Homotopy Theory of Algebraic Varieties and Wild Ramification

Información del proyecto

KAPIBARA

Identificador del acuerdo de subvención: 802787

DOI

10.3030/802787

Proyecto cerrado

Fecha de la firma de la CE 16 Agosto 2018

Fecha de inicio 1 Junio 2019

Fecha de finalización 31 Mayo 2026

Financiado con arreglo a

• EXCELLENT SCIENCE - European Research Council (ERC)

Coste total

€ 1 007 500,00

Aportación de la UE

€ 1 007 500,00

Coordinado por

INSTYTUT MATEMATYCZNY POLSKIEJ AKADEMII NAUK
Poland

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

Período documentado: 2023-12-01 hasta 2025-05-31

Resumen del contexto y de los objetivos generales del proyecto

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.

Trabajo realizado desde el comienzo del proyecto hasta el final del período abarcado por el informe y los principales resultados hasta la fecha

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.

Avances que van más allá del estado de la técnica e impacto potencial esperado (incluida la repercusión socioeconómica y las implicaciones sociales más amplias del proyecto hasta la fecha)

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.