Building upon the dynamic programming principle for set-valued functions arising from many applications, we will present a new notion of set-valued PDEs. The key component is a set-valued Ito formula, characterizing the flows on the surface of the dynamic sets. In the context of multivariate control problems, we establish the wellposedness of the set-valued HJB equations, which extends the standard HJB equations in the scalar case to the multivariate case. As an application, we discuss moving scalarization, constructed using the classical solution of the set-valued HJB equation. Additionally, we introduce the concept of set values for games under Nash equilibrium, along with the corresponding PDE, and explore its geometric properties. The talk is based on joint work with Jianfeng Zhang and ongoing work joint with Nizar Touzi and Jianfeng Zhang.

The expression "Parisi formula" refers to a variational formula postulated by Parisi in 1980 to give the limiting free energy of the Sherrington–Kirkpatrick (SK) spin glass.  The SK model was originally conceived as a mean-field description for disordered magnetism, and has since become a mathematical prototype for frustrated disordered systems and high-complexity functions.  In recent years, there has been an effort to extend the Parisi framework to various generalizations of the SK model, raising new physical questions met with fresh mathematical challenges.  In this talk, I will share some developments in this evolving story.  Based on joint works with Leila Sloman and Youngtak Sohn.

I will introduce basic concepts such as the concentration and fluctuation of Φ^4 Gibbs type measures from the perspectives of statistical physics, quantum field theory, and probability theory. The focus will be on the low temperature behavior and the thermodynamic limit of these probability measures, with particular attention to fluctuations around the soliton manifold.

Phylogenetic networks are a generalization of phylogenetic trees allowing for the representation of speciation and reticulate evolutionary events such as hybridization or horizontal gene transfer. The inference of phylogenetic networks from biological sequence data is a challenging problem, with many theoretical and practical questions still unresolved. In this talk, I will give an overview of the state of the art in phylogenetic network inference. I will then discuss a novel divide-and-conquer approach for inferring level-1 networks under the network multispecies coalescent model. I will end by discussing some open problems and avenues for future research.
Parts of this talk are based on joint work with Elizabeth Allman, Hector Baños, and John Rhodes.