In this lecture, we mainly discuss some problems in global geometry and geometric analysis of submanifolds. In particular, we proved the following theorems: (1) a mean value theorem for critical points of height functions of isometric immersion from closed Riemannian manifolds to $R^{n+p}$, which gives new proofs of the Gauss-Bonnet-Chern theorem, the Chern-Lashof theorem and the Chen-Willmore inequality; (2) an optimal convergence theorem for higher codimensional mean curvature flow with positive Ricci curvature in hyperbolic space. Contrast to the famous sphere theorems due to Perelman and Cheeger-Colding, our result implies the first optimal differentiable sphere theorem for submanifolds with positive Ricci curvature; (3) a generalized Li-Yau inequality for minimal submanifolds in Cartan-Hadamard manifolds with arbitrary codimension, which provides a strong evidence for that the Generalized Pólya Conjecture is true. References: [1] J. R. Gu and H. W. Xu, The sphere theorems for manifolds with positive scalar curvature, J. Differential Geom. 92 (2012), no. 3, 507-545. [2] L. Lei and H. W. Xu, New developments in mean curvature flow of arbitrary codimension inspired by Yau rigidity theory, Adv. Lect. Math. (ALM), 43 International Press, Somerville, MA, 2019, 327-348. [3] K. Shiohama and H. W. Xu, The topological sphere theorem for complete submanifolds, Compositio Math. 107 (1997), no. 2, 221-232. [4] K. Shiohama and H. W. Xu, Lower bound for $L^{n/2}$ curvature norm and its application, J. Geom. Anal. 7 (1997), no. 3, 377-386. [5] K. Shiohama and H. W. Xu, An integral formula for Lipschitz-Killing curvature and the critical points for height functions, J. Geom. Anal. 21 (2011), no. 2, 241-251. [6] H. W. Xu and E. T. Zhao, Topological and differentiable sphere theorems for complete submanifolds, Comm. Anal. Geom. 17 (2009), no. 3, 565-585. [7] Z. Y. Xu and H. W. Xu, A generalization of Pólya conjecture and Li-Yau inequalities for higher eigenvalues, Calc. Var. Partial Differential Equations 59 (2020), no. 5, Paper No. 182, 19 pp.