Advanced Mathematics Study Roadmap
Stage 1: Core Foundations
- Calculus I - Limits, derivatives, basic integrals
- Calculus II - Techniques of integration, sequences and series
- Calculus III (Multivariable Calculus) - Partial derivatives, double/triple integrals
- Linear Algebra - Vectors, matrices, systems of equations, eigenvalues
- Ordinary Differential Equations (ODEs) - First-order and second-order equations
Stage 2: Formal Thinking and Proofs
- Introduction to Proofs / Discrete Mathematics - Logic, set theory, induction, functions
- Real Analysis I - Rigorous foundations of calculus, limits, continuity
- Abstract Algebra I - Groups, rings, fields
- Complex Analysis - Complex functions, residues, contour integrals
Stage 3: Intermediate Structures and Applications
- Real Analysis II - Sequences of functions, metric spaces, convergence
- Abstract Algebra II - Field extensions, Galois theory
- Topology - Open sets, continuity, homeomorphisms
- Numerical Analysis - Algorithms for solving mathematical problems
- Partial Differential Equations (PDEs) - Heat, wave, and Laplace equations
Advanced Mathematics Study Roadmap
- Probability and Statistics (Advanced) - Measure theory, stochastic processes
Stage 4: Advanced Topics / Specializations
- Functional Analysis - Infinite-dimensional vector spaces, Banach and Hilbert spaces
- Differential Geometry - Curves, surfaces, manifolds
- Measure Theory - Lebesgue integration, sigma-algebras
- Category Theory - Foundations and abstract structures across math
- Algebraic Topology / Algebraic Geometry / Number Theory - Depending on interests
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