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

9401695 Hu这项建议涉及关于辛约化、代数曲面上向量丛的模空间和算术商的几何和拓扑问题。研究人员计划研究当改变定义基准(即辛形式)时辛约化的行为，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紧化在改变它们的定义数据(即充足因子)时的行为。最后，他打算比较算术商环向紧化的上同调与Baly-Borel-Satake紧化的交上同调。代数几何研究多项式的解的集合的几何。辛几何是由理论物理学推动的，它研究起源于物理学的一些数学模型的几何学。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。这些模型依赖于参数的选择(例如数字数据)。当我们改变参数时，找出模型的结构如何变化是很自然和重要的。对此的深入研究将澄清这些模型结构之间的神秘联系，从而对这两个领域的其他一些重要问题有新的认识。它将在物理学中有许多有趣的应用，因为辛几何是从物理模型发展而来的。***

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. ***