The goal of this project is the investigation of a topic in arithmetic algebraic geometry by algorithmic and experimental methods. Local models describe the étale-local structure of integral models of certain Shimura varieties, and therefore, as well as for other reasons, are of great interest in arithmetic geometry. However, in general their singularities are so complicated that it would be desirable to pass to a model with less severe singularities, in the best case to a semistable model. In general it is not known whether such a model exists. This is what we will investigate by explicit computations.
In cases of "small rank" computations (by the principal investigator, among others) have shown that a semistable resolution exists. In the general case there are candidates for semistable resolutions, for example by Genestier and Faltings, but so far (without using computers) their semistability could not be proved.
In addition, this and similar questions can also be investigated for other classes of schemes, for instance for certain degenerations of quiver Grassmannians.
For this project, I am looking for a PhD student. A corresponding position (75% E13 for up to 3 years) is available. Please distribute! Candidates should send a short application to me at email@example.com.