Logarithmic Donaldson-Thomas Theory

该奖项支持对理论粒子物理和数学之间相互作用的数学方面的研究。这种互动始于20世纪80年代S的弦理论的出现，并成为数学灵感的巨大来源，经常将迄今互不相连的研究领域联系在一起。这个项目中研究的Donaldson-Thomas不变量是某些D-膜的计数的数学版本，这些D-膜是弦理论中的高维对象，为开弦提供边界条件。这些计数的重要性来自于与其他数学对象和概念的众多关系，例如箭图表示法、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猜想的核心。这个项目将开发和测试唐纳森-托马斯不变量的对数框架理论。对数版本是为处理与除数相关的情况和研究退化族而设计的。这一扩展对于解释不变量、计算、新应用以及作为证明理论工具具有重要意义，从而极大地扩展了唐纳森-托马斯理论的范围。这一奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

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.