanders (FWO). From 2014 to 2019, he chaired the SIAM Activity Group on Orthogonal Polynomials and Special Functions.

Mini-Course: Multiple Orthogonal Polynomials—Theory and Applications

I will start by defining multiple orthogonal polynomials (type I and type II) and their relation with Hermite-Padé approximation. I will then give various properties, in particular the nearest neighbor recurrence relations and the Christoffel-Darboux formula. There is a Riemann-Hilbert problem that is very useful for getting asymptotics when the degree tends to infinity. Various examples of multiple orthogonal polynomials will be given. Applications will be discussed in number theory, random matrix theory, and other determinantal point processes.

 


Meeting Programme

  • Dates: December 19-20
  • Venue: Faculty of Physical Sciences, UCM, Room M2 (capacity: 60 people)
  • Programme:
    • Thursday, December 19:
      • 3:30 PM - 5:00 PM: IPs meeting
      • 5:00 PM - 6:00 PM: Opening plenary lecture ''Luis Vigil'' by Andrei Martínez-Finkelshtein, Flow of the zeros of polynomials under iterated differentiation
      • 6:00 PM - 6:30 PM: Coffee break
      • 6:30 PM - 7:10 PM: Diego Ruíz Antolín, Asymptotic and numerical approximations to the zeros of parabolic cylinder functions and numerical approximations to the zeros of parabolic cylinder functions
      • 7:15 PM - 7:55 PM: Iván Area, Digital-twin approach for mathematical modelling 
    • Friday, December 20:
      • 9:00 AM - 10:00 AM: Plenary lecture by Alberto Enciso, Spectral geometry and Bessel functions
      • 10:00 AM - 10:45 AM: María José Cantero, Khrushchev’s formula on the real line 
      • 10:45 AM - 11:30 AM: Ramón Orive, Optimal polynomial approximants. An electrostatic approach.
      • 11:30 AM - 12:00 AM: Coffee break | POSTER SESSION
      • 12:00 AM - 12:45 PM: Junior talk. Juan E F Díaz, Multiple type I polynomials in the Askey scheme 
      •  12:45 PM - 1:30 PM: Junior talk. Juan Antonio Villegas, Extension of multiple orthogonality to the bivariate case 
      • 1:30 PM - 3:30 PM: Lunch
      • 3:30 PM - 4:10 PM: Antonio J. Durán, Brenke polynomials with real zeros and the Riemann Hypothesis 
      • 4:15 PM - 4:55 PM: Juan Luis Varona, Machin’s type formulas for π
      • 5:00 PM - 5:30 PM: Coffee break
      • 5:30 PM - 6:30 PM: Closing plenary lecture ''Orthogonality and Applications'' by Walter Van Assche, Rational approximations for Catalan’s constant
      • 6:30 PM - 8:00 PM: ORTHONET researchers' assembly

Plenary lectures are 55 minutes long, with an additional 5 minutes for discussion, comments, suggestions, and questions. Invited talks are 40 minutes, followed by 5 minutes for discussion, comments, suggestions, and questions.

Speakers & corresponding  abstracts

Book of  abstracts

Abstract Compilation for the Talks

Poster session

Poster Session Announcement:
We are pleased to announce a Poster Session during the morning coffee break (11:30-12:00). Participants have two presentation options:

  • Vertical Format: Posters will be mounted on vertical panels.
  • Horizontal Format: Posters can be presented in manuscript form, displayed on tables.

Call for Proposals:
Please submit your poster proposals, including the title, abstract, and preferred format (vertical or horizontal), by December 1st 2024 to manuel.manas@ucm.es.

 


The ORTHONET VI Meeting

 

The main objective of the Meeting is to bring together researchers from ORTHONET, as well as those with common interests in the network’s themes. The event will begin with a meeting of the principal investigators from each ORTHONET project member, where they will review recent achievements, discuss new objectives, and explore strategies for future collaboration.

Following this, the main activitities will commence with a series of talks designed to present the latest discoveries in the field, offering fresh and engaging insights. These talks will also revisit important topics from new perspectives, ensuring that both cutting-edge research and foundational themes are explored in depth.

There will be a Opening Plenary Lecture Luis Vigil by Andrei Martínez-Finkelshtein, a Closing Plenary Lecture Orthogonality and Applications by Walter Van Assche and a Plenary Lecture by Alberto Enciso. The program is completed with two invited talks from junior researchers, six invited talks and a poster session.

The meeting will conclude with a general assembly of all participating researchers. This final meeting will provide an open forum to discuss key issues, share ideas, and shape the future direction of ORTHONET, ensuring that the network continues to foster innovation and collaboration in the coming years.

 

Alberto Enciso

After earning his degree (2003) and PhD (2007) in Physics from the Complutense University of Madrid, he spent time as a postdoctoral researcher at ETH Zurich. In 2010, he joined the Institute of Mathematical Sciences (ICMAT) in Madrid as a Research Professor for the Spanish National Research Council, where he now leads the Differential Equations and Applications Group and chairs the institute’s Scientific Committee.

His research sits at the crossroads of geometry, analysis, and mathematical physics. A large part of his work involves investigating geometrically-rich problems that arise in the context of differential equations within physics. A defining aspect of his approach is the ability to integrate 'rigid' estimates from analysis with the more 'flexible' arguments found in differential topology and dynamical systems.

Since 2015, he has served on the Scientific Committee of the Spanish Royal Mathematical Society, and from 2017 to 2021, he was a member of the Scientific and Technical Committee of the Spanish State Research Agency. His career has been marked by significant recognitions from the European Research Council (ERC), first with a Starting Grant in 2014, followed by a Consolidator Grant in 2019. 

Among the many honors he has received are the José Luis Rubio de Francia Prize from the RSME (2011), the Antonio Valle Prize from SEMA (2013), the Princess of Girona Foundation Research Award (2014), and the Barcelona Dynamical Systems Prize (2015). In 2021, he became the first mathematician to be a speaker at the European Research Council seminar. Since 2022 he is member of the Real Academía de Ciencias Exactas, Físicas y Naturales de España.

Plenary lecture: Spectral geometry and Bessel functions 

Bessel functions play a central role in the study of the eigenvalues of the Laplacian, both in domains in Euclidean space and in more general contexts. In this talk, we will review some open problems and recent results where Bessel functions are prominently featured.

 

 

 

Opening plenary lecture Luis Vigil by Andrei Martínez-Finkelshtein: Flow of the zeros of polynomials under iterated differentiation

Assume we have a sequence of polynomials whose asymptotic zero distribution is known. What can be said about the zeros of their derivatives? Especially if we differentiate each polynomial several times, proportional to their degree? This simple-to-formulate problem has recently attracted the attention of several researchers. The problem and its solution methods have exciting connections with free probability, random matrices, approximation theory on the complex plane, and nonlinear PDE, such as the inviscid Burgers equation. 

 

Closing plenary lecture Orthogonality and Applications by Walter Van Assche: Rational approximations for Catalan’s constant

Catalan’s constant  is closely related to π2 but so far it is not known whether or not this is an irrational number. One possible way to prove irrationality is to construct rational approximants that are better than possible for a rational number. We will give a few constructions for rational approximants but so far none of them is good enough to prove irrationality.

 


Fees & Inscription

Specific dietary requirements:

Participants with specific dietary requirements (such as allergies, gluten or lactose intolerance) or those who require vegan options are kindly asked to notify us by emailing manuel.manas@ucm.es.

School fees:

School fees include accommodation in shared double room in the student residence Emilia Pardo Bazan, one coffee break, and lunch.

If you need to extend your stay at the residence for the Meeting, please contact them directly.

Early bird (before November 17th): 100€

Standard: 150€

Meeting fees:

Meeting fees include coffee breaks and lunch.

Early bird (before November 17th extended until November 24th): 150€

Standard: 200€

Combo offer:

When a person registers early bird (extended until November 24th) for both the School and the Meeting, there will be a 50€ discount, and the total amount to be paid will be 200€.

Grants:

Students enrolled in the School who have paid the registration fee can apply for a grant that offers a 50€ refund. To apply, please send an email to manuel.manas@ucm.es, including a brief CV and a motivation letter.

Inscriptions here

 

Registration assistance:

For any questions regarding registration, please contact:

Unidad de Congresos | Fundación Complutense
+34 91394 6407 / 6511
congresosfg@ucm.es

 

 


The Venue

The Facultad de Ciencias Físicas of the Universidad Complutense de Madrid (UCM) is located at  Plaza de las Ciencias, Ciudad Universitaria.

Google Maps  

Location in Campus

Local transports

Construction of the current building commenced in 1930, but progress was interrupted by the onset of the Spanish Civil War, delaying its completion until the early 1940s. Initially, it housed the primary facilities for the present-day Faculties of Physical Sciences and Chemistry, where courses in Physics and Mathematics were offered. Following the establishment of a dedicated building for the Faculty of Mathematical Sciences, extensive renovations were undertaken on the original structure, culminating in their completion in 2016.

The Faculty of Physical Sciences traces its origins to the Physical Sciences division of the Faculty of Sciences at the University of Madrid, which was established in 1857. This division evolved from the Faculty of Philosophy, itself created through the amalgamation of the Imperial College and the Arts program at the Cisnerian University sited at Alcalá de Henares. Consequently, the Complutense University of Madrid is home to a wealth of historical resources in disciplines like astronomy, some of which date back to the university’s inception or even earlier, such as the Book of Knowledge of Astronomy. Additionally, a significant collection of antique physics teaching instruments can be found on display at the National Museum of Science and Technology, as well as on the first floor of the Faculty itself.

This year the Faculty celebrated its 50th anniversary. See here

For a brief historical account of the Faculty follow this link.

 

 



Organizing Committee

  • Óscar Ciaurri (Universidad de la Rioja, oscar.ciaurri@unirioja.es)
  • Manuel Mañas (Universidad Complutense, manuel.manas@ucm.es)
  • Francisco Marcellán (Universidad Carlos III, pacomarc@ing.uc3m.es)