Classifying Curves

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

Classifying and systematizing properties of mathematical objects can provide new insights about the objects and their nature. Similar projects have been underway since the birth of algebraic geometry in the 17th century. Basically, it deals with the study of geometric objects that correspond to solutions of polynomial equations. For example, solutions of first-degree equations in two variables form a line, whereas solutions to certain second-degree equations form a circle.

It soon became apparent that the task of classification is extensive, so many new mathematical tools have been developed over the years. About fifty years ago, an American mathematician, David Mumford, constructed the moduli spaces that are now studied broadly in algebraic geometry. They have become an important research tool for other fields in mathematics, such as number theory and algebraic topology, as well as for theoretical physics.

A moduli space can be thought of as a map. On an ordinary map, a point may correspond to a geographical location; similarly, points in a moduli space correspond to geometric objects, for example solutions to a polynomial equation. A moduli space is thus a geometric solution to the problem of classifying curves and other geometric objects.

Moduli spaces have a very complicated structure and many questions about their properties are still unanswered. In the planned project, moduli spaces of curves will be studied by the postdoctoral researcher who will join the algebraic geometry group at Stockholm University.