Towards 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”.