Category theory, a powerful mathematical framework, has transformed the way we approach complex systems. This seminar explores why we should study category theory in mathematics and how it helps us better understand diverse concepts by providing a new language for expressing relationships. Category theory looks at the big picture, ignoring specific details to uncover common structures and patterns in different areas. This approach gives us valuable tools to make sense of complex phenomena. Moreover, category theory acts as a bridge between seemingly unrelated subjects, showing how they are connected. This seminar makes a compelling case for the relevance of category theory, illustrating how it enriches our understanding and brings together different ideas, making mathematics and other fields more accessible.

We study the geometric ergodicity and the long time behavior of the Random Batch method for interacting particle systems, which exhibits superior numerical performance in recent large-scale scientific computing experiments. We show that for both the interacting particle system (IPS) and the random batch interacting particle system (RB-IPS), the distribution laws converge to their respective invariant distributions exponentially, and the convergence rate does not depend on the number of particles N, the time step τ for batch divisions or the batch size p. Moreover, the Wasserstein-1 distance between the invariant distributions of the IPS and the RB-IPS is bounded by O(√τ),  showing that the RB-IPS can be used to sample the invariant distribution of the IPS accurately with greatly reduced computational cost 

Many phenomena in biology are understood using mathematical models of PDEs and stochastic processes. In this paper, we aim to depict the intracellular phenomenon of Liquid-Liquid Phase Separation (LLPS) using a stochastic process, which can be a potential cause of human diseases in the field of current biology. LLPS is characterized by the aggregation of proteins, forming droplets that separate from the surrounding environment as liquid. In contrast to previous aggregation models that rely on partial differential equations, our population model is constructed using chemical reaction networks (CRNs) with the Markov Chain property as we analytically demonstrate that all reaction events follow exponential distributions. Moreover, this model explicitly incorporates the concept of the protein threshold, which is essential for the occurrence of LLPS. The central aim of this research is to provide valuable insights into the distinctive characteristics of LLPS in both two-dimensional (2D) and three-dimensional (3D) cellular environments.Additionally, we demonstrate that our model effectively describes previous experimental results, further enhancing our understanding of LLPS-related processes. 

We propose an asymptotic preserving (AP) numerical method for the three temperature radiative transfer equation in the framework of the PN method. A specially designed AP splitting scheme is proposed to decouple the radiation and electron system. An implicit and explicit numerical scheme is built with the higher-order expansion coefficients of the specific intensity in PN method treated explicitly and lower-order implicitly, which leads to an implicit system that can be solved at the computational cost of an explicit scheme. Several numerical examples validate the efficiency of this scheme 

Fermi-Dirac BGK equation is a quantum modification of classical BGK equation replacing the classical Maxwellian equilibrium to the Fermi-Dirac equilibrium. This equation has advantage in numerical computation cost compared to the original Boltzmann equation but has difficulty in highly nonlinear features. In this talk, I will introduce moments bound for Fermi-Dirac equation and show a global existence result using this estimate. 

This talk concerns a mean-field model for a network of neurons, which has been studied both from the probability perspective via the Mckean-Vlasov SDE, and from the PDE perspective through the Fokker-Planck equation.  A major mathematical issue for the model is the blow-up of the classical solution, which scientifically connects to the synchronization of a neuron network.

We propose a new generalized solution which allows blow-ups. The idea is to introduce a new timescale in which the blow-up event is "dilated and unfolded’’. We establish properties of the generalized solution including the characterization of blow-ups and the global well-posedness. The generalized solution provides a new perspective to understand the dynamics in the presence of blow-ups as well as the continuation of the solution after a blow-up. Joint work with Zhennan Zhou.

(p.s. Preprint available at https://arxiv.org/abs/2206.06972)