Homotopy Theory of Algebraic Varieties and Wild Ramification

Projektinformationen

KAPIBARA

ID Finanzhilfevereinbarung: 802787

DOI

10.3030/802787

Projekt abgeschlossen

EK-Unterschriftsdatum 16 August 2018

Startdatum 1 Juni 2019

Enddatum 31 Mai 2026

Finanziert unter

• EXCELLENT SCIENCE - European Research Council (ERC)

Gesamtkosten

€ 1 007 500,00

EU-Beitrag

€ 1 007 500,00

Koordiniert durch

INSTYTUT MATEMATYCZNY POLSKIEJ AKADEMII NAUK
Poland

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

Berichtszeitraum: 2023-12-01 bis 2025-05-31

Zusammenfassung vom Kontext und den Gesamtzielen des Projekts

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.

Arbeit, die ab Beginn des Projekts bis zum Ende des durch den Bericht erfassten Berichtszeitraums geleistet wurde, und die wichtigsten bis dahin erzielten Ergebnisse

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.

Fortschritte, die über den aktuellen Stand der Technik hinausgehen und voraussichtliche potenzielle Auswirkungen (einschließlich der bis dato erzielten sozioökonomischen Auswirkungen und weiter gefassten gesellschaftlichen Auswirkungen des Projekts)

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.