Moduli spaces of morphisms and sheaves

这个项目的PI将研究与层和态射的模空间相关的问题。关于层的模空间，主要的主题是：当曲线的模变化时，曲线上丛的模空间的“奇怪对偶”同构-特别是“奇怪对偶”到节点曲线和一般结构群的扩展; Theta丛，它们的截面，以及曲面上层的模空间上的“秩-水平”对偶（后者与Alina Marian合作）;计算三重高阶框架/非框架层的模空间上的相交理论不变量。关于稳定态射，并在以前的工作的基础上，作者将继续进一步研究稳定映射的模空间的拓扑和相关的重言式系统。上述项目旨在理解层和态射的各种模空间的拓扑、几何和交理论。其中一个目标是利用这些模空间所享有的相互联系。模空间的出现是为了解决分类和参数化问题，将具有相似结构的对象放在一起。关于这些对象的问题（例如计数问题，关于族的问题）通常涉及对相关模空间的详细研究。稳定层和稳定态射的模空间是代数几何中的两个主要例子。除了它们在该领域的重要性外，它们还与古典和现代数学的其他分支有着多种联系，如复分析，表示论，理论物理，辛几何，规范理论，微分几何，组合学和拓扑学。

The PI of this project will study questions related to the moduli spaces of sheaves and morphisms. Regarding the moduli spaces of sheaves, the main themes to be covered are: the 'strange duality' isomorphism for the moduli space of bundles on a curve as the curve varies in moduli - in particular, extensions of 'strange duality' to nodal curves and to general structure groups; the Theta bundles, their sections, and 'rank-level' dualities on the moduli spaces of sheaves on surfaces (the latter in collaboration with Alina Marian); computation of intersection theoretic invariants on the moduli space of higher-rank framed/unframed sheaves on threefolds. Regarding stable morphisms, and building on previous work, the author will pursue a further study of the topology of the moduli space of stable maps and of the associated tautological systems. The projects above are aimed to the understanding of the topology, geometry and intersection theory of various moduli spaces of sheaves and morphisms. One of the goals is to exploit the inter-connections these moduli spaces enjoy.Moduli spaces arise as answers to classification and parametrization problems, putting together objects with similar structure. Questions about these objects (e.g. counting problems, questions about families) often involve a detailed study of the relevant moduli spaces. The moduli spaces of stable sheaves and stable morphisms are two of the primary examples occurring in algebraic geometry. In addition to their importance within the field, they bear multiple connections with other branches of classical and modern mathematics such as complex analysis, representation theory, theoretical physics, symplectic geometry, gauge theory, differential geometry, combinatorics and topology.