Math Foundation Knowledge Base#

1. Logic & Axiomatic Foundations#

• Mathematical Logic, Set Theory, Axiomatic Systems
• Godel Incompleteness Theorems, Formal Proofs

2. Algebraic Structures#

• Groups, Rings, Fields, Modules, Category Theory
• Homological Algebra, Langlands Program

3. Geometry & Topology#

• Topology: continuity, homotopy, homology, manifolds
• Differential Geometry, Riemann Geometry
• Poincare Conjecture (solved by Perelman 2003)

4. Dynamics & Analysis#

• Real/Complex Analysis, Functional Analysis
• Differential Equations, Dynamical Systems, Calculus of Variations

5. Random & Statistics#

• Probability Theory, Stochastic Processes, Martingales
• Stochastic Differential Equations (SDE)
• Law of Large Numbers, Central Limit Theorem

6. Discrete & Computation#

• Combinatorics, Graph Theory, Number Theory
• Algorithm Complexity: P/NP/NP-Complete
• Cryptography: RSA, Elliptic Curve

Grand Unified Programs: Langlands Program#

Connects: Number Theory, Algebraic Geometry, Representation Theory, Analysis

Financial Mathematics Core#

• Black-Scholes: dS = mu dt + sigma dW
• Ito's Lemma, Girsanov Theorem
• Feynman-Kac Formula

Milestone Achievements#

• Poincare Conjecture (2003)
• Fermat's Last Theorem (1994)
• Green-Tao Theorem (2004)

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