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Among Poincare's numerous mathematical contributions is one related toÌýÌý
the local theory of differential equations. Near an equilibrium pointÌýÌý
of a system of ordinary differential equations, one can transform theÌýÌý
system to a "normal form", provided that certain "non-resonant"ÌýÌý
conditions are satisfied. In this talk, I will first introduce theÌýÌý
concept of "resonance", and then present Poincare's theorem on hisÌýÌý
normal form. Finally I will talk about the extension of Poincare'sÌýÌý
normal form to partial differential equations, which is currently anÌýÌý
active field of research.