We will discuss several recent results for aggregation-diffusion equations related to partial concentration of the density of particles. Nonlinear diffusions with homogeneous kernels will be reviewed quickly in the case of degenerate diffusions to have a full picture of the problem. Most of the talk will be devoted to discuss the less explored case of fast diffusion with homogeneous kernels with positive powers. We will first concentrate in the case of stationary solutions by looking at minimisers of the associated free energy showing that the minimiser must consist of a regular smooth solution with singularity at the origin plus possibly a partial concentration of the mass at the origin. We will give necessary conditions for this partial mass concentration to and not to happen. We will then look at the related evolution problem and show that for a given confinement potential this concentration happens in infinite time under certain conditions. We will briefly discuss the latest developments when we introduce the aggregation term. This talk is based on a series of works in collaboration with M. Delgadino, J. Dolbeault, A. Fernández, R. Frank, D. Gómez-Castro, F. Hoffmann, M. Lewin, and J. L, Vázquez.

Bio-Sketch:
José A. Carrillo，英国牛津大学数学系教授，欧洲人文与自然科学院院士、国际工业与应用数学学会会士，1996年获西班牙格拉纳达大学博士学位。他长期从事偏微分方程的数学理论与数值分析研究，在动力学方程、非线性与非局部扩散模型等方向取得了重要成果。他运用最优传输与熵方法深入研究偏微分方程的梯度流结构与奇异性，其研究广泛应用于颗粒介质、半导体、集体行为、生物系统等多个领域，同时发展了保持自由能耗散特性的非线性扩散数值格式。迄今已在 Inventiones Mathematicae、Duke Math J、Comm. Pure Appl. Math. 等国际顶尖数学期刊发表论文200余篇。现任国际工业与应用数学理事会执行委员、欧洲科学院数学部主任，并担任多个国际高水平数学期刊编委。因其卓越的学术贡献，曾获西班牙皇家科学院最高科学奖 Echegaray 奖章（2022）和意大利林琴国家科学院“Luigi Tartufari”国际数学奖（2024）。