Mock theta函数是当前组合数学、q-级数和数论的研究热点之一，该课题吸引了包括沃尔夫奖获得者Dyson教授和美国科学院院士Andrews教授在内的众多知名学者的研究兴趣。.本项目旨在研究mock theta函数的同余性质和组合性质。主要包括：（1）建立mock theta函数与经典的模形式理论的联系，证明非平凡非线性同余关系；刻画mock theta函数的self-similarity性质，发现余数不为0的同余关系；利用Newman理论和q-级数恒等式，建立包含任意素数的无穷族同余关系。（2）通过构造中间函数，将高阶不等式转化为多个低阶不等式，进而刻画mock theta函数的组合性质并证明相关猜想。.在项目实施过程中，我们期望不仅能够刻画mock theta函数的同余性质与组合性质，证明相关猜想，而且能进一步深化该领域与整数分拆、模形式理论的联系。

The theory of mock theta functions is one of the most active fields in contemporary combinatorics, q-series and number theory. Many well-known combinatorialists, including Professor Freeman Dyson who was awarded the Wolf prize and Professor George Andrews who is a member of the United States National Academy of Sciences, are interested in it. This project focuses on investigating the congruence properties and combinatorial properties for mock theta functions. It includes the following aspects: (1) By establishing the relations between mock theta functions and the theory of classic modular forms, we will prove nontrivial and nonlinear congruences for mock theta functions; By characterizing the "self-similarity" properties of mock theta functions, we will discover congruences which are congruent to a nonzero number; By employing Newman's theory and some q-series identities, we will establish infinite families of congruences which include any prime. (2) By constructing intermediate functions, we will transform inequalities with higher powers into several inequalities with lower powers, based on which, we shall characterize the combinatorial properties of mock theta functions and confirm some related conjectures. In the process of executing this project, we anticipate that we can not only characterize the congruences properties and combinatorial properties and confirm conjectures for mock theta functions, but also eventually intensify the bonds to the theory of partitions and modular forms.