Some problems in Algebraic Geometry and String Theory

在这个项目的调查是在几何和物理的界面。一方面，我们对基本力和物理粒子的几何性质的认识将得到扩展，就像爱因斯坦将引力解释为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提高我们对某些弯曲空间的几何知识。在过去的三十年里，这个数学研究的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，以及更普遍的科学和工程。博士生将接受培训，以协助这些调查的数学方面，为STEM劳动力增加高技能工人。除了弦理论中的问题外，还将研究受弦理论启发的代数几何中的几个问题。其中一组问题涉及到由弦理论驱动的三个重要的代数-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细扩展。已知其中一些理论描述等效数据，而其他理论仅在推测上等效。新的技术和例子将被调查，以扩大这些理论的影响。重要的目标是建立一些猜想，至少在某些情况下，并通过连接不同的方法，将弦理论的精炼BPS不变量的定义放在更坚实的数学基础上。在另一个方向上，将重新开始对环面变体的小量子积的研究，目的是回答一个关于其结构的问题，这个问题在20世纪90年代几乎得到了回答。最后，将研究f理论的几何物理字典及其在m理论和II型弦理论方面的对偶描述，目的是使字典更加精确，然后推导出代数几何和物理的应用。

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.