Geometry and Topology of Moduli Spaces of Parabolic Bundles, Toric Varieties, and Partial Flag Manifolds

菲利普·福斯教授将用泊松几何的方法研究某些代数族的拓扑。许多有趣的变种承认泊松结构，使得这种结构的辛叶的闭合相加地产生同调。此外，人们可以要求这些泊松结构的某些很强的不变性和渐近性质，以便通过研究相应的泊松调和形式来获得关于这些簇的(等变)上同调的完整信息。此外，如果所讨论的代数簇是光滑的，并且具有不变的Kaehler结构，使得Kaehler形式在某种意义上与Poisson结构相容，则可以利用辛商和Git商之间的对应关系，在该簇的Git商上找到这样的Poisson结构。此外，人们可以希望用类似的方法来研究周商。人们还希望在这种情况下获得新的可积系统，或者给已知的系统带来新的曙光。属于本项目兴趣范围的分支包括(部分)旗形流形、抛物线丛的模空间、环面分支等。这些主题与形式几何学和形变量子化之间也有联系。PI想要了解某些空间的拓扑(底层结构)，这些空间出现在不同的科学分支中，如几何和理论物理。Pi打算获得的结果很可能应用于拓扑学、共形论和量子场论，以及弦和镜像对称性。在这些努力中，PI计划应用代数和微分几何的方法以及正式的代数仪器。研究中的空间自然地配备了丰富的代数和几何结构。这些空间中的一些看起来像是非常经典的物体(如草莓)，而另一些则更复杂(如模空间)。国际和平研究所期待着进一步揭示这些结构，并将它们与有时跨越跨学科边界的已知现象联系起来。

DMS-0072520 Philip Foth The PI will study the topology of certain algebraic varieties by methods of Poisson geometry. Many interesting varieties admit Poisson structures such that the closures of the symplectic leaves for such structures additively generate the homology. Moreover, one can ask for certain quite strong invariance and asymptotic properties of thosePoisson structures so that the complete information about the (equivariant) cohomology of these varieties may be obtained by studying the corresponding Poisson harmonic forms. In addition, if the algebraic variety in question is smooth and has an invariant Kaehler structure such that the Kaehler form is compatible in a certain sense with the Poisson structure, then one can hope to find such Poisson structures on the GIT quotients of the variety using the correspondence between the symplectic and GIT quotients. In addition, one can hope to study the Chow quotients by similar methods. One also hopes to obtain new integrable systems in this context or throw a new light on already known ones. The varieties which fall into the scope of interest of this project include (partial) flag manifolds, the moduli spaces of parabolic bundles, toric varieties, and others. There is also a connection between these topics and formal geometry and deformation quantization. The PI would like to understand the topology (underlying structure) of certain spaces that appear in different branches of science such as geometry and theoretical physics. The results that PI intends to obtain are likely to have applications to topological, conformal, and quantum field theories, as well as strings and mirror symmetry. In these endeavors the PI plans to apply methods of algebraic and differential geometry and the formal algebraic apparatus. The spaces under investigation come naturally equipped with rich algebraic and geometric structures. Some of these spaces appear as quite classical objects (like grassmannians) and some of them are more sophisticated (like moduli spaces). The PI is looking forward to further unveiling these structures and relating them to known phenomena sometimes crossing interdisciplinary borders.