New discrete forms for continuous objects

Gabriele Balletti received his doctoral degree in mathematics from Stockholm University in 2018. Thanks to a grant from the Knut and Alice Wallenberg Foundation, he will hold a postdoctoral position with ProfessorFrancisco Santos at Freie Universität of Berlin, Germany.

The aim of the project is to obtain new insights about continuous objects, such as curves and surfaces, using methods from discrete mathematics. This means that continuous objects need to be described as composed of distinct elements which, in turn, make the questions suitable for computer processing.

Toric varieties are a special kind of surfaces studied in mathematics. They have a multitude of applications in areas such as theoretical physics, coding theory, algebraic statistics and geometric modeling, and they define a meaningful testing ground for general theories.

Although toric varieties are continuous objects, they can be associated with discrete objects that carry all the information about them. These discrete objects are geometrical shapes called polytopes, which are multidimensional relatives of the more familiar polygons, for example two-dimensional triangles and squares, or three-dimensional pyramids and cubes.

A polytope is like an ID card for the toric variety it represents: lots of data that is not directly accessible by simply looking at the toric variety can be easily read from its ID polytope. Additionally, polytopes can be effortlessly managed by a computer. Many unanswered questions about toric varieties can thus now be answered using computer-aided calculation methods.