Tautological motives of Shimura varieties and stratifications of derived F-zips
Final Report Abstract
The goal 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 obtain additional structure on the cohomology, which we can analyze geometrically. One aspect of the project was to strengthen the relationship between the geometry of these structures and number theory, extending the already existing theory. In this project, we have laid the groundwork for such an extension. In terms of the relationship between geometry and number theory, Wedhorn-Ziegler studied the geometry and cohomology of a space of G-zips, was first studied by Wedhorn-Pink-Ziegler and Moonen-Wedhorn. We studied this space of G-zips and its cohomology from a conceptually more general standpoint. As a result, the calculations of Wedhorn-Ziegler can be seen partially as a result of more general phenomena. The advantage of this perspective is that the methods can be generalized to other aspects of mathematics, such as the study of symmetries. Moreover, our approach allows us to take a first step toward leaving the case of positive characteristic. Motivated by this, we then generalized the theory of cohomology and the relations between them for our purposes. In a current project, we are using this to transfer our generalized calculation of the cohomology of the space of G-zips to the case of non-positive characteristic. Transferring calculations from the positive characteristic case to number-theoretic statements is a challenging task. This has been studied by mathematicians for a long time and can partly be seen as a motivation for Peter Scholze’s dissertation, for which he was awarded the Fields Medal in 2018, one of the highest honors in mathematics. To deal with such problems, one can work with so-called rigid spaces. These can be understood as follows: in classical geometry, one works with real numbers, where, for example, the numbers 1, 2, 3, . . . are unbounded, meaning the distance of the numbers from 0 becomes infinitely large. In rigid geometry, this is not the case, as the distance between the numbers 1, 2, 3, . . . is bounded. While this takes us of some of the intuition we gain from the real world, such geometry allows us to address number-theoretic problems. Inspired by current research, I have started studying cohomologies for rigid spaces together with Christian Dahlhausen. This is by no means a new idea, but the classical theory has limitations that we are attempting to overcome. One of the problems is that it is difficult to understand the relationship between classical geometry and geometry in positive characteristic, which we have studied. In my project with Christian Dahlhausen, we hope to resolve or at least alleviate this problem in the long term.
Publications
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Motivic homotopy theory of the classifying stack of finite groups of lie type, 2024.
Yaylali, Can
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Rational motives on pro-algebraic stacks, 2024.
Yaylali, Can
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Towards A1 -homotopy theory of rigid analytic spaces, 2024.
Dahlhausen, Christian & Yaylali, Can
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T-equivariant motives of flag varieties. Algebraic & Geometric Topology, 25(4), 2343-2367.
Yaylali, Can
