Exploring the theory of infinitely flexible tools

Anna Rodriguez Rasmussen will receive her doctoral degree in mathematics from Uppsala University in 2026. Thanks to a grant from the Knut and Alice Wallenberg Foundation, she will hold a postdoctoral position with Professor Sibylle Schroll at the University of Cologne, Germany. 

Representation theory concerns the understanding of abstract algebraic structures by describing them with more familiar tools from linear algebra, such as matrices and linear transformations. New properties of the objects can be revealed and, in many cases, calculations simplified through translating abstract problems to a more concrete context. The most fundamental objects dealt with in representation theory include algebras, meaning sets of objects equipped with binary operations such as addition and multiplication. The usual real numbers are a classic example of such an algebra. 

However, the algebras to be studied in the planned project are of a more sophisticated type, known as A_∞-algebras. Unlike a normal algebra, an A_∞-algebra also has successive corrections in addition to the basic operations, and the infinity symbol in the name indicates that the number of such corrections can, in principle, be unlimited.

This rich structure is what makes A_∞-algebras particularly powerful. They can reveal subtle connections between algebra and geometry that are otherwise difficult to discover. These methods have also gained importance far beyond their original context and have contributed to the formulation of stricter mathematical foundations for quantum theory and modern string theory.

In recent decades, A_∞-theory has developed towards a much broader scope. After initially being regarded as a technical aid in geometry and topology, it has also proven to be a versatile tool for purely algebraic and representation theoretic problems. These new applications for A_∞-structures are at the heart of this project.

Photo: Anna Rodriguez RasmussenUnknown