整数分拆中的秩函数与mock theta函数的研究

整数分拆是组合数学和数论的重要研究对象，其与概率论、代数学、表示论、模形式理论等多个数学分支有着密切的联系，包括国际著名的数学与物理学家戴森(F. Dyson)、美国科学院院士安德鲁斯(G.E. Andrews)等众多知名组合学家在此方面做出了丰富而卓越的研究成果。本项目将聚焦整数分拆中最基本、最核心的统计量秩函数(rank，crank等)与mock theta函数的研究，计划以生成函数、圆法、模形式理论等为工具，研究各种带约束条件的整数分拆中的秩函数等量和不等量关系与其凸凹性和对数凹性，建立或改进分拆秩函数及mock theta函数的系数等渐进公式，特别是给出一整类mock theta函数与某种分拆秩函数之间的等量关系，来系统诠释模形式理论在mock theta函数与分拆秩函数理论之间的重要纽带作用。

The theory of integer partitions is an important topic of combinatorics and number theory, and related to several branch of mathematics, like probability theory, algebra, theory of representations and modular forms. Many renowned combinatorial scientists who have made extensive and outstanding research in this area, including the famous mathematician and physicist F. Dyson, G.E. Andrews, a member of the American Academy of Science. This project studies the basic and central statistics (rank and crank, etc.) in integer partitions and mock theta functions. We plan to use generating functions, circle methods, modular form theory, etc. as tools to study the equality and inequality of rank functions and convexity and concavity of rank functions in integer partitions. Then, the proposer intends to establish or improve the asymptotic formula of rank functions and the coefficients of the mock theta functions, especially the equivalence relationship between a whole class of mock theta functions and some sort of rank functions, to systematically discover how important the modular forms plays important roles between mock theta functions and rank functions in integer partitions.