Mathematical Roadmap for Self-Study

📍 Stage 1: Foundations (Arithmetic → Algebra → Geometry)

✅ Arithmetic

• Study order: number systems → operations → fractions → decimals → percentages → ratios.

• Key theorems & rules:

  • Fundamental Theorem of Arithmetic (unique prime factorization).

  • Euclidean Algorithm (greatest common divisor).

  • Basic divisibility rules.

✅ Algebra

• Study order: variables → linear equations → inequalities → quadratic equations → functions → polynomials.

• Key theorems:

  • Zero Product Property.

  • Quadratic Formula.

  • Factor Theorem & Remainder Theorem.

  • Fundamental Theorem of Algebra (every nonconstant polynomial has a complex root).

✅ Geometry

• Study order: points & lines → angles → triangles → circles → congruence → similarity → area & volume.

• Key theorems:

  • Pythagorean Theorem.

  • Triangle Sum Theorem.

  • Basic Properties of Parallel Lines & Angles.

  • Euclid’s Elements (propositions on congruence, similarity).

📍 Stage 2: Intermediate Core (Linear Algebra → Calculus → Discrete Math)

✅ Linear Algebra

• Study order: vectors → matrices → systems of equations → determinants → vector spaces → eigenvalues/eigenvectors.

• Key theorems:

  • Rank-Nullity Theorem.

  • Cayley-Hamilton Theorem.

  • Spectral Theorem.

  • Linear Independence and Basis.

✅ Calculus

• Study order: limits → derivatives → integrals → series → multivariable calculus → vector calculus.

• Key theorems:

  • Intermediate Value Theorem.

  • Mean Value Theorem.

  • Fundamental Theorem of Calculus.

  • Green’s, Stokes’, and Divergence Theorems (advanced).

✅ Discrete Mathematics & Number Theory

• Study order: combinatorics → graph theory → modular arithmetic → prime numbers → Diophantine equations.

• Key theorems:

  • Fermat’s Little Theorem.

  • Chinese Remainder Theorem.

  • Euler’s Totient Theorem.

  • Pigeonhole Principle.

  • Principle of Inclusion-Exclusion.

📍 Stage 3: Advanced Core (Real/Complex Analysis → Abstract Algebra → Topology)

✅ Real Analysis

• Study order: sequences → limits → continuity → differentiability → integrability → series convergence.

• Key theorems:

  • Bolzano-Weierstrass Theorem.

  • Heine-Borel Theorem.

  • Mean Value Theorem (analysis form).

  • Lebesgue Dominated Convergence Theorem (advanced).

✅ Complex Analysis

• Study order: complex numbers → analytic functions → Cauchy-Riemann equations → contour integration → residues.

• Key theorems:

  • Cauchy Integral Theorem.

  • Cauchy Integral Formula.

  • Liouville’s Theorem.

  • Residue Theorem.

  • Maximum Modulus Principle.

✅ Abstract Algebra

• Study order: groups → subgroups → rings → fields → homomorphisms → isomorphisms.

• Key theorems:

  • Lagrange’s Theorem.

  • Isomorphism Theorems (for groups and rings).

  • Fundamental Theorem of Finite Abelian Groups.

  • Sylow Theorems.

✅ Topology

• Study order: open/closed sets → continuous maps → compactness → connectedness → metric/topological spaces.

• Key theorems:

  • Urysohn’s Lemma.

  • Tychonoff’s Theorem.

  • Brouwer Fixed Point Theorem.

  • Fundamental Group and Covering Spaces.

📍 Stage 4: Specialization Tracks (choose based on interest)

✅ Differential Geometry

• Gauss-Bonnet Theorem.

• Theorema Egregium.

✅ Algebraic Geometry

• Hilbert’s Nullstellensatz.

• Bezout’s Theorem.

✅ Functional Analysis

• Hahn-Banach Theorem.

• Banach Fixed Point Theorem.

• Open Mapping Theorem.

✅ Number Theory (advanced)

• Dirichlet’s Theorem on Primes.

• Prime Number Theorem.

✅ Category Theory

• Yoneda Lemma.

• Adjunctions.

🔑 Summary of Progression:

1. Build arithmetic & algebra fluency.

2. Strengthen geometric and analytic reasoning.

3. Advance into calculus, linear algebra, and number theory.

4. Go into rigorous real/complex analysis and abstract algebra.

5. Choose a specialization for research or application.