Mathematical Sciences: Geometry and Topology of Quotients in Algebraic and Symplectic Geometry

小行星9401695 这个建议涉及几何和拓扑问题的辛减少，模空间的向量丛代数曲面，算术代数。 研究人员计划研究辛约化在改变其定义基准（即辛形式）时的行为，并将其Gromov不变量与原始空间的等变Gromov不变量进行比较。 他还计划研究代数曲面上稳定向量丛的模空间的Gieseker和Uhlenbeck紧化在改变其定义数据（即充足因子）时的行为。 最后，他打算比较上同调的一个环面紧的算术商的交叉上同调的贝利-博雷尔-佐竹紧化。 代数几何学研究多项式的解的集合的几何学。 辛几何是由理论物理学发展而来的，它研究的是一些最初来自物理学的数学模型的几何。 目前的建议涉及辛几何和代数几何中某些重要模型的结构问题。 这些模型取决于参数的选择（如数值数据）。 当我们改变参数时，找出模型的结构如何变化是自然而重要的。 深入研究这一问题，将有助于澄清这两个模型结构之间的神秘联系，从而对这两个领域中的其他一些重要问题有新的认识。 它将有许多有趣的应用物理学，因为辛几何是发展出来的物理模型。 ***

9401695 Hu This proposal concerns geometric and topological questions on symplectic reductions, moduli spaces of vector bundles over algebraic surfaces, and arithmetic quotients. The investigator plans to study the behavior of a symplectic reduction when varying its defining datum (i.e. a symplectic form) and compare its Gromov invariants with the equivariant Gromov invariants of the original space. He also plans to study the behavior of the Gieseker and Uhlenbeck compactifications of the moduli space of stable vector bundles on an algebraic surface when varying their defining data (i.e. ample divisors). Finally, he intends to compare the cohomology of a toroidal compactification of an arithmetic quotient with the intersection cohomology of the Baily-Borel-Satake compactification. Algebraic Geometry studies the geometry of the set of solutions of a polynomial. Symplectic Geometry is motivated by Theoretical Physics, which studies the geometry of some mathematical models originally coming from Physics. The current proposal deals with some questions on the structures of certain important models in Symplectic and Algebraic Geometry. These models depend on a choice of parameters (such as numerical data). It is natural and important to find out how the structure of a model changes when we change the parameters. A deep result toward this will clarify the mysterious relation among the structures of these models, leading to new insight into some other important problems in the two fields. It will have numerous interesting applications to Physics, because Symplectic Geometry was developed out of physical models. ***