I will shortly list a series of topics that amount to what is essentially the first two years of an undergraduate degree.
I'd like to know what is considered best order in which to study these topics. I understand that some of these are related topics i.e. Analysis, so if like topics could be grouped together that would be appreciated. Any other suggestions that you feel are relevant would be great too.
1. Groups: Examples of groups, Lagrange's theorem, Group actions, Quotient groups, Matrix groups, Permutations.
2. Vectors and Matrices: Complex numbers, Vectors, Matrices, Eigenvalues and Eigenvectors.
3. Numbers and Sets: Introduction to number systems and logic, Sets relations and functions, the integers, elementary number theory, the real numbers, countability and uncountability.
4. Differential Equations: Basic calculus, first-order linear differential equations, nonlinear first-order equations, higher-order linear differential equations, multivariate functions: applications.
5. Analysis I: Limits and convergence, continuity, differentiability, power series, integration.
6. Probability: Basic concepts, axiomatic approach, discrete random variables, continuous random variables, inequalities and limits.
7. Vector Calculus: Curves in $R^3$, integration in $R^2$ and $R^3$, vector operators, integration theorems, Laplace's equation, Cartesian tensors in $R^3$.
8. Linear Algebra: self explanatory
9. Groups, Rings & Modules: self explanatory
10. Analysis II: Uniform convergence, uniform continuity and integration, $R^n$ as normed spaced, differentiation from $R^m$ to $R^n$, metric spaces, the Contraction Mapping theorem
11. Metric & Topological Spaces: Metrics, topology, connectedness, compactness
12. Complex Analysis: Analytic functions, contour integration and Cauchy's theorem, expanions and singularities, the residue theorem.
13. Complex Methods: Analytic functions, contour integration and Cauchy's theorem, Residue calculus, Fourier and Laplace transforms.
14. Geometry: Groups of rigid motions of Euclidean space, spherical geometry, Riemannian metrics, embedded surfaces in $R^3$, length and energy, the second fundamental form and Gaussian curvature.
15. Variational Principles: Stationary points for functions $R^n$, Functional derivatives, Fermat's principle, second variation for functionals.
16. Methods: Self-adjoint ODEs, PDEs on bounded domains, Inhomogenous ODEs, Fourier transforms.
17. Numerical Analysis: Polynomial approximation, computation of ordinary differential equations, systems of equations and least squares calculations.
18. Statistics: Estimation, hypothesis testing, linear models.