Vy över Stockholms universitet, Foto: Clément Morin/Stockholms universitet
Vy över Stockholms universitet, Foto: Clément Morin/Stockholms universitet

 

The overall subject of the project is the role that group theory plays in mathematics and, more specifically, determining the extent to which groups give rise to geometry. The main idea is to show that many more geometric objects are generated by groups than was previously believed.

Group theory has its roots in the early 1800’s and, ever since, groups have been omnipresent in literally all branches of mathematics. Their applications have also spread to other sciences such as physics, chemistry and biology. Several breakthroughs made in the late 1800’s paved the way for the 20th century’s mathematics (and physics). They demonstrated the unprecedented potential of group theory for organizing, classifying, and uniting mathematics. Was there anything left to say about the role of group theory in mathematics?

The fact is that the seeds of a completely different role for group theory in mathematics were already hinted at by the subject’s founding father: Évariste Galois. In his théorie de l'ambiguïté from the 1830’s (ambiguity theory, which later became known as Galois theory), he showed that groups were the source of much mathematics that had previously been studied without groups. In other words, groups also generate geometric objects.

“I am very happy and honored to receive this grant, and I am grateful to the Knut and Alice Wallenberg Foundation for its support. Since the proposal covers a very broad area of mathematics, to make progress on it requires technical knowledge in several different fields and understanding how these different fields interact with one another”, says Associate Professor Wushi Goldring.

During the second half of the last century, the groundbreaking programs of Alexandre Grothendieck and Robert Langlands and the work of Pierre Deligne and others have illustrated striking examples of objects -- originating from different areas -- generated by groups. However, a general theory which explains which objects are generated by groups -- and why -- is still lacking. It remains a mystery whether the examples of Deligne, Grothendieck and Langlands are restricted to a very special phenomenon, or whether they are prototypes for all geometric objects. It is this mystery that the proposal aims to solve.

“The grant will be used to a hire a postdoc who will have an expertise in an area where there is particular potential to discover new examples of geometry-generated-by-groups. By working with the postdoc, we will gain a better understanding of the different areas where geometry-by-groups is prevalent”, says Wushi Goldring.