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- Symmetry and harmony in focus for new mathematics research

# Symmetry and harmony in focus for new mathematics research

Modular forms, degenerate elliptic operators and broken symmetries. Three research projects at Mathematical Sciences receive funding when the Knut and Alice Wallenberg Foundation's mathematics programme distributes SEK 25 million to fifteen mathematicians. That should be enough to recruit three researchers from abroad for postdoctoral positions in Sweden. In addition, two recently defended scholarships receive a postdoctoral position abroad.

The mathematics programme has been instrumental in mathematics research in Sweden. Strong environments that have achieved international renown have evolved and attract leading researchers from around the globe.

“Advanced mathematics skills are an important foundation for many other sciences, as well as for the development of new technology. This is particularly visible in the rapid development of areas such as data-driven life science, AI and quantum engineering. If Sweden is to keep up with these developments, we must have mathematicians who are at the forefront,” says Peter Wallenberg Jr, chair of Knut and Alice Wallenberg Foundation.

Read the full press release from Knut and Alice Wallenberg Foundation here.

The researchers from Mathematical Sciences who received funding present their projects below.

## New theory when physics meets mathematics

### Associate professor Martin Raum

Physics has provided inspiration for development in mathematics ever since the seventeenth-century origins of mathematical analysis. This continues to the present day, with physics theories of everything, for example string theories, that will describe all known matter and all the forces of nature in one coherent theory. Modular forms play a vital role in the mathematics of string theories, and are also the basis of my planned project.

Modular forms are mathematical functions that fulfill particular symmetry conditions. Since their discovery, two hundred years ago, they have been vitally important to the development of number theory and other branches of mathematics. There are now many different modular forms and their generalisations. Finding a new class of modular forms is part of the planned project.

Modular forms also aid the understanding of many phenomena that involve geometric objects, including the small strings that are the foundation of string theory. When these strings collide, they can merge and form new strings. The probability of this is described by string amplitudes, which have shown to fit surprisingly well into an arithmetical framework that is closely related to motivic periods. One objective in this project is to use contributions from string theory in building a broader theory, one that will describe the connection between different motivic periods in families of geometric objects.

## Degenerate elliptic operators with broad applications

### Professor Andreas Rosén

Harmonic analysis is a branch of mathematics developed in the early nineteenth century, by French mathematician Joseph Fourier as part of his study of heat conduction. To arrive at a solution, he expressed general functions as infinite series of harmonic oscillations, the Fourier series. The planned project focuses on the development of harmonic analysis techniques, which will be used to study materials where the ability to conduct heat varies much in different directions.

Alongside the Fourier series, there are now many other ways of developing functions as infinite series of a set of basic functions. Wavelets, for example, revolutionised harmonic analysis when they were introduced in the 1980s. The impetus behind their development largely came from their applications – wavelets are now widely used to compress, store and recreate data in electronic engineering, image processing and signal processing.

There was a new breakthrough in the early twenty-first century, when harmonic analysis was further developed in close relation to, but beyond, wavelets. Andreas Rosén is an expert in these new methods and has applied them to elliptic partial differential equations. The planned study examines equations in which thermal conductivity is permitted to be infinite or zero at some points, and where knowledge is still incomplete. How much can the equation coefficients vary before the theory falls apart? And what happens in cases where thermal conductivity is both infinite and zero at the same point, but in different directions there?

## Hunting for broken symmetries

### Professor Genkai Zhang

The planned project focuses on representation theory, which is a study of abstract algebraic and geometric structures by representing them by simpler, more familiar linear structures in analytic setup. The ideas and methods are not only used in mathematics, but also in other areas such as theoretical physics. Among other things, representation theory is very instrumental in the Standard Model of particle physics.

Here, the key concept is symmetry. In our everyday lives, we usually think about symmetry as a mirror image, but there are many other symmetries that are less obvious and yet very vital to the advancement of mathematics. One can think of the situation when two children are playing at seesaw: The two kinds of diagonal and up-down symmetries can be better unified in a bigger symmetry group, whereas the symmetry is broken if we consider only one of them. The subject of this project comprises the rules for symmetry breaking (called branching rules by mathematicians) in representation theory.

Symmetries can be hard to discover. One way of finding them is to study their effects, i.e. their representations on spaces of states that are manifestations of symmetries. Such a study can be more manageable if the representation is restricted to a smaller group of symmetries; the rules for these restrictions are the branching rules. Within the project, researchers will look for and study branching rules for certain restricted representations using methods from analysis and geometry.

## Funding for international postdoctoral position

In addition, Antonio Trusiani and Milo Viviani, who both graduated with PhDs from Mathematical Sciences in 2020, have been awarded grants for postdoctoral positions at the Institut de Mathématiques de Toulouse, France, and the Scuola Normale Superiore, Pisa, Italy, respectively.