Rogers-Ramanujan type identities and Nahm sums

发布者:文明办发布时间:2024-07-02浏览次数:66

主讲人:王六权 武汉大学教授


时间:2024年7月2日15:30


地点:三号楼301室


举办单位:数理学院


主讲人介绍:王六权,2014年本科毕业于浙江大学,2017年博士毕业于新加坡国立大学,现为武汉大学教授。主要从事数论与组合数学领域的研究,研究课题多集中在q-级数、整数分拆、特殊函数、模形式理论等方面。迄今在《Advances in Mathematics》,《Transactions of the American Mathematical Society》、《Advances in Applied Mathematics》、《Journal of Number Theory》、《Ramanujan Journal》等期刊上发表学术论文40多篇,先后主持国家自然科学基金青年基金和面上项目各一项。


内容介绍:Let $r\geq 1$ be a positive integer, $A$ a real positive definite symmetric $r\times r$ matrix, $B$ a vector of length $r$, and $C$ a scalar. Nahm's problem is to describe all such $A,B$ and $C$ with rational entries for which $$F_{A,B,C}(q)=\sum_{n=(n_1,\dots,n_r)\in (\mathbb{Z}_{r\geq 0})^r} \frac{q^{\frac{1}{2}n^\mathrm{T} An+n^\mathrm{T} B+C}}{(q)_{n_1}\cdots (q)_{n_r}}$$ is a modular form. Zagier completely solved the rank one case. When the rank $r=2,3$, Zagier presented many examples of $(A,B,C)$ for which $F_{A,B,C}(q)$ appears to be a modular form. We present a number of Rogers-Ramanujan type identities involving double and triple sums, which give modular form representations for Zagier’s rank two and rank three examples. We will also discuss the modularity of some other generalized Nahm sums.