Homotopy Theory of Algebraic Varieties and Wild Ramification

Project Information

KAPIBARA

Grant agreement ID: 802787

DOI

10.3030/802787

Project closed

EC signature date 16 August 2018

Start date 1 June 2019

End date 31 May 2026

Funded under

• EXCELLENT SCIENCE - European Research Council (ERC)

Total cost

€ 1 007 500,00

EU contribution

€ 1 007 500,00

Coordinated by

INSTYTUT MATEMATYCZNY POLSKIEJ AKADEMII NAUK
Poland

Periodic Reporting for period 4 - KAPIBARA (Homotopy Theory of Algebraic Varieties and Wild Ramification)

Reporting period: 2023-12-01 to 2025-05-31

Summary of the context and overall objectives of the project

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.

Work performed from the beginning of the project to the end of the period covered by the report and main results achieved so far

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.

Progress beyond the state of the art and expected potential impact (including the socio-economic impact and the wider societal implications of the project so far)

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.