Questions Related to Curves on Complex Projective Manifolds

这个项目的目的是进一步了解的行为的家庭复杂的曲线上的一个复杂的射影流形X。这将是通过解决两个枚举问题（例如，如何可能有曲线与给定的程度和属和发病率？）和变形理论问题（例如，在X的一般变形中存在什么样的曲线变形？）。 攻击这些问题的一些重要工具包括Atiyah-Bott局部化定理和Kuranishi的形式形变理论。除了研究部分，研究人员将重新启动成功的WAGS（西方代数几何）会议。这项研究涉及计数问题。 例如，给定多变量多项式方程的解集，有多少特定类型的对象（例如直线）完全位于解集中？ 我们进一步研究的“形状”的集合的对象，即使在这种情况下，该集合的对象不再是有限的。这项研究与著名的现代物理学理论“弦理论”有关，该理论提出通过在六个独立方向上略微“加厚”时空来解决统一自然界基本力的问题。 我们的研究研究了宇宙在这六个新方向上的（微小）横截面的几何形状。

The aim of this project is to further the understanding of thebehavior of families of complex curves on a complex projective manifold X.This will be done by addressing both enumerative questions (e.g. How may curves are there with given degree and genus and incidences?) and deformation-theoretic questions(e.g. What deformations of curves lie in a generic deformation of X?). Some of the important tools for attacking such questions include the Atiyah-Bott localization theorem and Kuranishi's formal deformationtheory. In addition to the research component,the investigators will restart the successful WAGS(Western Algebraic Geometry) conferences.This research relates to counting problems. For example, given the set of solutions of a polynomial equation in several variables, how many objects of a certain type (e.g. lines) lie entirely inside the solution set? We further study the 'shape' of the collection of objects even in cases in which that collection of objects is no longer finite. This research is related to a celebrated modern theory in physics, called 'string theory,' which proposes to resolve the problem of unifying the fundamental forces of nature by slightly 'thickening' space-time in six independent directions. Our research studies the geometry of the (tiny) cross-section of the universe in those six new directions.