Unknown properties of abstract geometric objects

Associate Professor Sofia Tirabassi will receive funding from Knut and Alice Wallenberg Foundation to recruit an international researcher for a postdoctoral position at the Department of Mathematics, Stockholm University.

The aim of Sofia Tirabassi’s project is to develop new methods for studying algebraic geometry, a branch of mathematics that deals with the relationship between polynomial equations and their associated geometric objects – varieties. A simple variety is a circle with radius r, the geometric figure that is described by the equation x2 +y2 = r2. However, most varieties are much more complicated and are described by hundreds of equations and variables. Direct investigation of their geometric properties is almost inconceivable.

The main goal of the project is to develop new techniques for studying the geometry of varieties, with a special focus on algebraic varieties in positive characteristics. These are usually solutions to polynomial equations with coefficients in finite number systems. For example, a finite number system could be the set of the three possible remainders (0,1,2) that are left when an integer is divided by 3. For this set, two operations can be defined that meet arithmetic rules similar to those for ordinary addition and multiplication. The two operations, along with the coefficients, can then define a polynomial in this set. If there are polynomials, there are also polynomial equations and thus algebraic varieties.

To every algebraic variety one can associate its derived category. This is a tool that was introduced by the French mathematician Jean-Louis Verdier in the 1960s, which has led to many advances in research into varieties over the complex numbers. Derived categories have since become indispensable, even outside algebraic geometry – for example, they are used to study string theory and quantum physics. However, their role in the study of the geometry of varieties in positive characteristics is still largely unexplored. The goal of this project is to fill that gap.

Photo: Solfrid T. Langeland/UiB