Research project Unknown properties of abstract geometric objects

The most common and easy example of variety is perhaps that of  the circle of radius r, the geometric figure which can be described with the equation X²+Y²=r² in the XY plane. However there are varieties that are much more complicated, they might be described by hundreds of equations in as many variables. It is not possible to draw them and to investigate their property directly, as one would do for the properties of the circle, is almost unthinkable.

The main goal of this project is to develop new techniques to study the geometry of varieties, giving special attention to the so called positive-characteristic algebraic varieties. These  are typically zero loci of polynomials with coefficients in finite number systems. For example, on can consider the set of  the three possible remainders that can be obtained when dividing a whole number by 3. On this, one can define two operations which follow similar arithmetic rules to those of the usual addition and the usual multiplication. Once we have these two operations we can define polynomials with coefficients in this set. And if there are polynomials we have polynomial equations and hence algebraic varieties!

Now, to any algebraic variety, one can associate its derived category. This is a tool introduced by Verdier in the 60s which employs many advances in the research on varieties defined over the complex numbers. Today derived categories have also broader applications. For example they have brought many contributions to study the theory of strings and quantum physics. However their use to study the geometry  of positive-characteristic varieties is still widely unexplored. With the present project we aim to fix this gap.