Title: On Some Mathematics Related to the Interpolating Statistics Show Chinese title

Title: 关于与插值统计相关的某些数学问题

Abstract: Motivated by fractional quantum Hall effects, we introduce a universal space of statistics interpolating Bose-Einstein statistics and Fermi-Dirac statistics. We connect the interpolating statistics to umbral calculus and use it as a bridge to study the interpolation statistics by the principle maximum entropy by deformed entropy functions. On the one hand this connection makes it possible to relate fractional quantum Hall effects to many different mathematical objects, including formal group laws, complex bordism theory, complex genera, operads, counting trees, spectral curves in Eynard-Orantin topological recursions, etc. On the other hand, this also suggests to reexamine umbral calculus from the point of view of quantum mechanics and statistical mechanics. Show Chinese abstract

Abstract: 受分数量子霍尔效应的启发，我们引入了一个通用的统计空间，该空间在玻色-爱因斯坦统计和费米-狄拉克统计之间进行插值。 我们将插值统计与模糊微积分联系起来，并将其作为桥梁，通过变形熵函数的极大熵原理来研究插值统计。 一方面，这种联系使得能够将分数量子霍尔效应与许多不同的数学对象相关联，包括形式群定律、复 bordism 理论、复 genera、操作符、计数树、Eynard-Orantin 拓扑递归中的谱曲线等。 另一方面，这也提示从量子力学和统计力学的角度重新审视模糊微积分。