Homotopy Theory of Algebraic Varieties and Wild Ramification

Informacje na temat projektu

KAPIBARA

Identyfikator umowy o grant: 802787

DOI

10.3030/802787

Projekt został zamknięty

Data podpisania przez KE 16 Sierpnia 2018

Data rozpoczęcia 1 Czerwca 2019

Data zakończenia 31 Maja 2026

Finansowanie w ramach

• EXCELLENT SCIENCE - European Research Council (ERC)

Koszt całkowity

€ 1 007 500,00

Wkład UE

€ 1 007 500,00

Koordynowany przez

INSTYTUT MATEMATYCZNY POLSKIEJ AKADEMII NAUK
Poland

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

Okres sprawozdawczy: 2023-12-01 do 2025-05-31

Podsumowanie kontekstu i ogólnych celów projektu

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.

Prace wykonane od początku projektu do końca okresu sprawozdawczego oraz najważniejsze dotychczasowe rezultaty

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.

Innowacyjność oraz oczekiwany potencjalny wpływ (w tym dotychczasowe znaczenie społeczno-gospodarcze i szersze implikacje społeczne projektu)

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.