Qi'an Guan's research area is several complex variables in mathematics which studies the properties and structures of holomorphic functions of several variables, and also called complex analysis of several variables. Because the properties of holomorphic functions are largely affected by the geometric and topological properties of their domains of definition, the research is involved not only the studying of local properties, but also of global properties.
In the research of several complex variables, the methods from partial differential equations, algebraic geometry, complex geometry, topology, Lie groups and other areas are widely used. The research of function theory of several complex variables also stimulates the development of these research fields. For example, Qikeng Lu proved "Lu Qikeng Theorem" named after him; Yum-Tong Siu proved the deformational invariance of plurigenera of projective algebraic manifolds; Xiangyu Zhou proved the extended future tube conjecture which was listed as a main unsolved problem in Encyclopaedia of Mathematics.
In cooperation with Professor Xiangyu Zhou, Qi'an Guan solved the optimal L2 extension problem by proposing new ideas and methods, established the optimal L2 extension theorem which unifies the previous various famous L2 extension theorem, and found its unexpected connections with a lot of open problems in different fields and solved them.
Multiplier ideal sheaves play a central role in algebraic geometry. Guan and Zhou proved the strong openness conjecture on multiplier ideal sheaves posed by Professor Demailly, which was thought to be "rather inaccessible". The strong openness conjecture is a bottleneck problem in the development of several complex variables and complex geometry. Many mathematicians have obtained important results under the assumption that the conjecture holds.
They also proved important conjectures posed by Demailly-Koll r and by Jonsson-Mustata, originated from complex geometric analysis and algebraic geometry.