Nonlinear Partial Differential Equations, Concentration Phenomena, and Applications

该提案侧重于对非线性偏微分方程组(NPDES)中集中现象的理论和跨学科研究。在反应、扩散和几何的驱动下，集中现象成为微分几何(Yamabe问题中的气泡)、数学生物学(斑点和条带)、化学和物理中各种现象的中心特征之一，这些主题是NPDES领域中最重要和最活跃的领域之一。在理论方面，我们希望在椭圆型方程的整体解的分类、几何化程序等基本问题上取得实质性进展，并在关键领域保持我们的优势；在应用方面，我们的研究将在数学生物学、材料科学、图像处理和社会科学方面探索新的前沿。我们的目标是在纯数学和应用数学的一些关键研究领域脱颖而出，特别是成为国际领先的应用分析和NPDES中心。

The proposal focuses on theoretical and interdisciplinary studies of concentration phenomena in Nonlinear Partial Differential Equations (NPDEs). Driven by reaction, diffusion and geometry, concentration phenomena appear as one of the central features in various phenomena in differential geometry (bubbling in Yamabe problem), mathematical biology (Spots and Stripes), chemistry and physics, These topics are one of the most important and active areas in the field of NPDEs. On the theoretical side, we expect to make substantial progress on fundamental problems such as classifying entire solutions and geometrization program of elliptic equations, and maintain our edge in key areas; on the applied side, our investigations will explore new frontiers in mathematical biology, material science, image processing and social sciences. We aim to excel in some of key research areas in pure and applied mathematics, in particular to become a leading centre for applied analysis and NPDEs internationally.