Dr Karol Duda, a graduate of the Faculty of Applied Mathematics at the Silesian University of Technology, received a distinction in the International Award named after Stefan Banach for his outstanding doctoral thesis in mathematical sciences.

Dr Karol Duda is a graduate of Mathematics at the Faculty of Applied Mathematics of our University. He also worked for a year at this faculty before moving to the doctoral school of the University of Wrocław.

- I remember these studies well, especially since my wife is also a graduate of this field, and we met during our studies - recalls Dr Karol Duda. - Also, in terms of scientific work, it was a good time for me, in particular cooperation with my main promoter - prof. Aleksander Iwanow. I was very fortunate that Prof. Iwanow moved to the Silesian University of Technology from the University of Wrocław during my undergraduate studies. The cooperation with him, lasting since my master’s degree studies, had a huge impact on my scientific development.

The graduate of our University admits that the individual Study Program implemented at the Silesian University of Technology was also important for him.

- It allowed to adapt the study program to the needs of a student. Thanks to this, I was able to gain knowledge in my chosen field of mathematics, regardless of which specialty and subjects other students chose to study. Some of the subjects I attended were conducted especially for me – recalls Dr Karol Duda.

The title of the awarded doctoral dissertation is “Dynamics and computability in Geometric Group Theory.” Its promoters were Prof. Dr Hab Aleksander Iwanow and Dr Damian Osajda. It concerns two areas of research in geometric group theory.

- In the first part of the dissertation, the computable aspects of mediality are examined. This is a continuation of research conducted in a master’s thesis written at the Silesian University of Technology. The topic is closely related to the famous Banach-Tarski paradox, which states that a three-dimensional sphere can be "cut" into a finite number of parts, and then - using only shifts and rotations - assemble two spheres of the same radius as the radius of the output sphere. - The results of this part of the dissertation concern the existence of algorithms determining the paradoxical distributions of various structures, as well as algorithms related to the so-called von Neumann geometric hypothesis, i.e. determining the division of certain spaces into so-called d-regular trees.

The second part concerns locally elliptical actions of groups on the so-called complexes of small deletions, i.e. one of the examples of space with non-positive curvature. 

- The main result of this part of the dissertation is that groups with the presentation of small deletions do not have infinite torsion subgroups - explains Dr Karol Duda. - This result was particularly highlighted by the reviewers of the dissertation. This is an answer to a problem that has been open for many years The result itself is a conclusion from a more general assertion proved in the dissertation, about the local-global preservation of fixed points for the activities of groups on complexes of small deletions.  In other words, let’s take a certain set of actions in a certain space. If the actions are "decent", i.e., they are reflections or rotations, and we have local fixed points, i.e. each of these actions stabilizes a point of space, then there is a global fixed point, stabilized by all these actions.

The laureate admits that the award in the competition is not a surprise to him.

- Over the last two years, many mathematicians in my field have expressed very positive opinions about my results, so I hoped that the jury of the competition would also appreciate them,” comments the graduate of the Silesian University of Technology. This is a huge success for me, especially since in the coming years, this award will significantly increase the chances of my applications for grants and postdoc positions before my results are published in prestigious journals, which often takes several years in the case of mathematics.