Professor Douglas Arnold receives 2025 Frontiers of Science Award

MINNEAPOLIS / ST. PAUL (2/28/2025) – The International Congress of Basic Science recently selected a paper by McKnight Presidential Professor Douglas Arnold and collaborator Kaibo Hu for the 2025 Frontiers of Science Award, recognizing it as a major breakthrough in the field of numerical analysis. The paper – titled “Complexes from Complexes” – was published in Foundations of Computational Mathematics in May 2020.

Many of the fundamental laws of physics are expressed as systems of partial differential equations (PDE), the properties of which have crucial physical implications. These systems of PDE are often best understood via an associated hybrid analytic/algebraic object called a differential complex, consisting of certain spaces of functions and certain differential operators connecting them. The best known example is the de Rham complex, which is intimately related to Maxwell's equations of electromagnetism and has been intensely studied. However, other physical systems, such as fluid flow, elastic deformation of solids, heat transfer, and gravitation require different differential complexes which must be constructed and whose properties must be established. In their award-winning paper, Arnold and Hu introduce a systematic algebraic procedure, inspired by the Bernstein-Gelfand-Gelfand resolution from the representation of Lie algebras, which takes two or more complexes that are assumed to have already been analyzed (such as the de Rham complex) and constructs from them a new complex which is quite distinct from the old ones. Moreover, they show how to obtain the essential properties of the new complex from the corresponding properties of the starting complexes. The paper's approach to construction of new complexes from old ones applies not only at the continuous level, but also at the discrete level, in which the function spaces and differential operators are approximated by finite dimensional spaces and operators acting on them. In this way, the construction and analysis presented in “Complexes from Complexes” can be used to guide the construction of stable, convergent numerical methods, in some cases for the first time. This is currently a very active area of research internationally.

Prof. Arnold joined the University of Minnesota in 2001 as a faculty member in the School of Mathematics and as director of the Institute for Mathematics and its Applications, a post he held until 2008. In the following year he was named a McKnight Presidential Professor, awarded a Guggenheim Fellowship, and elected a foreign member of the Norwegian Academy of Arts and Letters. In 2009 and 2010, he went on to serve as the president of SIAM. Throughout his career, Arnold has been involved in editorial work and in issues surrounding scholarly publication, and has served on the editorial boards of many of the top journals in numerical analysis. Since 2021, he has served as the editor-in-chief of the journal Acta Numerica. A number of the numerical algorithms devised by Arnold have become widely-used tools in areas such as structural engineering, fluid mechanics, and climate modeling. In 2023, he was awarded the Peter Henrici Prize in recognition of his significant contributions to the field of numerical analysis. Arnold is possibly best known as the creator of the Finite Element Exterior Calculus, a new approach to the numerical solution of PDE, which he initiated in 2002 in a plenary address at the International Congress of Mathematicians. Another current research interest of Arnold’s is the behavior of waves in disordered media. He pursues this work through the interdisciplinary Simons Collaboration on Localization of Waves, for which he is one of the Principal Investigators.

The Frontiers of Science Award (FSA) was inaugurated in 2023 with support from the International Congress of Basic Sciences (ICBS), with the sponsorship from the City of Beijing and the Yanqi Lake Beijing Institute of Mathematical Sciences and Application. It is awarded to a recent paper in mathematics, theoretical physics, or theoretical computer science recognized for a major breakthrough in its field. FSA recipients are invited to the ICBS to accept the award in person in July 2025 in Beijing. This is the second consecutive year in which a School of Mathematics faculty member has been awarded an FSA.