理想逼近是由申请人及其合作者在正合范畴的框架下提出和建立的理论，在同调代数和表示论的研究中有重要的应用。另一方面，模型结构是近年来的研究热点，它在表示论和代数几何的研究中扮演着重要的角色。本项目将借鉴经典逼近理论中的研究方法，结合三角范畴和表示论中的方法，进一步发展理想逼近理论，并将其应用于研究三角范畴和群代数的phantom理想和ghost理想，讨论phantom数和ghost数的上界和下界，系统地发展计算phantom数和ghost数的方法，并且在复形的同伦范畴中研究生成假设。此外，我们还将讨论理想逼近理论和模型结构的联系，利用理想余挠对研究Gillespie猜测。最后，我们将应用理想逼近理论和模型结构来研究telescope猜测。

Ideal approximation theory is formulated by the applicant and his collaborators in the setting of an exact category which has important applications in homological algebra and representation theory. On the other hand, the model structure is a hot topic of much recent research, and it plays an important role in representation theory and algebraic geometry. In this project, we will use methods from classical approximation theory, combining with methods from triangulated category and representation theory, to develop ideal approximation theory further. We then apply ideal approximation theory to investigate phantom ideals and ghost ideals of triangulated categories and group algebras, analyze upper bounds and lower bounds of phantom numbers and ghost numbers, develop methods to compute phantom numbers and ghost numbers systematically, and study the generating hypothesis in homotopy cateogories of complexes. We will also draw the connection of ideal approximation theory and model structures, use ideal cotorsion pair to study the Gillespie conjecture. Finally, we will apply ideal approximation theory and model structures to study the telescope conjecture.