Differential Geometry Topics: Tensors, forms, vector fields Flows of Vector Fields, Flow Coordinates Derivatives, Lie Derivatives, Jacobian Lie Bracket Tangent Vectors & the Tangent Bundle Vector Calculus (all classical theorems) Coordinate Charts, Parameterizations, Atlas Embedded Submanifolds Regular and Critical Values of a function Lagrange Multipliers Volume/Area forms Pullback, Push-forward Stereographic Projection (circle, sphere, hyper-sphere) Contraction of forms with Vector Fields Closed Forms Exact Forms Cartan’s Formula Change of Coordinates 1-Parameter groups of Diffeomorphisms Partition of Unity Theorems: Regular Value Theorem Rank Theorem Implicit Function Theorem Inverse Function Theorem Stokes’ Theorem (classical and general) Green’s Theorem Divergence Theorem Algebraic Topology Topics: Classification of Compact Surfaces Euler Characteristic Connect Sum Homology, Cohomology Groups Fundamental Group Singular/Cellular/Simplicial Homology Mayer-Vietoris Long Exact Sequence Diagram Chasing Degree of Maps from S^n to S^n Orientability, Compactness Top Level (Co) Homology Reduced (Co) Homology Relative Homology Homotopy (Invariance) Deformation Retract Retract Excision Kunneth Formula Factoring Maps Fundamental Theorem of Algebra Theorems: Brower Fixed Point Theorem Poincare’ Lemma Poincare’ Duality deRham Theorem Van-Kampen Theorem Covering Space Theory Topics: Covering maps Free Actions Proper Discontinuous Action Universal Cover Correspondence Between Covering Spaces and Subgroups of the Fundamental Group of the Base. Lifting Paths Homotopy Lifting Property Deck Transformations The Action of the Fundamental Group Normal/Regular Cover Complex Analysis Topics: Contour Integration Conformal Maps Conformal Equivalence Meromorphic, Holomorphic, Analytic Linear Fractional Trasformations Radii of Convergence of Power Series Poles Singularities, and removable Singularities Residues Cauchy Riemann Equations Laurent Series Trig Functions Complex Numbers (arithmetic, computation, polar,…) Theorems: Louville’s Theorem Cauchy’s Residue Theorem Jordan’s Lemma M-L Estimate Maximum Modulus Principle