PACM Graduate Students

Qinxin Yan- PACM Graduate 4th year student advised Mete Soner.

Title: Mean field system and its applications

Abstract: Mean field optimal control problems arise in various applications, including financial mathematics and machine learning, where a large number of interacting agents are approximated by a mean field representation. In this talk, I will present a framework for analyzing viscosity solutions of Hamilton-Jacobi-Bellman (HJB) equations in the space of probability measures. This extension of classical viscosity theory to infinite-dimensional spaces enables the formulation of comparison principles, ensuring uniqueness of the solutions. I will then discuss a neural network-based numerical method for solving mean field optimal control problems. The algorithm’s convergence is rigorously established using the viscosity solution framework, providing a solid theoretical foundation for its applicability in complex systems. The talk will highlight both the analytical and computational aspects of mean field models. This is joint work with Prof. H. Mete Soner and Prof. Josef Teichmann.

Shouda Wang- PACM Graduate 4th year student advised by Ramon van Handel.

Title: Monotonicity of mixed volumes

Abstract: It is obvious that for any convex body A containing another convex body B, we have volume(A)>=volume(B), and equality holds if and only if A=B. This monotonicity inequality has a useful generalization to mixed volumes (which is a rich class of integral geometric quantities including volume, surface area, mean width, etc.). However, the equality cases become much more involved as soon as one considers non-smooth convex bodies. In this talk I will give a brief introduction to mixed volumes and then explain exactly how the boundary structure comes into play for causing equal mixed volumes. This is based on joint work with my advisor Ramon van Handel. 

Hezekiah Grayer- PACM Graduate 5th year student advised by Peter Constantin.

Title: On radiative collisionless plasma

Abstract: I will discuss the global will-posedness problem for the Vlasov-Maxwell system, and my investigations of this model when the effect of radiative damping is included. For the Vlasov-Maxwell system, it is only known that there is regularity conditional on moments of the distribution function being controlled. I will discuss the effect of radiation damping on the dynamics of these moments, and its implications for the global well-posedness problem. This joint work with Peter Constantin.