Wallenberg grants to six mathematicians
Six of the eighteen grants awarded to prominent mathematicians within the Knut and Alice Wallenberg Foundation's mathematics program go to Stockholm University
The mathematics program is a long-term investment by the Knut and Alice Wallenberg Foundation. During the years 2014–2029, SEK 650 million will be allocated to Swedish mathematical research. Every year, both younger and more senior mathematicians are recruited to Sweden. At the same time, young Swedish mathematicians will have the opportunity to travel the world on a postdoctoral position. Including this round, 152 researchers have received grants within the program. Swedish universities with a department of mathematics have the opportunity to nominate candidates for the program. These are then evaluated by the Royal Swedish Academy of Sciences.
In this year's round, support is granted to 18 mathematicians. Of these, six go to Stockholm University. (Read articles about all six of these researchers below.)
International postdoctoral positions
Four out of six researchers who are granted postdoctoral positions at universities abroad and support for two years after returning to Sweden are at Stockholm University. These are the PhD students:
Josefien Kuijper, for service at the University of Toronto in Canada
Stefan Reppen, for service at the University of California, Berkeley in the United States
Robin Stoll, for service at the University of Cambridge in the United Kingdom
Louis Hainaut, for service at the University of Chicago in the United States.
Foreign visiting researchers
Six established researchers from abroad are recruited as visiting professors at Swedish universities. Two of these are recruited to Stockholm University. They are:
Professor Rita Pardini, Universitá di Pisa in Italy (applied for by Professor Sofia Tirabassi)
Professor Mario Wüthrich, ETH Zürich in Switzerland (applied for by Professor Mathias Millberg Lindholm).
In addition, five researchers have been awarded funding to recruit a researcher from abroad for a postdoctoral position in Sweden.
Read about all 18 researchers and their research
Understanding geometry through elementary building blocks
Josefien Kuijper will receive her doctoral degree in mathematics from Stockholm University in 2024. Thanks to a grant from Knut and Alice Wallenberg Foundation, she will hold a postdoctoral position with Professor Elden Elmanto at the University of Toronto.
Algebraic geometry is the study of varieties, which are geometric shapes that can be described by algebraic equations. A simple variety is a circle with the radius r, which is described by the equation x2+y2=r2. An important tool for studying algebraic varieties are cohomology theories; instruments that turn algebraic varieties into simpler objects, such as groups or vector spaces. They are essential for understanding the algebraic and geometric information that is contained in algebraic varieties.
The circle is an example of an algebraic variety that is both smooth and complete, meaning that the geometric shape does not cross itself or show other singular behavior; and that it is closed and bounded. Smooth and complete varieties are better understood than arbitrary ones. Luckily, methods such as compactification and desingularization can be used to relate an arbitrary variety to varieties that are smooth and complete.
In the proposed project, Josefien Kuijper will use these principles, of “building arbitrary varieties from smooth and complete ones”, to understand cohomology theories better, and construct new ones. When is a cohomology theory uniquely determined by its behavior on varieties that are smooth and complete? What is the minimum amount of data needed to pin down a cohomology theory?
These questions are inspired by the work of Grothendieck and Deligne in the previous century. But modern tools, such as the language of infinity-categories, have already given rise to new perspectives and exciting results. Another aim of the project is to investigate how these new tools, combined with old principles, shed light on other concepts that date back to Grothendieck: the algebraic K-theory of the category of varieties, and six-functor formalisms.
Towards a mathematical theory of everything
Stefan Reppen will receive his doctoral degree in mathematics from Stockholm University in 2024. Thanks to a grant from Knut and Alice Wallenberg Foundation, he will hold a postdoctoral position with professor Sug Woo Shin at the University of California, Berkeley.
Many complex advances in mathematical research stem from the desire to solve an equation. However, it is often impossible to do this by working directly with the equation itself, and instead solutions are linked to abstract geometric objects, algebraic varieties. In turn, these are used as building blocks for even more sophisticated geometric objects called Shimura varieties.
The Shimura varieties are families of algebraic varieties; they have become the focus of modern number theory due to their unexpected and crucial role in proving number theory’s most famous problem – Fermat’s Last Theorem, which dates from the 17th century. The French mathematician claimed to have a proof that the equation xn + yn = zn has no solutions for any integer value of n greater than 2. The real proof came first in 1995 and, in addition to Shimura varieties, modular forms were also used to prove the almost 400-year-old theorem.
Modular forms, which are a kind of functions on Shimura varieties, are used in this project to explore specific aspects of the varieties’ fundamental geometric structure. Ultimately, these studies contribute to finding deep connections between geometry and number theory, which is the aim of one of the largest initiatives in modern mathematics – the Langlands Program – sometimes called “the mathematical theory of everything”.
Graph complexes – a useful tool in algebraic topology
Robin Stoll will receive his doctoral degree in mathematics from Stockholm University in 2024. Thanks to a grant from Knut and Alice Wallenberg Foundation, he will hold a postdoctoral position with professor Oscar Randal-Williams at the University of Cambridge.
Algebraic topology uses tools from algebra to investigate abstract geometric objects, such as a plane, a sphere or a torus (i.e. the surface of a donut). All three of these two-dimensional examples are so-called manifolds, which, together with analogues in higher dimensions, are the central objects of study in topology.
Over the past few centuries, the study of manifolds has played a vital role in various areas of mathematics and theoretical physics. However, many of their properties remain unknown, and the aim of this postdoctoral project is to explore new methods for understanding certain abstract manifolds with an odd number of dimensions.
The question of classifying manifolds and their symmetries has driven the development of algebraic topology throughout the twentieth century. Two manifolds are said to belong to the same class if they can be deformed into each other without any tearing or glueing. Understanding the collection of all manifolds belonging to one such class is of particular importance. However, this problem has proved extremely difficult to tackle.
In this subject, it is important whether the dimension is even or odd. For manifolds in even dimensions, much progress was made over the last decade. The project will explore a new approach to investigate the odd dimensional case, using certain combinatorial structures called graph complexes.
Comparing transformations of topological spaces
Louis Hainaut will receive his doctoral degree in mathematics from Stockholm University in 2024. Thanks to a grant from Knut and Alice Wallenberg Foundation, he will hold a postdoctoral position with professor Benson Farb at the University of Chicago.
Algebraic topology uses algebra to measure topological properties and has proven to be effective in classifying topological objects by topological invariants. Invariants are quantities that do not change when topological objects undergo continuous deformation without breaking, such as stretching or twisting. Topological objects are the spaces where geometry takes place and are blind to quantitative geometry like angles, lengths, areas, and volumes.
A basic topological invariant is homology. The idea of homology was developed for the analysis of the holes in geometric objects. It originates from Euler's polyhedron formula, which was named after the Swiss mathematician Leonhard Euler. In the mid-eighteenth century, he proved that the sum of the number of vertices and sides of a polyhedron, such as a cube or a tetrahedron, is always equal to the number of edges plus two.
The concept of homology developed over the next few centuries, eventually proving to have extremely broad applications in mathematics. Of particular interest for the planned project are configuration spaces and studies of the relationship between transformations of topological spaces to themselves and transformations of their related homology groups.
Recognising the correct variety
Rita Pardini is a professor at the University of Pisa in Italy. Thanks to a grant from Knut and Alice Wallenberg Foundation, she will be a visiting professor at the Department of Mathematics, Stockholm University.
Geometric objects derived from polynomial equations are studied in algebraic geometry, which is the subject of the planned project. These objects are called algebraic varieties and come in two main types – projective and quasi-projective.
The most studied projective varieties include abelian varieties; these also have the most applications in our everyday lives. For example, modern cryptography is based on one-dimensional Abelian varieties (also known as elliptic curves); these underly the open internet protocols SSH, PGP and TLS, which guarantee the security of all data communication. Transactions with Bitcoin and other cryptocurrencies are secured using cryptography based on elliptic curves.
The quasi-projective varieties are much less explored, and the project’s aim is to expand some known methods for classifying projective varieties to the quasi-projective case. A previous paper co-authored by the invited professor Pardini examined which quasi-projective varieties also fulfil the requirements for inclusion in another category, semi-abelian: If a quasi-projective variety dresses, looks and behaves like a semi-abelian variety, is it a semi-abelian variety? In other words, what are the key determining properties of the quasi-projective varieties? Previously, there were no methods for answering this and similar questions about quasi-projective varieties, and completely new approaches are needed to deal with them.
Fairness and discrimination in insurance pricing
Mario Wüthrich is a professor at ETH Zürich in Switzerland. Thanks to a grant from Knut and Alice Wallenberg Foundation, he will be a visiting professor at the Department of Mathematics, Stockholm University.
The project is about using mathematical methods to address issues surrounding fairness and discrimination in the insurance industry. How should these issues be handled when it comes to pricing of insurance?
There are already laws that prohibit companies from using protected attributes, such as gender, when pricing insurance. However, simply ignoring the protected attributes is not enough, as they can still be inferred through a variety of non-protected variables that are now collected in large quantities, so called big data. This inference is called indirect discrimination. Issues of discrimination, both direct and indirect, are already regulated by law, but definitions in the legal texts are not mathematically rigorous and leave room for interpretation.
In practice, insurance prices will rarely be completely discriminatory or completely avoid discrimination, and an important part of this research project is the development of methods to reliably measure the degree of fairness and discrimination in insurance pricing. This is of particular interest when it comes to e.g. identifying disadvantaged groups of individuals.
Another part of the project deals with the fact that the insurance business is conducted in a competitive market, which means that mathematical risk-based insurance premiums are adjusted for business considerations. This type of adjustment opens up new mathematical questions about fairness and avoiding discrimination.
Last updated: April 9, 2024
Source: Communications Office