Complex Numbers and Algebraic Completeness

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Will AI solve future math proofs?

Every time you type a password, buy something online, or send an encrypted message, you’re trusting an assumption about prime numbers: that they don’t hide an exploitable pattern. Modern cryptography depends on primes behaving “randomly enough,” yet many fundamental questions about primes remain unproven.

In this episode, I’m joined by Fields Medalist Terence Tao to explore what mathematicians can prove, what they strongly suspect, and what could change if unexpected structure appeared in the primes. We discuss pseudorandomness and why it matters for encryption, the twin prime conjecture, and how quantum computing reshapes what is feasible in computation and security.

We also get into AI and mathematics: why large language models can sound convincing even when unreliable, how AI can help with idea generation and literature recall, and why verification and proof assistants will matter if AI is to contribute to real mathematical progress. Along the way, Tao explains proof techniques like proof by contradiction, why complex numbers and the square root of minus one are so central, and how high-dimensional geometry breaks low-dimensional intuition.

Finally, Tao shares a real-world example of how math breakthroughs translate into technology: compressed sensing, which has enabled much faster MRI scans by reconstructing images from far less data.

Timestamps:

00:00 Discrepancy Theory Explained

09:35 Induction: Science and Mathematics

14:38 "Proof Concept Through Play"

18:43 "Complex Numbers and Completeness"

25:36 Prime Numbers and Cryptography

29:51 "Computability and Complexity in Mathematics"

35:04 AI Discovers New Knot Theory Insights

42:02 "Elegance in Nature's Laws"

47:12 "AI as Complementary Research Tools"

51:27 "Humility in Pursuit of Proofs"

54:46 "Multiple Approaches to Mathematics"

01:03:04 Rethinking Reality and Physics

01:07:14 "Origins of Compressed Sensing"

01:10:17 "Shannon Bound and Information Limits"

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