9:30-10:30 July 6, 2020 Li-Yau inequality and Gaussian estimate for the heat kernel on graphs. 刘双（中国人民大学）
Tencent Meeting ID：954 492 804, Code：24680
Abstract: Studying the heat semigroup, we prove Li-Yau inequality for bounded and positive solutions of the heat equation on graphs. These are proved under the assumption of the curvature-dimension inequality CDE’(n,0), which can be considered as a notion of curvature for graphs. We further show that non-negatively curved graphs also satisfy the volume doubling property. From this we prove a Gaussian estimate for the heat kernel, along with Poincaré and Harnack inequalities.
8:30-9:30 June 21, 2020 The heat equation on graphs: unbounded Laplacians, stochastic completeness, intrinsic metrics and volume growth. Radoslaw Wojciechowski（CUNY）
Zoom Meeting ID： 666 263 17658 Code: 246802
Abstract: We will discuss the existence and uniqueness (aka stochastic completeness) of solutions of the heat equation on infinite graphs. This will require us to introduce the case of unbounded Laplacians. Furthermore, in order to get volume growth criteria for stochastic completeness which are analogous to the manifold setting, we will need to introduce the concept of an intrinsic metric.
9:30-10:30 June 14, 2020 Finding the global graph partition via convex relaxation: spectral clustering and community detection. 凌舒扬（上海纽约大学）
Tencent Meeting ID：500 106 721, Code：24680
Abstract: Spectral clustering has become one of the most widely used clustering techniques when the structure of the individual clusters is non-convex or highly anisotropic. Yet, despite its immense popularity, there exists fairly little theory about performance guarantees for spectral clustering. This issue is partly due to the fact that spectral clustering typically involves two steps which complicated its theoretical analysis: first, the eigenvectors of the associated graph Laplacian are used to embed the dataset, and second, k-means clustering algorithm is applied to the embedded dataset to get the labels. This paper is devoted to the theoretical foundations of spectral clustering and graph cuts. We consider a convex relaxation of graph cuts, namely ratio cuts and normalized cuts, that makes the usual two-step approach of spectral clustering obsolete and at the same time gives rise to a rigorous theoretical analysis of graph cuts and spectral clustering. We derive deterministic bounds for successful spectral clustering via a spectral proximity condition that naturally depends on the algebraic connectivity of each cluster and the inter-cluster connectivity. Moreover, we demonstrate by means of some popular examples that our bounds can achieve near-optimality. Our findings are also fundamental to the theoretical understanding of kernel k-means. Numerical simulations confirm and complement our analysis.
March 05-18, 2017 David Cushing and Norbert Peyerimhoff from Durham University visit and talk in 3-10国家数学与交叉科学中心合肥分中心报告 and 3-8国家数学与交叉科学中心合肥分中心报告 with titles "Bakry-\'Emery curvature functions of graphs" and "Sectional curvature of polygonal complexes with planar substructures".
December 29-30, 2016 Peter L. Guo (郭龙) from Nankai University (南开大学) visits and talks in GAP Seminar on "Some combinatorial structures involving Young tableaux "