Linear Algebra and Topology: review of vector space, subspace, linear operators, image, kernels, dual vector space, inner product, tensors, topological space, continuous map, neighborhood, connectedness, compactness, homeomorphism and topological invariants, Euler characteristics; Homology Group: definition of group, Abelian and non-Abelian groups, cyclic group, simplexes and simplicial complexes, oriented simplexes, chain, cycle, and boundary group, homology group, general properties, Betti numbers, Euler-Poincare theorem; Homotopy Group: paths, loops, homotopy, fundamental group, general properties, examples, fundamental groups of polyhedral, higher homotopy group, example in physics; Manifolds: definition, example, differentiable maps, vectors, oneforms, tensors, flows and Lie derivatives, differential forms, exterior derivative, integration, Lie group and Lie algebra, orbits, isotropy group, adjoint representation; de Rham Cohomology Groups: Stokes theorem, definition of de Rham cohjomology group, duality, Poincare Lemma, Poincare duality, Kunneth formula; Riemannian Geometry: Riemannian and pseudo-Riemannian manifolds, metric tensor, parallel transport, connection, covariant derivative, the metric connection, the Riemann tensor, Ricci tensor and curvature, LeviCivita connection, holonomy, isometries, Killing vectors, conformal Killing vectors, non-coordinate basis, Cartan’s structure equation, differential forms and Hodge theory, harmonic form and de Rham cohomology group; Complex Manifolds: definition, holomorphic maps, complexifications, almost complex structures, complex differential forms, Hermitian metric, Kahler form, Kahler manifold, Kahler differential geometry, Harmonic form, almost complex manifold, orbifolds; Fiber Bundles: tangent bundles, definition of fiber bundles, bundle maps, pullback, vector bundles, sections, cotangent and dual bundles, principal bundles, connection on principal bundles, holonomy, curvature, Ambrose-Singer theorem, covariant derivative on associated bundles, gauge theories; Characteristic Class: Chern-Weil homomorphism, invariant polynomials, Chern classes, properties, Chern character, Pontrjagin and Euler classes, Chern-Simons forms, Stiefel-Whitney classes.