# AI Crisis in Math: Navier-Stokes, Riemann Hypothesis & Complexity Theory | Lex Fridman Podcast #503

## Executive summary

The discussion analyzes the current 'crisis moment' in mathematics, triggered by AI systems solving historically difficult problems like the Navier-Stokes equations, the quasi-Riemann hypothesis (quasi-RH), and the unique games conjecture. Experts debate whether this represents the end of human mathematical endeavor or a new 'golden age.' Key themes include the shift toward formal, machine-checkable proofs using tools like Lean and Mathlib, the critical distinction between pure mathematical insight (the 'why') and mere problem-solving (the 'what'), and the profound implications for fields like fluid dynamics and cryptography.

## Key takeaways

- AI's Impact on Mathematical Proofs: AI systems are capable of solving problems that were previously thought to require decades of human effort. The formalization of Fermat's Last Theorem (FLT) into 13 million lines of Lean code demonstrated the power of autoformalization, moving mathematics toward a 'crowdsourced' model where proofs are built on verifiable, general libraries (Mathlib).
- The Nature of Progress: Semigroup vs. Convex Hull: Experts debate if AI progress is merely interpolating known human knowledge (within a 'convex hull') or if it is genuinely extrapolating by combining existing ideas in novel ways (a 'semigroup' action). The consensus is that human insight—asking the right questions—remains crucial for defining new boundaries.
- Fluid Dynamics and Navier-Stokes: The Navier-Stokes equations, which model the conservation of momentum for fluids, are fundamental to engineering. The difficulty lies not just in writing the equations (which is simple) but in solving them and predicting complex phenomena like turbulence, which requires advanced simulation and understanding of physical boundaries (e.g., molecular scale).
- The Role of Human Insight: The value of a mathematician shifts from being the sole solver of theorems to being the conceptual architect. The most valuable human contributions are seen as building new theories, making new definitions, and asking the right, guiding questions, rather than just producing a final result.

## Technical details

- Navier-Stokes Equations: A partial differential equation describing how the velocity field of a fluid changes in space and time, based on the conservation of momentum (F=ma). The challenge lies in predicting complex phenomena like turbulence and determining if the equations admit solutions that blow up in finite time.
- Riemann Hypothesis (RH) / Quasi-RH: The RH concerns the distribution of prime numbers, specifically predicting that the non-trivial zeros of the Riemann zeta function lie on the line where the real part equals 1/2. The recent proof of the quasi-RH established a vertical strip where no zeros exist, representing a major step toward the full hypothesis.
- Formal Verification and Mathlib/Lean: Lean is an interactive theorem prover, and Mathlib is the associated library for formalizing mathematics. This process allows mathematical results to be checked down to the axioms, ensuring 'sorry-free' proofs and enabling a systematic, reproducible, and crowdsourced approach to mathematics.
- Complexity Theory: The discussion touches on the 'solved axiom': the hypothesis that any problem solvable by human interaction with AI could also be solved by AI alone. Experts debate the validity of this axiom, arguing that human value lies in the exploratory process and conceptual leaps.

## Practical implications

- **Fluid Dynamics/Aerospace:** Improved ability to predict and simulate turbulence and perform inverse design (e.g., optimizing wing shapes) will lead to massive efficiency gains in transport and energy sectors.
- **Cryptography:** The difficulty of factoring large numbers (the basis of modern public-key encryption) highlights the urgent need for developing post-quantum cryptographic bases (e.g., lattice-based methods).
- **Software/Research:** The shift toward formal verification (Lean/Mathlib) suggests that future academic and engineering research will increasingly require code-based, verifiable specifications rather than solely relying on natural language proofs.
- **AI Development:** The discussion emphasizes that the most valuable AI applications may involve synthesizing knowledge across disparate fields (e.g., combining theorems from number theory and fluid dynamics) rather than solving single, isolated problems.

## Topics

Theoretical Computer Science, Number Theory, Fluid Dynamics, Artificial Intelligence, Formal Methods, Complexity Theory, Lean, Mathlib, Navier-Stokes Equations

Source: https://www.youtube.com/watch?v=ZAzWPE9CItg