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Logarithmic Donaldson-Thomas Theory

对数唐纳森-托马斯理论

基本信息

批准号：
1903437
负责人：
Bernd Siebert
金额：
$ 59.85万
依托单位：
University of Texas at Austin
依托单位国家：
美国
项目类别：
Continuing Grant
财政年份：
2019
资助国家：
美国
起止时间：
2019-06-15 至 2024-05-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1903437&HistoricalAwards=false
关键词：
Logarithmic Donaldson Thomas Theory

项目摘要

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.

该奖项支持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，经常将迄今为止不相关的研究领域联系起来。本项目研究的Donaldson-Thomas不变量是某些d膜计数的数学版本，d膜是弦理论中的高维对象，为开放弦提供边界条件。这种计数的重要性来自于与其他数学对象和概念的众多关系，例如颤振表示、瞬态、稳定性条件和Gromov-Witten不变量。donaldson - thomas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。该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有关的情况和研究退化族而设计的。这一扩展对于不变量的解释、计算、新应用以及作为证明理论工具都具有重要意义，从而极大地扩展了Donaldson-Thomas理论的范围。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。