This is a survey article based on the content of the plenary lecture given by Jos\'e A. Carrillo at the ICIAM23 conference in Tokyo. It is devoted to produce a snapshot of the state of the art in the analysis, numerical analysis, simulation, and applications of the vast area of aggregation-diffusion equations. We also discuss the implications in mathematical biology explaining cell sorting in tissue growth as an example of this modelling framework. This modelling strategy is quite successful in other timely applications such as global optimisation, parameter estimation and machine learning.

翻译：本文是基于José A. Carrillo在东京ICIAM23全体会议上所作主旨报告内容的综述文章。文章旨在对聚集-扩散方程这一广阔领域在理论分析、数值分析、模拟仿真及应用研究方面的最新进展进行概览性总结。我们还将讨论其在数学生物学中的意义，并以组织生长中细胞分选过程作为该建模框架的实例进行说明。该建模策略在其他前沿应用领域（如全局优化、参数估计和机器学习）中也取得了显著成功。