Notes

• Notes on Heat Kernel Estimates in Ricci Flow

These notes discuss heat kernel estimates in the setting of Ricci flow, with an emphasis on analytic techniques and applications in geometric analysis. They cover heat equation methods, comparison arguments, and related tools for studying curvature and geometric flows. The notes are intended as supplementary material for students and researchers interested in Ricci flow, heat kernel methods, and geometric analysis. Comments and corrections are welcome.

• Riemannian Geometry

These notes are based on the graduate course Riemannian Geometry taught at the University of Science and Technology of China in Spring 2025. They cover Riemannian metrics, Levi-Civita connections, geodesics, exponential maps, curvature, Jacobi fields, comparison geometry, submanifold geometry, and classical global results such as the Hopf-Rinow, Cartan-Hadamard, and Bonnet-Myers theorems. The notes were taken by Wenxin He and Yichen Yao, and have been edited for readability and consistency. They are intended as introductory material and may still contain typos or inaccuracies. Comments and corrections are welcome.

• Geometric Analysis

These notes are based on the graduate course Geometric Analysis taught at the University of Science and Technology of China in Spring 2026. They cover analytic tools in Riemannian geometry, curvature identities, Hodge theory and Weitzenbock formulas, variational and conformal methods including the Einstein-Hilbert functional and the Yamabe problem, minimal hypersurfaces, comparison geometry, Sobolev inequalities, the heat equation and heat kernel estimates on Riemannian manifolds, and basic topics in Cheeger-Colding theory such as Gromov-Hausdorff convergence, almost splitting, and volume convergence. The notes were compiled from course materials taken by Yichen Yao. Comments and corrections are welcome.