Tautological motives of Shimura varieties and stratifications of derived F-zips

• Applicant: Dr. Can Yaylali
• Subject Area: Mathematics
• Term: since 2023
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 524431573

The Aim of this project is to understand number theoretical problems via geometry. The idea is to assign geometric objects to zero sets of polynomials and analyze these geometric structures. To understand geometry, it is often important to attach certain numbers to these objects, which we call invariants. An important invariant for us the so called cohomology. In this project we are particularly interested in the case of positive characteristic, i.e. we set a fixed prime number to zero. To get a better feel, one can image the clock, where do not set a prime number to zero but the number 12. Under these assumptions, we get an extra structure on the cohomology, which we can analyze geometrically. One aspect of these structures was analyzed in the PhD-thesis of the applicant. Our goal is to extend the results of the applicant's PhD-thesis and get a basis for further studies. Furthermore, we want to strengthen the relation between the geometry of these structures and number theory. In this way, we generalize the known theory. The above mentioned extension can also be used to concretely calculate unknown invariants that appear in other places in number theory. Further, to understand the PhD-thesis of the applicant, one has to understand a theory, that has been studied for a few years, but still has a lot of gaps. We want to apply our new gained knowledge to close some of these gaps and make the theory more accessible. In the future, we want to use all of these results to finally move away from the positive characteristic case.

• DFG Programme: WBP Fellowship
• International Connection: France
• Host: Dr. Vincent Pilloni