Workshop on Equivariant Gromov-Witten Theory and Symplectic Vortices; July 2009, Luminy, France

AbstractAward：DMS-0835558首席研究员：克里斯托弗伍德沃德代数几何，辛几何和低维拓扑结构已经彻底改变了近年来引进的全纯曲线技术。 本次研讨会，将于2009年7月6日至10日在CIRM（Luminy）会议中心举行，关注全纯曲线定义的不变量的等变推广。 在Gromov-Witten理论中，Givental等人在稳定映射模空间上的等变上同调类的积分意义下，对等变不变进行了广泛的研究。 研讨会的重点将是以下更复杂的版本equivariantGromov-Witten理论：研究模空间的映射到堆栈理论商，说，一个光滑的投影品种由一个还原群。 根据定义，这样的映射由全纯主丛和一段相伴丛组成。从辛的角度来看，这种对的合适模空间被称为{\emsymplectic vortices}的模空间。 该研讨会是围绕这一特定方向组织的，目的是汇集研究人员在代数和辛几何谁没有以前的互动。 另一方面，研讨会将促进对全纯映射、规范理论和群作用相互作用的更广泛的讨论。 该理论的应用是预期的，例如，不同品种的Gromov-Witten理论之间的关系，以及规范理论不变量之间的关系，如Floer-Donaldson不变量和它们的辛类似物。 规范理论是研究数学和物理中的局域性的理论，最著名的例子是电磁学。 在代数几何中，全纯曲线技术的目的是通过理解空间中的某些曲线类来理解空间，在辛几何中，全纯曲线技术的目的是理解动力系统的周期轨道。 目标簇的对称性可以作为一种计算工具，也可以与全纯曲线相结合来定义新的不变量。研讨会将集中在代数几何，辛几何和低维拓扑的最新发展和新兴应用。 研讨会将使一些在这一领域工作的研究生和博士后研究员受益，他们与其他研究相关问题的小组几乎没有互动。由组织者控制的会议网页是www.math.rutgers.edu/mathctw/Luminy，而由会议中心CIRM控制的网页是iswww.cirm.univ-mrs.fr/web.ang/liste_rencontre/Rencontres2009/Renc346/Renc346.html。

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.