Some problems in Algebraic Geometry and String Theory

这个项目的研究是几何学和物理学的结合。 在一个方向上，我们对物理学基本力和粒子的几何性质的知识将得到扩展，就像爱因斯坦解释引力是空间和时间曲率的表现一样。 在另一个方向上，物理思想和见解将被应用于推进我们对某些弯曲空间的几何知识。 在过去的30年里，数学研究的这个一般领域已经产生了许多重要的应用到数学的其他领域，以及有时令人惊讶的应用到物理学，科学和工程更普遍。 博士生将接受培训，以协助这些调查的数学方面，增加高技能的工人到STEM劳动力。除了弦理论问题外，还将研究弦理论启发的代数几何问题。其中一组问题涉及到弦理论所激发的三个重要的代数几何曲线计数不变量的研究：Gopakumar-Vafa（或BPS）不变量，Donaldson-Thomas不变量和Pandharipande-Thomas不变量，以及它们的动机和精炼扩展。这些理论中的一些被认为描述了等价的数据，而另一些则仅仅是理论上等价的。 新的技术和例子将被调查，以扩大这些理论的范围。 重要的目标是建立一些结构，至少在某些情况下，并通过连接不同的方法，将弦理论的精确BPS不变量的定义置于更坚实的数学基础上。在另一个方向上，对复曲面的小量子产物的研究将重新开始，目标是回答一个在20世纪90年代几乎得到回答的关于其结构的问题。最后，几何物理字典的F-理论和它的双重描述的M-理论和II型弦理论将调查的目标，使字典更精确，然后推导应用代数几何和物理。

The investigations in this project are at the interface of geometry and physics. In one direction, our knowledge of the geometric nature of the fundamental forces and particles of physics will be expanded, much as Einstein explained gravity as a manifestation of the curvature of space and time. In another direction, physical ideas and insights will be applied to advance our knowledge about the geometry of certain curved spaces. For the past thirty years, this general area of mathematical investigation has produced many important applications to other areas of mathematics, as well as sometimes surprising applications to physics, and to science and engineering more generally. Doctoral students will be trained to assist in mathematical aspects of these investigations, adding highly skilled workers to the STEM workforce.Several problems in algebraic geometry inspired by string theory will be investigated, in addition to problems in string theory. One set of problems involves the study of three important algebro-geometric curve-counting invariants motivated by string theory: Gopakumar-Vafa (or BPS) invariants, Donaldson-Thomas invariants, and Pandharipande-Thomas invariants, as well as their motivic and refined extensions. Some of these theories are known to describe equivalent data, while others are only conjecturally equivalent. New techniques and examples will be investigated to extend the reach of these theories. Important goals are to establish some of the conjectures, at least in certain situations, and to put the definition of the refined BPS invariants of string theory on a more firm mathematical foundation by connecting the differing approaches. In another direction, the investigation of the small quantum product of a toric variety will be reopened with the goal of answering a question about its structure that was almost answered in the 1990s. Finally, the geometry-physics dictionary for F-theory and its dual descriptions in terms of M-theory and type II string theory will be investigated with the objectives of making the dictionary more precise and then deducing applications to both algebraic geometry and physics.