Sebastian Debus

Title of the PhD thesis:

"Combinatorics of Reflection Groups and Real Algebraic Geometry"

Abstract:

Real algebraic geometry studies sets defined by a finite system of real polynomial equalities and inequalities. The algorithmic study of such semialgebraic sets provides also solutions to algorithmic problems arising in optimization, robotics, computer vision, automated theorem proving, and many more. A central topic in this area is the study of the cone of nonnegative polynomials. Verifying that a given polynomial is nonnegative is an NP-hard problem even for quartics. However, it turns out to be algorithmically much more feasible to verify if a given polynomial admits a representation into a sum of squares of polynomials, and such a decomposition provides a certificate for nonnegativity. Therefore, understanding the sets of sums of squares and nonnegative polynomials provides applications to various fields such as polynomial optimization and graph theory. The few cases of equalities between the sets of sums of squares and nonnegative polynomials in different numbers of variables and degrees have been classified already by Hilbert. Those do not necessarily transfer to equivariant situations, i.e., if the polynomials are invariant by the action of a group.
In this thesis, tropicalization and the combinatorics of reflection groups are exploited to examine the cones of invariant nonnegative forms and sums of squares forms, and to study invariant systems of equations.

The thesis is published and available in Munin

Supervisors

• Professor Cordian Riener, Institute for mathematics and statistics, UiT (main supervisor)
• Associate professor Hugues Verdure, Institute for mathematics and statistics (co supervisor)

Evaluation committee

• Professor Claus Scheiderer, Universitet of Konstantz. (1. Opponent)
• Professor Marta Panizzut, Max Planck Institute for Mathematical Sciences, Berlin (2. Opponent)
• Professor Trygve Johnsen, IMS, UiT (Intern member and leader of the committee)

Links to the trial lecture and defense will be possible to open when the live stream begins. If you haven`t clicked the link to the folder before it begins, refresh the web browser for them to become visible. If you have clicked the link to the trial lecture or defense before it has started, it will open automatically when the stream begins. 

The trial lecture starts at 11.00 November 18th

Trial lecture

The defense starts at 14.15 november 18th

Defense