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Homotopy Theory of Algebraic Varieties and Wild Ramification

Objective

The aim of the proposed research is to study the homotopy theory of algebraic varieties and other algebraically defined geometric objects, especially over fields other than the complex numbers. A noticeable emphasis will be put on fundamental groups and on K(pi, 1) spaces, which serve as building blocks for more complicated objects. The most important source of both motivation and methodology is my recent discovery of the K(pi, 1) property of affine schemes in positive characteristic and its relation to wild ramification phenomena.

The central goal is the study of etale homotopy types in positive characteristic, where we hope to use the aforementioned discovery to yield new results beyond the affine case and a better understanding of the fundamental group of affine schemes. The latter goal is closely tied to Grothendieck's anabelian geometry program, which we would like to extend beyond its usual scope of hyperbolic curves.

There are two bridges going out of this central point. The first is the analogy between wild ramification and irregular singularities of algebraic integrable connections, which prompts us to translate our results to the latter setting, and to define a wild homotopy type whose fundamental group encodes the category of connections.

The second bridge is the theory of perfectoid spaces, allowing one to pass between characteristic p and p-adic geometry, which we plan to use to shed some new light on the homotopy theory of adic spaces. At the same time, we address the related question: when is the universal cover of a p-adic variety a perfectoid space? We expect a connection between this question and the Shafarevich conjecture and varieties with large fundamental group.

The last part of the project deals with varieties over the field of formal Laurent series over C, where we want to construct a Betti homotopy realization using logarithmic geometry. The need for such a construction is motivated by certain questions in mirror symmetry.

Host institution

INSTYTUT MATEMATYCZNY POLSKIEJ AKADEMII NAUK
Net EU contribution
€ 1 007 500,00
Address
Ul. Sniadeckich 8
00-656 Warszawa
Poland

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Region
Makroregion województwo mazowieckie Warszawski stołeczny Miasto Warszawa
Activity type
Higher or Secondary Education Establishments
Non-EU contribution
€ 0,00

Beneficiaries (1)

INSTYTUT MATEMATYCZNY POLSKIEJ AKADEMII NAUK
Poland
Net EU contribution
€ 1 007 500,00
Address
Ul. Sniadeckich 8
00-656 Warszawa

See on map

SME

The organization defined itself as SME (small and medium-sized enterprise) at the time the Grant Agreement was signed.

Yes
Region
Makroregion województwo mazowieckie Warszawski stołeczny Miasto Warszawa
Activity type
Higher or Secondary Education Establishments
Non-EU contribution
€ 0,00