勷勤数学•专家报告

题      目：Modularity of Nahm sums and Rogers-Ramanujan type identities

报  告  人： 王六权 教授  (邀请人：尤利华)

                                   武汉大学

时      间： 7月1日  10：00-11:00

地     点：数科院东楼302

报告人简介:

       王六权，武汉大学数学与统计学院教授，主要从事组合数学（q-级数、整数分拆）与数论（模形式、同余）领域的研究。迄今在Adv. Math.、Trans. Amer. Math. Soc.、J. Combin. Theory Ser. A、Adv. Appl. Math.、J. Number Theory等期刊上发表论文50多篇，先后主持国家自然科学基金青年基金、面上项目、国家重点研发计划青年科学家项目各一项，2021年入选国家级青年人才计划。

摘      要：

       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 find all such $A,B$ and $C$ with rational entries for which the Nahm sum

$$f_{A,B,C}(q)=\sum_{n=(n_1,\dots,n_r)\in (\mathbb{Z}_{r\geq 0})^r}\frac{q^{\frac{1}{2}n^\mathrm{T}An+n^\mathrm{T}B+C}} {(q;q)_{n_1}\cdots (q;q)_{n_r}}$$ is modular. Such modular Nahm sums are usually expected to be characters of certain twodimensional rational conformal field theories, and they have important connections with combinatorics and the representation theory of Lie algebras. Zagier solved Nahm's problem in the rank one case. For rank $r=2,3$, he presented many examples of $(A,B,C)$ for which $f_{A,B,C}(q)$ appears to be modular. We will report on some of our recent progress on Nahm's problem. In particular, we will present a number of Rogers-Ramanujan type identities that establish the modularity of many Nahm sums.

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