摘要： 

In 1887, Hoelder proved that the Gamma Function, defined by the difference equation y(x+1) = x y(x), satisfies no nonzero polynomial differential equation with complex coefficients. In the last several years Galois theories have been developed that reprove this result and allow one to characterize when functions satisfying certain linear differential or difference equations also satisfy auxiliary difference or differential equations. These Galois theories allow one to reduce such kinds of questions to questions concerning linear differential or difference groups, that is groups of matrices whose entries are functions satisfying a fixed set of differential or difference equations. I will give an introduction to the theory of these groups and the related Galois theories and survey recent results applying these theories to questions of functional transcendence.

报告人简介： 

Michael F. Singer 现为美国北卡罗来纳州立大学数学系教授。 主要从事微分与差分代数及其相关符号计算问题研究。Singer 教授是微分伽罗瓦理论的知名学者，也是含参微分方程伽罗瓦理论的开拓者之一。与 van der Put合著的《Galois Theory of Linear Differential Equations》是该领域最经典的参考书。与 Schecter 合作在平面向量场方面的工作发表于 Acta Mathematica. 与姚期智等人合作在稀疏插值方面也作出了深刻的结果。2001年至2002年，Singer教授是加州大学伯克利分校的数学科学研究所 (MSRI) 的副所长。2012年，Singer教授当选为美国数学会会士。