Abstract: In this lecture series, we shall present the essence, basic theories and applications for bifurcation and phase transitions. We shall try to make the talks suitable for beginning graduate students, with no prior knowledge in the related fields needed. There are three parts of the talks, which are somewhat independent. The talks are based mainly on recent joint work with Tian Ma.
Part I is devoted to the critical phenomena, basic bifurcation and transition theories and tools, as well as applications to thermodynamic systems.
Part II is on transitions and stability of classical and geophysical fluid flows, focusing on the Burgers equation, the Benard convection and the boundary-layer separations of incompressible flows.
Part III is on quantum phase transitions (QPTs) and mechanism of quantum condensates (Bose-Einstein condensation, superfluidity, and superconductivity). In particular, we shall 1) introduce a new field theoretical interpretation of quantum mechanical wave functions, 2) derive a mechanics of quantum condensates, 3) derive the field and topological phase transition equations for condensates, and 4) make connections to the quantum phase transition, as a topological phase transition.
