题 目：Interplay between Advanced Mathematics and Physics
In this talk, I shall present one or two examples demonstrating the symbiotic interplay between advanced mathematics and physics -- the development of new mathematics under close links to the physics and in return its application to the physics for a better understanding and new predictions to the underlying physical phenomena.
The first example is the gas-liquid transition, one of the most basic problems in equilibrium phase transitions. In the pressure-temperature phase diagram, the gas-liquid coexistence curve terminates at a critical point C, also called the Andrews critical point. It is, however, still an open question why the Andrews critical point exists and what is the order of transition going beyond this critical point. To answer this basic question, a new dynamic model is established, and is consistent with the van der Waals equation in steady state level. With this dynamic model, we are able to derive a theory on the Andrews critical point C: 1) the critical point is a switching point where the phase transition changes from the first order with latent heat to the third order, and 2) the liquid-gas phase transition going beyond Andrews point is of the third order. This clearly explains why it is hard to observe the liquid-gas phase transition near the critical point. The study is based on the development of a new dynamic transition theory with the philosophy to search the complete set of transition states. The theory has been applied to a wide range of nonlinear problems.
If time permits, the second example is the development of a new geometric theory for incompressible flows with applications to boundary-layer separation of fluid flows. This is a joint work with Tian Ma.