Recognising the correct variety

Rita Pardini is a professor at the University of Pisa in Italy. Thanks to a grant from Knut and Alice Wallenberg Foundation, she will be a visiting professor at the Department of Mathematics, Stockholm University.

Geometric objects derived from polynomial equations are studied in algebraic geometry, which is the subject of the planned project. These objects are called algebraic varieties and come in two main types – projective and quasi-projective.

The most studied projective varieties include abelian varieties; these also have the most applications in our everyday lives. For example, modern cryptography is based on one-dimensional Abelian varieties (also known as elliptic curves); these underly the open internet protocols SSH, PGP and TLS, which guarantee the security of all data communication. Transactions with Bitcoin and other cryptocurrencies are secured using cryptography based on elliptic curves.

The quasi-projective varieties are much less explored, and the project’s aim is to expand some known methods for classifying projective varieties to the quasi-projective case. A previous paper co-authored by the invited professor Pardini examined which quasi-projective varieties also fulfil the requirements for inclusion in another category, semi-abelian: If a quasi-projective variety dresses, looks and behaves like a semi-abelian variety, is it a semi-abelian variety? In other words, what are the key determining properties of the quasi-projective varieties? Previously, there were no methods for answering this and similar questions about quasi-projective varieties, and completely new approaches are needed to deal with them.

Photo: Barbara Agostini