Descrizione del progetto
Far progredire la comprensione della teoria di Gromov-Witten e della geometria enumerativa
In matematica, gli invarianti di Gromov-Witten sono numeri razionali utilizzabili per contare curve algebriche in grado di soddisfare condizioni prestabilite in determinate varietà algebriche. Finanziato dal programma di azioni Marie Skłodowska-Curie, il progetto LOGEO si propone di applicare gli invarianti di Gromov-Witten per affrontare questioni appartenenti a una vasta gamma di aree della matematica: teorie del conteggio dei fasci, simmetria speculare e teoria dei moduli delle curve. I ricercatori si avvarranno inoltre della geometria logaritmica, una variante moderna della geometria algebrica che è stata sviluppata per risolvere due problemi fondamentali, ovvero la compattificazione e la degenerazione, e ha fatto progredire in modo significativo le conoscenze in questi settori. I risultati del progetto dovrebbero rivoluzionare il campo della geometria enumerativa, migliorando la comprensione del onteggio delle curve.
Obiettivo
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.
Campo scientifico
Parole chiave
Programma(i)
Argomento(i)
Meccanismo di finanziamento
MSCA-IF - Marie Skłodowska-Curie Individual Fellowships (IF)Coordinatore
2311 EZ Leiden
Paesi Bassi