自上个世纪80年代以来，几何中的概念和研究方法逐渐被推广到图上，促进了图在理论上和应用上的快速发展。近二十年来，随着曲率概念在离散空间上的推广，图上的研究得到了新的突破。曲率是一个局部概念，曲率下界具有分析的作用，能推导出空间的整体几何性质。本项目将研究以下两个主要课题。课题一：图上的泛函不等式。Sobolev型不等式是估计热核和特征值的重要工具，并且有助于研究图上偏微分方程。等周不等式揭示了空间的重要几何性质，并且与Sobolev型不等式关系密切。本课题的主要目标是在非负曲率图上得到等周不等式和相应热核估计；课题二：有向图上的Ricci曲率和相关结论。无向图在物理、化学与社会学中有着广泛应用，但是在生物学中，比如神经元网络，有向图是更为准确的数学模型。近期，曲率概念也被引入有向图上。本课题的目标是得到非负曲率有向图上的特征值估计和研究有向图上的超压缩性质。

Since the 1980s, the concepts and methods from geometry have been introduced to graphs, which has brought the rapid development of graphs in theory and application. With the popularization of curvature on graphs, the research has achieved a new breakthrough in the last twenty years. Curvature is a local concept, its lower bound can be used to derive the global geometric properties. In this project, we will study two main topics. One is functional inequalities. Sobolev type inequalities are important tools for estimating heat kernel and eigenvalues, and are helpful for studying PDEs on graphs. Isoperimetric inequality reveals the important geometric properties of spaces. These two inequalities are closely related. The main goal of this project is to obtain isoperimetric inequality and heat kernel estimate on non-negatively curved graphs. The other is Ricci curvature and the relevant results on directed graphs. Undirected graphs are widely used in Physics, Chemistry and Sociology. However, in Biology, such as neural networks, directed graphs are more accurate mathematical models. Recently, the concept of curvature has also been introduced in directed graphs. The goal of this project is to obtain eigenvalue estimates under the assumption of nonnegative curvature and study ultracontractivity on directied graphs.