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Three ideas in open Gromov Witten theory

The questions motivating symplectic geometry, from classical mechanics to enumerative algebraic geometry, have been studied for centuries. Many recent advances in the field have stemmed from the theory of J-holomorphic curves, an... ver más

30/06/2020

THE HEBREW UNIVERS...

1M€

Presupuesto del proyecto: 1M€

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THE HEBREW UNIVERSITY OF JERUSALEM No se ha especificado una descripción o un objeto social para esta compañía.

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Financiación concedida El organismo FP7 notifico la concesión del proyecto el día 2020-06-30 No tenemos la información de la convocatoria

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Líder del proyecto

THE HEBREW UNIVERSITY OF JERUSALEM No se ha especificado una descripción o un objeto social para esta compañía.

TRL 4-5

Presupuesto del proyecto 1M€

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Descripción del proyecto The questions motivating symplectic geometry, from classical mechanics to enumerative algebraic geometry, have been studied for centuries. Many recent advances in the field have stemmed from the theory of J-holomorphic curves, and in particular Gromov-Witten theory. The past 25 years of research have produced a fairly detailed picture of what can be expected from classical, closed Gromov-Witten theory. However, closed Gromov-Witten theory by itself lacks an interface with Lagrangian submanifolds, one of the fundamental structures of symplectic geometry. The nascent open Gromov-Witten theory, in which Lagrangian submanifolds enter as boundary conditions for J-holomorphic curves, provides such an interface. The goal of the proposed research is to broaden and systematize our understanding of open Gromov-Witten theory. My strategy leverages three connections with more established fields of research to uncover new aspects of open Gromov-Witten theory. In return, open Gromov-Witten theory advances the connected fields and reveals links between them. First, the closed and open Gromov-Witten theories are intertwined. Representation theoretic structures in closed Gromov-Witten theory admit mixed open closed extensions. Further, real algebraic geometry gives rise to a large variety of Lagrangian submanifolds providing an important source of intuition for open Gromov Witten theory. In return, open Gromov-Witten theory techniques advance Welschinger's real enumerative geometry. Finally, open Gromov-Witten theory plays a key role in mirror symmetry, a conjectural correspondence between symplectic and complex geometry originating from string theory. In particular, open Gromov-Witten invariants appear in the construction of mirror geometries. Moreover, under mirror symmetry, Lagrangian submanifolds correspond roughly to holomorphic vector bundles. Well understood functionals associated to holomorphic vector bundles go over to open Gromov-Witten invariants.

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