Classifying von Neumann algebras—the mathematical language of quantum mechanics—is one of the great unsolved problems in mathematics. From 18 to 22 May 2026, ICMAT is bringing together fifty international specialists to explore a promising strategy: using harmonic analysis to ‘listen’ for differences that other methods have failed to detect.
Ágata Timón (ICMAT-CSIC)
During the week of 18–22 May, ICMAT is hosting the conference ‘Harmonic analysis and von Neumann algebras’, which will bring together around fifty specialists from various national and international academic institutions. The event, focusing on the intersection of these two cutting-edge areas of mathematics, is part of the ICMAT’s Severo Ochoa “Giles Pisier-Mikael de la Salle” Laboratory.
On the one hand, von Neumann algebras are “the mathematical language that describes systems with a great many variables and symmetries”, according to Javier Parcet, research professor at the Spanish National Research Council (CSIC) at ICMAT and co-organiser of the event. “Especially in quantum mechanics, where quantum observables are self-adjoint operators in a Hilbert space.” For decades, mathematicians have been working to classify these objects, which also appear when studying random processes, complex dynamical systems or the geometric properties of non-Abelian groups. “The idea of classifying them is similar to that of ordering the chemical elements in a periodic table: it is not enough to know that there are many different structures; we want to understand which ones are essentially the same, which have unique properties, and what fundamental types there are,” explains Parcet, who has been invited as a speaker to the 2026 International Congress of Mathematicians (ICM) in Philadelphia to discuss, precisely, his findings on this topic.
This classification involves not only ordering abstract objects, but also gaining a better understanding of the mathematical architecture of complex systems. “The classification has led to the discovery of different ‘levels of infinity’ and unexpected behaviours in mathematics and physics. Furthermore, it connects very different areas—such as geometry, probability, analysis, quantum computing and number theory—showing that they share deep underlying structures,” explains Parcet.
In recent years, von Neumann algebras have also proved to have a very fruitful relationship—which is the focus of the conference—with another area of mathematics: harmonic analysis. “Harmonic analysis studies how to decompose packets of information—sounds and any other ‘signal’ expressed as a function—into simpler components (waves), traditionally in Euclidean spaces or their compact quotients. Von Neumann algebras provide the language to describe these functions in more complex contexts, where the product of two functions does not commute, that is, AxB is distinct from BxA,” says Parcet.
“The model is non-Abelian groups, mathematical structures where order matters (such as in the rotations and symmetries of an object in space or the permutations of a set),” he says. “There, classical Fourier analysis is insufficient and the information packets are decomposed using operators on Hilbert spaces (representations of the group),” he adds. Once again, von Neumann algebras organise the “deep structure of non-Abelian geometries”, summarises Parcet. “And harmonic analysis attempts to ‘listen’ to these geometries through decompositions, which generalise the Fourier series in classical harmonic analysis. This bridge is important in theoretical physics and various fields of modern mathematics”, he concludes.
Can the classification of von Neumann algebras be ‘heard’?
In 2011, a major paper by Vincent Lafforgue and Mikael de la Salle (also an organiser of the event) laid the foundations for a new strategy, using Fourier analysis, to tackle one of the big questions about von Neumann algebras: the rigidity conjecture. “Rigidity is a phenomenon that dates back to group theory. In von Neumann algebras associated with groups, the fundamental question is ‘how much of the original group’s structure is retained by the associated algebra?’ and, in that context, rigidity can be defined as the amount of information retained,” explains Parcet.
In 1982, Alain Connes was awarded the Fields Medal for demonstrating, amongst other things, that certain groups, known as discrete amenable groups (groups that do not admit paradoxical Banach-Tarski-type decompositions), give rise to essentially indistinguishable von Neumann algebras. “Conversely, there are ‘superrigid’ groups, whose von Neumann algebras recall their complete structure. Thus, von Neumann algebras are fascinating structures ranging from very flexible to completely rigid,” he explains.In 1980, Connes wondered whether the von Neumann algebras of the group SLn(Z) (which are n × n matrices with integer entries and a determinant of 1) are not isomorphic for different values of n. “A somewhat more general version of this statement is known as Connes’s rigidity conjecture, which today represents one of the most significant challenges in operator algebras and the geometric theory of groups,” explains Parcet.
“The work of Sorin Popa and many other mathematicians so far in the 21st century has marked a second phase of development in the field, but their methods (which are predominantly ergodic) have not yet succeeded in resolving the problem,” notes Parcet. For this reason, the radically different perspective introduced by Haagerup and refined in its Lp form by Lafforgue and La Salle in 2011 is, today, a promising direction. “What we are trying to do is ‘listen’ to the differences between such von Neumann algebras through the Lp approximation properties of their Fourier series,” says Parcet.
“In recent years we have found several pieces of evidence suggesting the possible existence of invariants related to non-commutative forms of Bochner-Riesz and Kakeya-type inequalities, on which we are actively working. In addition, we remain very interested in aspects of harmonic analysis on free groups, non-commutative tori and other von Neumann algebras. That is why it is tremendously interesting to organise this conference, to bring both communities together,” concludes Parcet.
Translation provided by an automatic translation tool.