With background from last semester, we will explore different aspects of Geometry and introduce basic notions and ideas to various directions. The class will emphasize on students’ independent studies and presentations. We will discuss vector bundles, connections in vector bundles, the moving frame method, De Rham Cohomology and Harmonic Differential Forms, Yang-Mill Functional and Yang-Mill equations, Chern Classes, the covariant derivatives of tensors and the rules on exchanging order of derivatives, the Laplacian of 2nd fundamental form (in general co-dimension and manifolds), Simon’s identity and its applications, the 1st and 2nd variation formula of area (also the restricted case that encloses fixed volume), minimal surfaces, some applications of the 2nd variation formula of area, spin and spin^c structures, Dirac operator and Weitzenbock formulas, the proof of Toponogov Theorem, properties of Killing vector fields, the Bochner method, Symmetric Spaces, various notions of convergence for Riemannian manifolds. At suitable stages of the class, we will also briefly introduce some important developments in Geometry such as Donaldson theory, Seiberg-Witten equations, positive mass theorems, Chern-Simon forms, Morse Theory and Floer Homology, Geometric Analysis, Curvature flow and the resolve of Poincare conjecture.