## Project Details

### Description

AbstractAward: DMS-0835558Principal Investigator: Christopher WoodwardAlgebraic geometry, symplectic geometry and low-dimensionaltopology have been revolutionized in recent years by theintroduction of holomorphic curve techniques. This workshop, tobe held July 6-10, 2009, at the CIRM (Luminy) conference center,concerns equivariant generalizations of invariants defined byholomorphic curves. In Gromov-Witten theory, equivariantinvariants have been extensively studied by Givental and others,in the sense of integrals of equivariant cohomology classes overmoduli spaces of stable maps. The focus of the workshop will beon the following more sophisticated version of equivariantGromov-Witten theory: the study of moduli spaces of maps to thestack-theoretic quotient of, say, a smooth projective variety bya reductive group. By definition, such a map consists of aholomorphic principal bundle, together with a section of theassociated bundle. From the symplectic point of view, suitablemoduli spaces of such pairs are known as moduli spaces of {\emsymplectic vortices}. The workshop is organized around thisspecific direction, with an aim to bring together researchers inalgebraic and symplectic geometry who have had no previousinteraction. On the other hand, the workshop will promotediscussion of the interaction of holomorphic maps, gauge theory,and group actions more broadly. Applications of the theory areexpected for instance, to the relation between Gromov-Wittentheories of different varieties, and to the relation between thegauge-theoretic invariants such as Floer-Donaldson invariants andtheir symplectic analogs.We propose a workshop which will bring together gauge theory andholomorphic curves. Gauge theory is the study of local symmetryin mathematics and physics; the most famous example iselectromagnetism. Holomorphic curve techniques aim, in algebraicgeometry, to understand spaces by understanding the certainclasses of curves in them, or in symplectic geometry, tounderstand the periodic orbits of dynamical systems. Symmetry ofthe target varieties may be used as a computational tool, andalso combined with holomorphic curves to define new invariants.The workshop will focus on recent developments and emergingapplications to algebraic geometry, symplectic geometry, andlow-dimensional topology. The workshop will benefit a number ofgraduate students and postdoctoral fellows working in this areawho have had little interaction with other groups working onrelated problems.The conference web-page controlled by the organizers iswww.math.rutgers.edu/~ctw/Luminy, while that controlled by theconference center CIRM iswww.cirm.univ-mrs.fr/web.ang/liste_rencontre/Rencontres2009/Renc346/Renc346.html.

Status | Finished |
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Effective start/end date | 10/1/08 → 9/30/09 |

### Funding

- National Science Foundation (National Science Foundation (NSF))

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