Logarithmic Donaldson-Thomas Theory

该奖项支持理论粒子物理学和数学之间相互作用的数学方面的研究。这种相互作用始于20世纪80年代弦理论的出现，并已成为数学灵感的巨大来源，经常连接到迄今为止脱节的研究领域。在这个项目中研究的唐纳森-托马斯不变量是某些D-膜计数的数学版本，D-膜是弦论中为开弦提供边界条件的高维对象。这种计数的重要性来自于与其他数学对象和概念的大量关系，例如双线性表示、瞬子、稳定性条件和Gromov-Witten不变量。唐纳森-托马斯型不变量也是物理学家和数学家提出的一长串问题和挑战的核心。本项目将开发和测试唐纳森-托马斯不变量的对数框架理论。对数版本的目的是治疗的情况相对于一个除数和研究退化的家庭。这个扩展对于解释不变量、计算、新的应用以及作为一个证明理论的工具都是非常重要的，从而极大地扩展了Donaldson-Thomas理论的范围。这个奖项反映了NSF的法定使命，并且通过使用基金会的知识价值和更广泛的影响评审标准进行评估，被认为是值得支持的。

This award supports research on the mathematical side of the interaction between theoretical particle physics and mathematics. This interaction started with the advent of string theory in the 1980's and has become a huge source of inspiration in mathematics, often connecting hitherto disconnected areas of research. The Donaldson-Thomas invariants under investigation in this project are a mathematical version of the count of certain D-branes, which are higher-dimensional objects in string theory providing boundary conditions for open strings. The importance of such counts comes from the multitude of relations to other mathematical objects and notions, such as quiver representations, instantons, stability conditions, and Gromov-Witten invariants. Donaldson-Thomas-type invariants are also central to a long list of questions and conjectures posed both by physicists and mathematicians.This project will develop and test a theory of a logarithmic framework for Donaldson-Thomas invariants. The logarithmic version is designed for treating situations relative to a divisor and for studying degenerating families. This extension is of importance for the interpretation of the invariants, for computations, for new applications, and as a proof-theoretic tool, thus greatly expanding the scope of Donaldson-Thomas theory.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.