Dr. Karim Adiprasito, BMS and FU Berlin alumnus, is one of the four recipients of this year's Dissertation Prize awarded by the “Ernst-Reuter-Gesellschaft” at Freie Universität Berlin. His dissertation, supervised by Günter M. Ziegler, builds a bridge between differential geometry and the theory of polyhedra.

The core part of Adiprasito’s thesis deals with the problem of the degrees of freedom in the realization of polyhedra. In two dimensions one is looking for coordinates that allow one to realize all possible shapes of a polygon with n vertices. Each of these vertices is determined by 2 coordinates, and these can be varied or perturbed independently, so the number of “degrees of freedom” is 2n for each polygon with n vertices.

In three dimensions, it has been claimed by Legendre 1791 and proved by Steinitz 1922 that the number of “degrees of freedom” is e+6, where e is the number of edges of the polyhedron. This in particular implies that if we set a limit for the number N of degrees of freedom, then there are only finitely many types of polyhedra for which all possible shapes can be described by N coordinates. In particular, there are only four types of polyhedra for which all realizations are the same (up to a transformation of coordinates).

In the 1960s, the geometers Geoffrey C. Shephard und Micha A. Perles conjectured that a similar result should hold in any higher dimension: there are only finitely many different types of polyhedra for which the possible coordinates are unique up to transformation of coordinates. 

In the main part of his dissertation, Adiprasito shows that this is not true: First he constructs an infinite series of four-dimensional polyhedra that all have less than 100 degrees of freedom. Furthermore, from these he constructs an infinite series of polyhedra of dimension 69 which have only one realization up to coordinate transformations. The proof for this is a technical tour-de-force. On the way to the breath-taking construction, he develops very original new methods and techniques.

Adiprasito was born in Aachen in 1988 and completed mathematics courses at TU Dortmund already as a highschool student. After spending one year as a visiting student at the Indian Institute of Technology in Bombay, he obtained a diploma degree in Mathematics at TU Dortmund in 2010. During his doctoral studies in Berlin (2010-2013), he also visited UC Berkeley and the Hebrew University in Jerusalem.

Every year the “Ernst-Reuter-Gesellschaft” honors four outstanding doctoral theses with its Dissertation Prize. All award-winning dissertations achieved the best possible grade of “summa cum laude” (with distinction). The prizes will be given at an award ceremony on 4 December, "Ernst-Reuter-Day", the founding day of the Freie Universität Berlin. The award comes with a prize money of 5000 Euro.

Congratulations Karim!