Prize for Dr. Gebhard Martin
Friends of the TUM: Doctoral Prize 2018
Martin studied within the TopMath program and completed his doctorate under the supervision of Christian Liedtke, Professor for Algebra. He successfully defended his doctoral thesis "Automorphisms of Enriques Surfaces" in 2018 with the best result "summa cum laude". His interest in Enriques surfaces were already apparent within his Master's thesis "On Extremal Enriques Surfaces", which was similarly honored with the TopMath Award 2016.
Complex Enriques Surfaces
In 1896, the mathematician Federigo Enriques constructed the first example of a complex Enriques surface. These algebraic surfaces are not rational and generically have an infinite number of symmetries, or so-called automorphisms. But the more complicated and specific the configuration of smooth, rational curves on an Enriques surface is, then the more asymmetrical the surface itself is.
However, the automorphism group of an Enriques surface can nonetheless be infinite, as Gino Fano proved in 1944. Ultimately, the researchers Shigeyuki Kondo and Viacheslav Nikulin succeeded in classifying Enriques surfaces with finite automorphism groups over the complex numbers.
Automorphisms of Enriques surfaces – a question of characteristic
In his thesis "Automorphisms of Enriques surfaces" Martin classifies Enriques surfaces with finite automorphism groups completely. He generalises the results of Kondo and Nikulin, as well as other results on symmetry groups of Enriques surfaces over the complex numbers to arbitrary fields. Hereby he concentrates primarily on fields with positive characteristics.
The characteristic is an invariant of the field, which is the number of times the 1 in this field must be added to itself in order to become 0. If that never happens - for example over the complex numbers - then the field has characteristic 0.
What is particularly interesting in this case: whilst the classification in larger characteristics, at least 7, is the same as over the complex numbers and is only minimally different from the smaller characteristics 3 and 5, when dealing with the characteristic 2 a completely different list of Enriques surfaces with finite automorphism groups are the result. Within each characteristic a classification of such Enriques surfaces is nonetheless possible.
Since April 2018, Dr. Gebhard Martin is a postdoctoral fellow of the Rheinische Friedrich-Wilhelms-University in Bonn. There he continues to research algebraic surfaces and the resulting questions on singularity theory and higher-dimensional geometry.