Arranging and sorting algebraic structures

Lisa Nicklasson will receive her doctoral degree in mathematics from Stockholm University in 2020. Thanks to a grant from Knut and Alice Wallenberg Foundation, she has been given a postdoctoral position with Professor Aldo Conca, Università di Genova, Italy.

Modern abstract algebra developed at the beginning of the last century, thanks to efforts to create a more general theory of mathematics. Algebra was thus reframed, moving from a theory of equations to a theory of algebraic structures. This theory is now so comprehensive that it has branched into many different areas.

Lisa Nicklasson’s project includes multiple studies of objects called graded algebras. A graded algebra is a structure with elements that are sorted by degree. For example, x can be assigned degree 1, x2 degree 2, and so on. This algebra can then be divided into different spaces, where elements of the same degree are in the same space. The size of these spaces forms a number series, called the Hilbert series. The Hilbert series can be used as a tool to measure the size and growth of a graded algebra.

One of the questions in this project is to discover what structure a certain type of algebra should have to make the numbers in the Hilbert series as small as possible. Another direction is to introduce properties in the algebra that limit its structure, rather than limiting the size of the algebra. One example of such a restriction is the Koszul property, which was discovered in the 1970s. Since then, several sub-classes of Koszul algebras have been discovered.

Trying to describe exactly how the various classes of Koszul algebras relate to each other is also part of the project. An important first step in answering the various questions in the area being studied is the use of computer calculations to investigate multiple interesting examples of graded algebras.

 Photo: Stefan Lagerstam