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Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI | Lex Fridman Podcast #472
@ Lex Fridman
06/15/25
Takeaways
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Terence Tao
06/15/25
@ Lex Fridman
I'm often referred to as the Mozart of math, having won the Fields Medal and the Breakthrough Prize in mathematics, and contributed groundbreaking work across a wide range of fields in mathematics and physics.
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Terence Tao
06/15/25
@ Lex Fridman
In undergraduate education, you learn about really hard problems like the Riemann Hypothesis and the Twin Primes Conjecture, but what's interesting are the problems that exist on the boundary between solvable and unsolvable.
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Terence Tao
06/15/25
@ Lex Fridman
As a PhD student, the CA problem caught my eye, which I worked on a lot in my early research. It originated from a puzzle by the Japanese mathematician Soji Kato in 1918, involving a needle on a plane that needs to execute a U-turn using minimal area.
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Terence Tao
06/15/25
@ Lex Fridman
The Navier-Stokes equations govern fluid flow for incompressible fluids like water, and the Navier-Stokes regularity problem asks whether a smooth velocity field can ever concentrate to the point of becoming infinite, which would indicate a singularity.
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Terence Tao
06/15/25
@ Lex Fridman
In recent years, the consensus has shifted towards the belief that singularities can form under certain special initial configurations of water, although this has not yet been established.
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Terence Tao
06/15/25
@ Lex Fridman
The Clay Foundation offers a million-dollar prize for solving one of the seven Millennium Prize Problems, one of which is the Navier-Stokes problem, and only one of these problems has been solved so far.
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Terence Tao
06/15/25
@ Lex Fridman
Mathematicians care about whether something holds true 100% of the time, unlike most fields where 99.99% suffices. This is crucial in fluid dynamics, where we need to understand if there are special initial states that could lead to blow-up scenarios.
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Terence Tao
06/15/25
@ Lex Fridman
The Navier-Stokes equations are supercritical, meaning that at smaller scales, the transport terms dominate over the viscosity terms, leading to unpredictable and turbulent behavior.
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Terence Tao
06/15/25
@ Lex Fridman
The phenomenon of supercriticality versus criticality and subcriticality is a key qualitative feature that distinguishes equations that are predictable from those that are not, such as in fluid dynamics.
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Terence Tao
06/15/25
@ Lex Fridman
The Game of Life, a cellular automaton, demonstrates how simple local rules can lead to complex emergent behavior, similar to how fluid dynamics can exhibit complex structures under certain conditions.
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Terence Tao
06/15/25
@ Lex Fridman
Through engineering, we can create clever constructions that lead to interesting behaviors in fluid dynamics, similar to how glider guns and gates operate in the Game of Life.
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Terence Tao
06/15/25
@ Lex Fridman
In mathematics, we often start with hypotheses and explore their consequences, unlike other disciplines where conclusions drive the inquiry.
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Terence Tao
06/15/25
@ Lex Fridman
The interplay between theory and experiment is crucial in science, as observations can guide theoretical models and vice versa.
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Terence Tao
06/15/25
@ Lex Fridman
Mathematics has a theoretical and experimental component, with recent advancements allowing for more experimental approaches in the field.
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Terence Tao
06/15/25
@ Lex Fridman
The central limit theorem explains why many phenomena in nature follow a Gaussian distribution, demonstrating the concept of universality in mathematics.
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Terence Tao
06/15/25
@ Lex Fridman
Mathematics can help validate models by identifying weaknesses, such as systemic correlations in economic models that may lead to failures.
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Terence Tao
06/15/25
@ Lex Fridman
Mathematics is about finding connections between different fields, and this process has been ongoing throughout history, such as the unification of geometry and number theory.
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Terence Tao
06/15/25
@ Lex Fridman
The space of all proofs has a shape, and we can rank them based on different axes like elegance and length.
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Terence Tao
06/15/25
@ Lex Fridman
The beauty of mathematics often lies in its simplicity, as seen in elegant equations like Euler's identity, which connects fundamental constants.
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Terence Tao
06/15/25
@ Lex Fridman
The exponential function models growth and decay, while pi relates to rotation. The equation e^(iπ) + 1 = 0 connects these concepts, unifying geometry and dynamics.
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Terence Tao
06/15/25
@ Lex Fridman
The Hamiltonian is a central concept in both classical and quantum mechanics, illustrating how perspectives in physics can evolve over time.
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Terence Tao
06/15/25
@ Lex Fridman
The laws of physics are translation invariant, meaning if I move 10 steps to the left, I experience the same laws of physics as if I was here, which corresponds to the conservation of momentum.
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Terence Tao
06/15/25
@ Lex Fridman
There's a fundamental connection between symmetry and conservation laws in physics, which holds true in both classical and quantum mechanics.
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Terence Tao
06/15/25
@ Lex Fridman
We haven't unified quantum mechanics and general relativity yet because we don't know what the fundamental objects are, and we need to rethink our notions of space and time.
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Terence Tao
06/15/25
@ Lex Fridman
I believe it's possible to unify general relativity and quantum mechanics, as history shows that physics has often moved towards unification, like how Maxwell unified electricity and magnetism.
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Terence Tao
06/15/25
@ Lex Fridman
Physics is a victim of its own success; general relativity and quantum mechanics are so effective that they cover 99.9% of observations, making it hard to find deviations to combine them.
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Terence Tao
06/15/25
@ Lex Fridman
When physicists need mathematics, they often rely on precursors developed by mathematicians, as seen when Einstein sought help from mathematicians for curved space.
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Terence Tao
06/15/25
@ Lex Fridman
The theories that explain the universe often involve the same mathematical objects that solve mathematical problems, indicating a deep connection between the two fields.
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Terence Tao
06/15/25
@ Lex Fridman
Mathematics allows you to change the rules of the problem, which is not possible in other fields like engineering, where fixed requirements exist.
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Terence Tao
06/15/25
@ Lex Fridman
To solve a difficult math problem, first find a version that simplifies the problem by turning off most difficulties, focusing on just one at a time.
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Terence Tao
06/15/25
@ Lex Fridman
Lean programming enables collaboration on proofs at an atomic level, allowing mathematicians to work together more effectively than with traditional pen and paper methods.
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Terence Tao
06/15/25
@ Lex Fridman
In collaboration, there's often a brainstorming process where unrealistic assumptions are made to explore potential solutions before breaking the problem into smaller sub-problems.
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Terence Tao
06/15/25
@ Lex Fridman
With lean programming, you can create a blueprint for a problem, allowing for divide and conquer approaches where separate parts can be worked on independently.
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Lex Fridman Cast
06/15/25
@ Lex Fridman
Multiplication is commutative, meaning X * Y is always equal to Y * X, and it is also associative, so X * Y * Z equals X * (Y * Z). However, not all operations obey the same laws; for instance, X * X is not always equal to X.
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Lex Fridman Cast
06/15/25
@ Lex Fridman
We generated about 4,000 possible laws of algebra that certain operations can satisfy, and we investigate which laws imply others. For example, commutativity does not imply associativity, as there are operations that obey commutativity but not associativity.
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Lex Fridman Cast
06/15/25
@ Lex Fridman
The project involves analyzing 22 million pairs of algebraic laws to determine if one law implies another, providing proofs or counterexamples as necessary. While many problems are solvable by undergraduate students, a few are quite challenging.
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Lex Fridman Cast
06/15/25
@ Lex Fridman
The project required scaling up efforts, as no single person could verify 22 million proofs. It became feasible with the use of lean programming and AI assistance, and we are close to completing the project, having settled all but two problems.
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Lex Fridman Cast
06/15/25
@ Lex Fridman
We had about 50 contributors to this project, which is significant in mathematics, and we are documenting contributions in a matrix format to recognize everyone's input.
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Lex Fridman Cast
06/15/25
@ Lex Fridman
The beauty of lean projects is that they automatically generate data, allowing us to track contributions and assess performance without relying solely on traditional metrics.
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Lex Fridman Cast
06/15/25
@ Lex Fridman
AI systems like DeepMind's AlphaProof are trained using reinforcement learning on both successful and failed formal proofs, demonstrating the potential for AI to assist in solving high-level mathematical problems.
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Lex Fridman Cast
06/15/25
@ Lex Fridman
The difficulty of proving mathematical problems increases exponentially with the number of steps involved in the proof, making it challenging for AI systems to maintain accuracy.
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Lex Fridman Cast
06/15/25
@ Lex Fridman
Mapping natural language to formal programming languages is extremely difficult, as formal languages require precise syntax, while natural language is more fault-tolerant.
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Lex Fridman Cast
06/15/25
@ Lex Fridman
AI struggles with generating long-form proofs because it often produces outputs that appear correct but contain subtle errors that are difficult to detect.
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Lex Fridman Cast
06/15/25
@ Lex Fridman
Humans possess a 'mathematical smell' that helps them identify flawed proofs, a quality that AI has yet to replicate effectively.
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Lex Fridman Cast
06/15/25
@ Lex Fridman
If AI can develop a sense of viability for proof strategies, it could potentially collaborate with human mathematicians, enhancing the problem-solving process.
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Lex Fridman Cast
06/15/25
@ Lex Fridman
AI could assist in generating conjectures by identifying connections between seemingly unrelated mathematical concepts, which is a significant aspect of mathematical discovery.
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Lex Fridman Cast
06/15/25
@ Lex Fridman
The point conjecture, a significant problem in mathematics, involves determining whether a three-dimensional space with certain properties can be continuously deformed into a sphere.
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Lex Fridman Cast
06/15/25
@ Lex Fridman
The Navier-Stokes equations are sophisticated nonlinear equations that are challenging to solve, particularly in three dimensions where singularities can form.
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Lex Fridman Cast
06/15/25
@ Lex Fridman
Grigori Perelman transformed the supercritical Navier-Stokes problem into a critical problem by introducing concepts like reduced volume and permanence entropy, which simplified the analysis of singularities.
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Lex Fridman Cast
06/15/25
@ Lex Fridman
When faced with a problem that seems insurmountable, I often switch to a different problem or modify the existing one to find a new approach.
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Lex Fridman Cast
06/15/25
@ Lex Fridman
It's important to recognize when a problem is blocking progress; if it's the only issue, it's worth fighting through, but if there are multiple problems, it may be time to give up.
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Lex Fridman Cast
06/15/25
@ Lex Fridman
Mistakes can be productive; sometimes, the initial false dawn of a solution can lead to breakthroughs after much effort, as was the case in a project I worked on for two years.
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Lex Fridman Cast
06/15/25
@ Lex Fridman
The twin prime conjecture posits that there are infinitely many pairs of prime numbers that differ by two, but proving it remains elusive due to the sparsity of twin primes as numbers increase.
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Lex Fridman Cast
06/15/25
@ Lex Fridman
The twin prime conjecture is difficult because while there are infinitely many primes, twin primes become increasingly rare as numbers grow larger, complicating proof strategies.
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Lex Fridman Cast
06/15/25
@ Lex Fridman
The parity barrier in number theory limits the density of primes, making it a significant obstacle in proving conjectures like the twin prime conjecture.
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Lex Fridman Cast
06/15/25
@ Lex Fridman
The Riemann hypothesis, which concerns the distribution of prime numbers, is a fundamental problem in mathematics that remains unsolved, with significant implications for number theory.
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Lex Fridman Cast
06/15/25
@ Lex Fridman
The Collatz conjecture, which involves iterating a simple process on natural numbers, is easy to state but remains unsolved, illustrating the complexity of seemingly simple mathematical problems.
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Terence Tao
06/15/25
@ Lex Fridman
Cryptography relies heavily on number theory, particularly on the behavior of prime numbers, to create encryption methods that transform readable text into something indistinguishable from random noise, making it nearly impossible to crack mathematically.
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Terence Tao
06/15/25
@ Lex Fridman
If our core belief about the randomness of prime numbers is proven wrong, it could reveal patterns that undermine many cryptographic systems, leading to significant doubts about their security.
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Terence Tao
06/15/25
@ Lex Fridman
The P vs NP problem is technically possible to solve, but evidence suggests it is unlikely that P equals NP, similar to the Riemann Hypothesis, where the evidence leans heavily towards 'no'.
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Terence Tao
06/15/25
@ Lex Fridman
In mathematics, there are many obstructions to solving problems, and computer scientists have been effective in ruling out certain approaches through no-go theorems, indicating that some problems may be undecidable.
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Terence Tao
06/15/25
@ Lex Fridman
Winning the Fields Medal can make a mathematician part of the establishment, leading to increased expectations and responsibilities, which can constrain their freedom to explore and innovate.
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Terence Tao
06/15/25
@ Lex Fridman
While the Fields Medal can inspire young mathematicians, it's important to respect those like Grigori Perelman, who prioritize their principles over recognition and awards.
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Terence Tao
06/15/25
@ Lex Fridman
The way we educate mathematics often fails to accommodate different learning styles, leading many to abandon the subject before finding their unique approach.
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Terence Tao
06/15/25
@ Lex Fridman
Students struggling with math should seek enrichment opportunities outside the classroom, such as math competitions and online resources, to foster their interest and skills.
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Terence Tao
06/15/25
@ Lex Fridman
The mathematical community is a collective intelligence that surpasses any individual mathematician, as seen in collaborative platforms like MathOverflow, where diverse expertise contributes to solving complex problems.
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Terence Tao
06/15/25
@ Lex Fridman
Human civilization's progress shows that problems once deemed difficult can become trivial over time, highlighting the potential for future advancements that we may not yet envision.
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Terence Tao
06/15/25
@ Lex Fridman
The future of mathematics and technology is promising, with the potential for new discoveries and innovations that could transform our understanding and capabilities.