Unexpected relationships revealed by abstract algebra

Dr Gustavo Jasso will receive funding from Knut and Alice Wallenberg Foundation to recruit an international researcher for a postdoctoral position at the Centre for Mathematical Sciences, Lund University.

The project concerns algebraic structures in representation theory, which is a way of describing abstract algebraic objects using other, more familiar, objects from linear algebra. The aim is to recognise new properties and sometimes simplify calculations of more abstract algebraic structures. 

The project studies objects called A∞-algebras. These were discovered in the 1960s, during studies of the equations that arise in families of geometric operations of higher and higher dimensions.  The A∞-algebras reveal subtle algebraic and geometric relationships that are otherwise shrouded in mystery.

Interesting examples of A∞-algebras arise in geometry from the study of Grassmannians. A Grassmannian can be compared to the cartesian coordinate system, where each point on a plane is determined by its two coordinates, x and y. Grassmannians have their own coordinate systems, where the coordinate functions fulfil equations that were discovered by German mathematician Julius Plücker in the nineteenth century. 

Almost two hundred years later, in the early 2000s, new algebraic structures, cluster algebras, were introduced. They provide a new perspective on Grassmannians’ coordinate systems. The objective of this project is to investigate what the algebraic relationships provided by A∞-algebras say about the cluster algebras associated with Grassmannians.