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Curve Counting and Log Geometry

Project description

Advancing understanding of Gromov-Witten and enumerative geometry

In mathematics, Gromov-Witten invariants are rational numbers that could count algebraic curves meeting prescribed conditions in given algebraic varieties. Funded by the Marie Skłodowska-Curie Actions programme, the LOGEO project aims to apply Gromov-Witten invariants to address questions in a broad range of mathematical areas: sheaf counting theories, mirror symmetry and moduli theory of curves. Researchers will also use logarithmic geometry, a modern variant of algebraic geometry developed to deal with two fundamental problems – compactification and degeneration – that has significantly advanced knowledge on these areas. Project results are expected to break new ground in enumerative geometry and increase understanding of curve-counting.

Objective

The Gromov--Witten invariants of a space X record the number of curves in X of a given genus and degree which meet a given collection of cycles in X. Gromov--Witten theory is an extremely active field of research, and through its technical challenges attracts some of the most talented researchers at the interface of geometry with physics, who have made a lot of progress here over the last 20 years. We propose a program to apply Gromov--Witten theory to questions from a broad range of areas of mathematics: from sheaf counting theories, from mirror symmetry, and from the moduli theory of curves. The key new ingredient here is the recent significant advance in our understanding of these theories using logarithmic (log) geometry, which is a modern variant of algebraic geometry, developed to deal with two fundamental and related problems: compactification and degeneration. We will investigate solutions to these problems in interlinked areas of algebraic geometry, and use them to obtain major advances in Gromov--Witten theory. Building on the success of our previous work on log Gromov--Witten theory, we propose a program to 1) construct a computationally effective log geometric extension of sheaf counting theories, 2) develop new techniques to enumerate curves in Deligne-Mumford stacks (orbifolds) and to construct mirrors to such stacks, and; 3) investigate stability in the moduli spaces of curves along with original new connections to quiver-stability theories. Completion of these projects, will break new ground in enumerative algebraic geometry, and even if not all of the overall goals are achieved it will be a cornerstone in understanding curve-counting in different setups via modern log geometric techniques.

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Coordinator

UNIVERSITEIT LEIDEN
Net EU contribution
€ 253 052,16
Address
Rapenburg 70
2311 EZ Leiden
Netherlands

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Region
West-Nederland Zuid-Holland Agglomeratie Leiden en Bollenstreek
Activity type
Higher or Secondary Education Establishments
Links
Other funding
€ 0,00

Partners (1)