Biochemical reaction networks and gene regulatory networks in cells are prototypical examples of complex systems, characterized by highly nonlinear and stochastic, multilevel dynamical interactions. Gaining a deep understanding of the stochastic dynamics and thermodynamic principles governing biochemical reaction networks not only helps elucidate the intrinsic mechanisms underlying cell fate decisions and the onset and progression of diseases, but also provides new theoretical paradigms for the study of complex systems. This line of research has become one of the forefront interdisciplinary areas, bridging mathematics, physics, biology, chemistry, statistics, and intelligent science. In this talk, I will present our recent research progress in this area, with the hope of stimulating further discussion and inspiring new ideas.

Bubbling is a form of singularity formation that commonly arises in critical partial differential equations. In the context of dispersive equations, this phenomenon is closely related to what is commonly known as soliton resolution: the asymptotic decomposition of solutions into a sum of several solitons (each modulated by time dependent parameters) and a radiation term. This talk focuses on classifying the asymptotic behavior of these modulation parameters, thereby providing a more refined understanding of bubbling dynamics. Modulation analysis serves as both the key methodology and the guiding philosophy. I will present several results in this direction.

Given two knots in a 3-manifold M (e.g. in Euclidean 3-space) the Gordian distance is defined as the minimal number of crossing changes needed to turn one knot into the other. I will discuss the relationship between the Gordian distance in Euclidean 3-space and Gordian distance in general 3-dimensional manifolds.