講座內(nèi)容：In this talk, I will introduce a relatively newglobal bifurcation theory for Fredholm operators, due to Fitzpatric,Pejsachowicz and Rabier, that generalizes Rabinowitz theorem. Itdoes not require the conversion of the PDE to a compact perturbation of theidentity; and as a special case of the theory, by adding a Fredholmnesscondition on the nonlinear operator, the Crandall-Rabinowitz local bifurcationtheorem becomes a global bifurcation theorem. I will also introducethe unilateral bifurcation theorem, and sufficient conditions for a ellipticsystem with general boundary conditions to be Fredholm with 0 index, due toJunping Shi and myself.

Finally, I will show how to apply the globalbifurcation theory to a class of chemotaxis systems, obtaining steadystates with striking features such as spikes and transition layers.