Periodic Reporting for period 1 - HEF (Higher Epsilon-Factors for Higher Local Fields)
Reporting period: 2016-10-17 to 2018-10-16
Algebra on the other hand provides a solid framework for number theory, many curious arithmetic phenomena that mathematicians observed (sometimes for centuries) found satisfactory explanations in rigorous algebraic terms. A famous example is Deligne’s proof of the Weil conjectures. From an arithmetic perspective, these conjectures are concerned with the number of solutions to equations over finite fields. E.g. rather than searching for integral solutions to the equation x^2 + y^2 = z^2, we could search for triples of integers x, y, z, such that the equality holds up to the multiple of a prime number. The Weil conjectures predict that the number of solutions over finite fields is deeply connected to the topological properties of the geometric shape given by solutions to the same system of equations over the complex numbers. Deligne’s proof uses hard algebraic methods, namely the full strength of Grothendieck’s étale cohomology.
Recently it has become evident that the border between algebra and analysis is not as clear-cut as expected. A web of analogies emerged linking differential equations to arithmetic phenomena related or analogous to the Weil conjectures and the surrounding mathematics. Furthermore, a new proof of the Weil conjectures, due to Kedlaya, utilises the theory of differential equations over so-called p-adic fields (in the guise of F-isocrystals) rather than Grothendieck’s étale cohomology.
The content of this project also lies in this transition zone between algebra and analysis. Inspired by these considerations above, the project studied systems of differential equations (as they arrive in algebraic geometry) from an arithmetic perspective. This leads to (as we believe) an interesting mix of algebraic, categorical and analytical methods and sheds new light on the structures governing differential equations. The main objective was to continue the development of study of epsilon factors for systems of differential equations by Deligne, Beilinson—Bloch—Esnault, and Patel, and explore connections to the (arithmetic) theory of F-isocrystals and the theory of Higgs bundles (algebro-geometric, yet related to mathematical physics).
We reached the following conclusions: there exists a formalism epsilon factors for differential equations in higher dimensions, suggesting a similar formalism in arithmetic contexts (Galois representations, F-isocrystals). Furthermore, the theory of epsilon factors for differential equations relies on constructions related to the theory of Higgs bundles. In joint work with Wyss and Ziegler, the fellow proved an open conjecture in the theory of Higgs bundles, due to Hausel—Thaddeus. Furthermore, in order to provide a bridge between differential equations and arithmetic applications, the fellow studied in joint work with Prof. Esnault so-called rigid systems of differential equations and could prove several results implied by a conjecture of Prof. Simpson, including integrality of monodromy for cohomologically rigids, and the existence of an F-isocrystal structure for rigid systems of differential equations.
The importance to society of a project in pure mathematics is notoriously hard to evaluate. Similar mathematics with relation to arithmetic geometry has proven useful in areas such as encryption and mathematical physics. But we aren’t aware of any potential applications of our work in these areas. On a different level, we believe that the presence of this project at FU Berlin has positively contributed to the mathematical community in Berlin, Germany, and more generally in Europe."