The objects of study of this project are geometric objects described by polynomial equations, for instance a sphere or an ellipse, and they are called algebraic varieties.
A broad objective of Algebraic Geometry is to classify algebraic varieties up to isomorphisms. This is, however, much too hard, already for surfaces, and one therefore aims to classify them up to so-called birational maps, which are isomorphisms between Zariski-open dense subsets of algebraic varieties. One can view birational maps as an analogue of symmetries, only that one is allowed to remove points or curves.
The Minimal Model Program is a way to simplify algebraic varieties with birational maps, but it remains to classify the outputs up to birational maps between themselves.
Certain types of output varieties have very few birational maps, like varieties of general type, and other types may have plenty of birational maps, like Mori fibre spaces. This project studies birational maps between Mori fibre space, and in general they can be very wild. The Sarkisov program states that any birational map between Mori fibre spaces is a composition of so-called Sarkisov links; these are chunks of birational maps that seem controllable. They are listed in dimension two and in the case of toric threefolds. The aim of the project is to list them reasonably in any dimension.
A second objective of the project is to study birational maps between real algebraic varieties that extend continuously at every real point; they are called biregulous maps. They are hard to study and currently not much is known about the group of biregulous maps of the plane, and even less is known in higher dimension. The aim of the project is to describe properties of the group of biregoulous maps of the projective space of dimension two and higher.
