This article presents an in-depth educational overview of the latest mathematical developments in coupled cluster (CC) theory, beginning with Schneider's seminal work from 2009 that introduced the first local analysis of CC theory. We offer a tutorial review of second quantization and the CC ansatz, laying the groundwork for understanding the mathematical basis of the theory. This is followed by a detailed exploration of the most recent mathematical advancements in CC theory.Our review starts with an in-depth look at the local analysis pioneered by Schneider which has since been applied to analyze various CC methods. We then move on to discuss the graph-based framework for CC methods developed by Csirik and Laestadius. This framework provides a comprehensive platform for comparing different CC methods, including multireference approaches. Next, we delve into the latest numerical analysis results analyzing the single reference CC method developed by Hassan, Maday, and Wang. This very general approach is based on the invertibility of the CC function's Fr\'echet derivative. We conclude the article with a discussion on the recent incorporation of algebraic geometry into CC theory, highlighting how this novel and fundamentally different mathematical perspective has furthered our understanding and provides exciting pathways to new computational approaches.

翻译：本文系统综述了耦合簇理论中数学发展的最新成果，从施耐德2009年开创性引入耦合簇理论局部分析的奠基性工作展开论述。我们首先以教学性视角回顾第二量子化与耦合簇假设，为理解该理论的数学基础奠定根基，继而深入探讨耦合簇理论中数学方法的最新突破。本文从施耐德开创的局部分析方法论切入，阐释其如何被应用于分析多种耦合簇方法；随后讨论奇里克与拉斯特迪乌斯发展的基于图的耦合簇方法框架，该框架为包括多参考态方法在内的不同耦合簇方法提供了统一的比较平台；进而深入剖析哈桑、马代与王提出的基于单参考态耦合簇方法的数值分析新成果，该通用方法建立在耦合簇函数Fréchet导数的可逆性基础之上。最后，我们探讨了代数几何与耦合簇理论的交叉融合，突出这种全新数学视角如何深化理论认知，并为新型计算方法开辟令人振奋的发展路径。