Periodic Reporting for period 3 - KAPIBARA (Homotopy Theory of Algebraic Varieties and Wild Ramification)
Reporting period: 2022-06-01 to 2023-11-30
An "algebraic variety" is a geometric object defined in an algebraic way (namely, described using polynomial equations). Its "homotopy type" measures qualitative properties of its shape while discarding irrelevant geometric information. In the project, we study the homotopy types of algebraic varieties and related geometric structures. Such a study may yield sophisticated information about such shapes, with important applications in algebraic geometry, number theory, and mathematical physics.
Our main results deal with so-called non-Archimedean analytic spaces. We are able to study their shapes (homotopy types) from several different perspectives, called the "Betti homotopy type" and the "pro-etale fundamental group". The latter provides us with a means of understanding "loops" (closed paths) on such complicated spaces.
In the project we aim at gaining new insights into homotopy theory of algebraic varieties and non-archimedean analytic spaces. While typically pure mathematicians do not openly state their expected results before they are proven, let us write a few sentences nevertheless. In one direction, we plan to study several variants of fundamental groups of such spaces. For "complex non-archimedean analytic spaces", i.e. rigid-analytic spaces over Laurent series field C((t)), we plan to develop a version of the "Riemann-Hilbert correspondence", which relates representations of the fundamental groups to differential equations. This might serve as a lead-in into a non-archimedean variant of Hodge theory. We also plan to obtain results about fundamental groups of p-adic analytic spaces. In a different direction, we shall investigate singularities of meromorphic differential equations using non-archimedean methods.