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

PI将用泊松几何的方法研究某些代数变体的拓扑结构。许多有趣的变种都承认泊松结构，使得这种结构的辛叶的闭包加性地产生同调性。此外，我们可以要求这些泊松结构具有某些相当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这些变体的（等变）上同调的完整信息。此外，如果所讨论的代数变量是光滑的，并且具有不变的Kaehler结构，使得Kaehler形式在一定意义上与泊松结构相容，则利用该变量的辛商与GIT商之间的对应关系，可以期望在该变量的GIT商上找到这样的泊松结构。此外，人们可以希望用类似的方法来研究周氏商。人们还希望在这种情况下获得新的可积系统，或者对已知的系统提出新的见解。本课题研究的种类包括（部分）旗流形、抛物束的模空间、环面种类等。这些主题与形式几何和变形量化之间也有联系。PI想要了解某些空间的拓扑结构（底层结构），这些空间出现在不同的科学分支中，比如几何和理论物理。PI打算获得的结果很可能应用于拓扑，共形和量子场论，以及弦和镜像对称。在这些努力中，PI计划应用代数和微分几何方法以及形式代数仪器。所研究的空间自然地配备了丰富的代数和几何结构。其中一些空间看起来非常经典（如格拉斯曼），而另一些则更复杂（如模空间）。PI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.