Five Wallenberg grants to mathematicians at Stockholm University

This year, 17 mathematicians will share SEK 28 million in grants within the framework of the Knut and Alice Wallenberg Foundation's mathematics program. The mathematics program is a long-term investment by the Knut and Alice Wallenberg Foundation (KAW). With this year's grants, a total of 134 researchers have been granted grants since 2014. The program is a collaboration between the foundation and the Royal Swedish Academy of Sciences, which evaluates the candidates nominated by universities around Sweden.

Six researchers receive postdoctoral positions at universities abroad and support for two years after returning to Sweden. Two of these are at the Department of Mathematics, Stockholm University. They are PhD students Erik Lindell and Thomas Blom.

Five researchers receive funding to recruit a researcher from abroad for a postdoctoral position in Sweden. Docent Wushi Goldring at the Department of Mathematics, Stockholm University receives one of these grants.

Six established researchers from abroad are recruited as visiting professors at the Swedish universities. Professor Gianpaolo Scalia Tomba, Università di Roma Tor Vergata in Italy and Professor Stefan Schwede at Universität Bonn in Germany are recruited as visiting professors at Stockholm University.

Erik Lindell: Elusive symmetries in geometric spaces

Erik Lindell will receive his doctoral degree in mathematics from Stockholm University in 2023. Thanks to a grant from the Knut and Alice Wallenberg Foundation, he will hold a postdoctoral position with Professor Nathalie Wahl at the Department of Mathematical Sciences, University of Copenhagen.
 

Photo: Vilhelm Book

Within algebraic topology, tools from algebra are used to investigate abstract geometric spaces, which are important study objects in the branch of mathematics called topology. Eighteenth-century mathematician Leonhard Euler can be regarded as the grandfather of algebraic topology, thanks to his famous formula for the relationship between the number of faces, vertices, and edges of a polyhedron.

Algebraic methods in studies of geometric objects were refined throughout the nineteenth century and culminated in the outstanding work of Frenchman Henri Poincaré. His fundamental conjecture from 1904 says that every closed three-dimensional object without holes is the surface of a four-dimensional ball. Proving Poincaré’s conjecture took almost a century and many new ideas and mathematical tools.

One tool in algebraic topology is the symmetries of topological spaces that preserve some properties of these spaces. Two symmetries can also be repeated sequentially to form a third. The set of symmetries thus has the algebraic structure of a group, leading to rich algebraic theories such as mapping class groups.

The upcoming project is particularly interested in using algebraic methods to investigate low dimensional topological objects such as graphs and surfaces, as well as the internal relationships between the algebraic structures associated with such objects. The study of mapping class groups and homology is fundamental to low dimensional topology. Some symmetries leave the homology unchanged, and this part of the mapping class group is called the Torelli group. So far, little is known about the Torelli groups and new methods are necessary to investigate them more closely. 

Thomas Blom: Better approximations of abstract spaces

Thomas Blom will receive his doctoral degree in mathematics from Stockholm University in 2023. Thanks to a grant from the Knut and Alice Wallenberg Foundation, he will hold a postdoctoral position with Professor Jesper Grodal at the Department of Mathematical Sciences, University of Copenhagen in Denmark.
 

Algebraic topology is a branch of mathematics in which abstract objects called topological spaces are studied using methods from algebra, and has many applications both inside and outside mathematics. For example, algebraic topology is used in modern physics’ string theory and in the theory of general relativity. At the same time, ideas from algebraic topology are utilised in many other areas of mathematics, such as topological data analysis, algebraic geometry and combinatorics.

The planned project primarily deals with the theoretical side of algebraic topology. One part of the project is to further develop a method to approximate a space using simpler spaces. The intention is to advance the understanding of Goodwillie calculus, which is a technique for making such approximations that was developed by the American mathematician Tom Goodwillie in the 1990s. Within the project, Thomas Blom hopes to build upon Goodwillie calculus and further develop it in the context of equivariant homotopy theory, which is the study of spaces with symmetries.

Another part of the project is to apply Goodwillie calculus to the evasion path problem in topological data analysis, where the question is whether, in a room with movement sensors, it is possible to move from one point to another without being detected.  

Wushi Goldring : Building bridges using symmetries

Associate Professor Wushi Goldring will receive funding from the Knut and Alice Wallenberg Foundation to recruit an international researcher for a postdoctoral position at the Department of Mathematics, Stockholm University.

Over the past fifty years, there has been extensive research in mathematics aimed at bringing together two apparently different areas, using a programme by the mathematician Robert Langlands. Its aim is to unite algebraic number theory and the associated algebraic geometry with a specific type of function in mathematical analysis that has special symmetries. Goldring’s project is part of this programme, which is so deep and complex that it is expected to occupy mathematicians for many years to come.

Perhaps the most famous result within this programme has a direct relationship to the equation written about by Pierre de Fermat in 1637: “I have discovered a truly marvellous demonstration of this proposition that this margin is too narrow to contain.”

In 1993, after seven years of hard work, Andrew Wiles shocked the world with a proof of what is generally known as Fermat’s Last Theorem. The proof was based upon a known reduction of Fermat’s Last Theorem to a problem about elliptic curves and some special functions in mathematical analysis. Wiles’ sensational solution to this problem was based on a new method developed jointly with his student Richard Taylor. The "Taylor-Wiles method" reinvigorated the Langlands programme.

However, the myriad of results proved using the Taylor-Wiles method over the past three decades is still only the tip of a gigantic iceberg made up of similar relationships that are expected to apply if Langlands predictions are correct. In accordance with the Langlands programme, the aim of the planned project is to prove that some numbers associated with special analytic functions are actually algebraic, by using symmetries shared by objects that appear to be very different.

Gianpaolo Scalia Tomba: The mathematics of contagion

Gianpaolo Scalia Tomba is a professor at the University of Rome Tor Vergata, Italy. Thanks to a grant from the Knut and Alice Wallenberg Foundation, he will be a visiting professor at the Department of Mathematics, Stockholm University.

The recent global coronavirus pandemic has had a great influence on the statistical analyses and mathematic models used for the spread of contagious diseases. The pandemic has also brought up numerous new research questions about the effects of preventive measures in the spread of infections, such as total or partial lockdowns, remote working, closed schools, vaccinations and other actions. The purpose of this project is to provide a deeper mathematical understanding of some of these issues. Apart from purely scientific interest and a better analysis of what has already happened, mathematical studies of the pandemic also provide increased preparedness for future outbreaks of infectious diseases.

One important quantity in models for the spread of diseases is the generation time, which describes the typical time between being infected and infecting others. The difficulty is that generation time varies; it is usually shorter at the start of an epidemic and is affected by the efforts made to stop the spread of infection – for example, if sick people are isolated, the generation time is shortened. The planned project includes finding methods for improved estimation of the generation time.

A more complex issue is how the development of the disease is affected by new variants of the virus that fully or partially replace the old one, often overturning previous predictions. Whether a new virus variant will take over from an old one may depend on how contagious the old and the new viruses are, whether the two variants provide full or partial immunity, and how effectively a vaccine can slow the spread of each variant. In the theoretical analysis, the two virus variants are assumed to have different properties but have the same potential to take over.

Stefan Schwede: A global perspective on symmetries

Stefan Schwede is a professor at the University of Bonn in Germany. Thanks to a grant from the Knut and Alice Wallenberg Foundation, he will be a visiting professor at the Department of Mathematics, Stockholm University.
 

Foto: Universität Bonn

Symmetries and regular patterns are everywhere around us – from the symmetry of a human face to the repeated cycle of the seasons, the intricate structure of crystals and the deep symmetry of the fundamental laws of physics. Symmetry also plays a central role in mathematics. One of mathematics’ main tasks is to discover symmetry and extract information from it.  

Mathematics encodes the concept of symmetries in the notion of a group. Roughly speaking, a group in mathematics means the collection of symmetries that an object may possess. For example, the hexagonal symmetry of a snowflake will form a group with six elements.

This project advances a global approach to the study of symmetries. The philosophy here is that one does not just focus on the symmetries of one object at a time, or on one group at a time, but considers simultaneously all possible groups, and all possible types of symmetries, as parts of one big picture. The intent is to extract information about the symmetries of an individual object by studying its interactions with many other objects.

The global approach entails a new way of looking at symmetrical objects and has already led to dramatic progress in our understanding of the relationships between geometric shapes, which is also the subject of this research.  

More information on the Knut and Alice Wallenberg Foundation web.