Recently, Professor Wang Kaizhi from the School of Mathematical Sciences delivered a report titled "Weak KAM Theory and Mean Field Game Systems." at Zhiyuan College. The report delved into the profound intersections of dynamical systems, partial differential equations, and game theory, systematically presenting the development and significant value of these theories in cutting-edge applications.

Wang began with the foundational theories of Hamiltonian systems, reviewing the formation and core ideas of Weak KAM (Kolmogorov-Arnold-Moser) theory. He explained how this theory, within the framework of compact manifolds, introduces the concept of weak KAM solutions—equivalent to viscosity solutions—and proves the existence and uniqueness of the effective Hamiltonian. This provides a powerful mathematical tool for characterizing the asymptotic behavior of system solutions.

Wang then focused on Mean Field Games (MFGs), a crucial topic in differential game theory, he outlined the development of MFG theory and elaborated on its mathematical modeling approach, which couples a Hamilton–Jacobi equation with a Fokker–Planck equation to describe the collective behavior of a vast number of rational agents. Wang highlighted the theory's broad application prospects in economics, engineering sciences, and complex systems modeling.

A key insight of the report was the revelation of the intrinsic connection between Weak KAM theory and first-order MFGs. Wang pointed out that the weak KAM method, originating from dynamical systems, offers a new perspective and analytical framework for handling first-order systems with lower regularity.

Wang also shared his team's recent research progress, including extending the theoretical framework from compact manifolds to non-compact spaces and domains with boundaries, and employing tools from contact Hamiltonian systems to study generalized mean field game models.

During the Q&A session, attendees engaged in in-depth discussions with Wang on topics such as the uniqueness of weak KAM solutions and potential applications of MFGs. He provided detailed analyses using specific mathematical constructs and application cases, encouraging young scholars to build a solid foundation in dynamical systems and PDE theory while actively expanding their interdisciplinary visions to seek innovative breakthroughs at the crossroads of mathematical theory and practical application. Wang’s report vividly demonstrated the deep integration of fundamental mathematical theory with frontier application fields.