Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI | Lex Fridman Podcast #472

Terence Tao discusses the Kakeya problem, a geometric puzzle about rotating a needle in minimal space, and its surprising connections to partial differential equations, number theory, geometry, and combinatorics. He explains how insights from the Kakeya problem relate to understanding wave propagation and the potential for singularities in physical systems. This leads to a detailed exploration of the Navier-Stokes equations, which govern fluid flow, and the million-dollar Millennium Prize Problem concerning their existence and smoothness. Tao explains the concept of 'finite time blowup' and how his work on an averaged Navier-Stokes equation, which he engineered to blow up, helps mathematicians understand why current proof approaches for the original equations might fail. He likens his engineering of a blowup to constructing an electronic circuit, showing how certain assumptions about energy transfer can lead to blowup.

Tao elaborates on the concept of 'supercriticality' in partial differential equations, explaining that in Navier-Stokes, at small scales, transport forces (nonlinearity) are much stronger than dissipation (viscosity), making the equations difficult to control and predict, unlike 'critical' or 'subcritical' equations. He then presents a highly imaginative idea: using fluid dynamics to build a 'liquid computer' or 'fluid robot' that could perform computations. Inspired by Conway's Game of Life, Tao suggests that if fluid mechanics could support such computational structures, it might offer a roadmap for demonstrating a blowup in the actual Navier-Stokes equations. He acknowledges the practical difficulties, comparing it to analog computing and error correction, but emphasizes its theoretical possibility.

Tao discusses the dichotomy between structure and randomness in mathematics, noting that while most mathematically generated objects appear random, identifying those with patterns allows for focused analysis. He introduces 'inverse theorems' that test for structure, creating a dichotomy where objects are either random or related to something structured. He illustrates this with Szemeredi's Theorem on arithmetic progressions in sufficiently dense sets of numbers, contrasting structured sets (like odd numbers) with random ones, and highlights the intuitive but often counterintuitive aspects of infinity with the 'infinite monkey theorem.' He explains how mathematical formalisms idealize concepts like infinity, simplifying analysis but also introducing pitfalls that require careful reasoning, such as in the reordering of infinite series.

Tao distinguishes between mathematics, physics, and engineering by their relationship to models, observations, and reality. He positions mathematics as concerned with the internal consistency and consequences of models, while science uses models to explain observations of the real world. He discusses the interplay between theory and experiment in science, extending this to mathematics itself, which has historically been more theoretical but is increasingly embracing experimental components, especially with computers. He explains the 'unreasonable effectiveness of mathematics' in physics through the concept of 'universality,' where complex macro-scale systems emerge from micro-scale interactions and can be described by simple models with few parameters. The central limit theorem is given as an example of universality, explaining the prevalence of the bell curve, and he cautions about its limitations, drawing a parallel to the 2008 financial crisis.

Tao explains how mathematical concepts like the Hamiltonian help unify seemingly disparate areas of physics, like classical and quantum mechanics, by identifying underlying symmetries and conservation laws. He discusses the challenge of unifying general relativity and quantum mechanics, attributing it to a lack of fundamental mathematical objects to describe spacetime at tiny scales. He then delves into his work on Einstein's field equations, specifically the 'wave maps equation,' which models fields on spacetime. He explains how he used a 'gauge transformation' to simplify the complex nonlinear dynamics of this critical equation, visualizing it as stabilizing a flow of 'sheaves of wheat.' This leads to a discussion of the Poincare Conjecture, a millennium prize problem concerning the topological classification of 3D spaces. He explains how Grigori Perelman solved it using Richard Hamilton's groundbreaking 'Ricci flow,' a process that smooths out curved spaces. Perelman's critical contribution was to introduce new quantities that transformed the problem from a 'supercritical' to a 'critical' one, making the nonlinearities more manageable and allowing for the classification of singularities, leading to the solution.

Tao shares his approach to solving difficult mathematical problems, which he describes as 'cheating strategically.' This involves simplifying problems by making unrealistic assumptions, solving them in reduced forms, and gradually reintroducing complexities. He likens this to martial arts choreography where the hero fights one enemy at a time. He also discusses his preference for pen and paper, visualized with personal doodles on blackboards, though he's increasingly using computers for exploration. Transitioning to collaboration, Tao describes the evolution of mathematical collaboration, from traditional methods to modern tools like Lean. He emphasizes the initial 'brainstorming' phase where unrealistic assumptions allow for exploratory thinking. He details his collaboration with Ben Green on the Green-Tao theorem, highlighting the negotiation of hypotheses to make progress.

Terence Tao explains Lean, a formal proof programming language that not only computes answers but also generates verifiable proofs. He contrasts working with Lean to explaining a proof to a 'pedantic colleague,' highlighting the need for explicit type declarations. He discusses the integration of AI with Lean, particularly large language models (LLMs) assisting with 'lemma search' in Mathlib, a massive library of mathematical facts. He notes that while LLMs can suggest relevant components, their output is often unreliable, requiring significant human oversight. Despite these challenges, Tao sees Lean as a tool that enhances reproducibility, collaboration, and the management of complex proofs, potentially leading to a 'phase shift' in how mathematics is done, akin to the adoption of LaTeX. He describes a collaborative project where 50 mathematicians used Lean to formalize 22 million algebraic problems, showcasing the potential for large-scale, distributed mathematical research.

Tao discusses the current limitations of AI in mathematics, particularly its struggle with combinatorial explosions in complex proofs and its tendency to make subtle, 'stupid' errors that are difficult for humans to detect. He emphasizes that while AI can replicate some mathematical skills, it currently lacks the 'mathematical smell' or intuition that guides human mathematicians towards fruitful paths. He speculates that AI might gain this 'smell' through reinforcement learning, similar to AlphaZero's capabilities in chess and Go. Tao predicts that within this decade, AI will generate meaningful and potentially correct conjectures between previously unrelated mathematical concepts. He highlights the challenge of training AI for mathematical discovery, referencing the lack of data on 'failed' conjectures or the trial-and-error process of human research. He playfully suggests that AI needs to 'go through grad school' to learn from mistakes and develop intuition. The discussion then shifts to the unsolved Millennium Prize Problems.

Tao discusses the profound mystery of prime numbers, often called the 'atoms of mathematics.' He explains that numbers can be generated additively (by adding one) or multiplicatively (by multiplying primes), and while each operation is understood individually, their combination leads to immense complexity. This is exemplified by the twin-prime conjecture (infinitely many primes differing by two), which is notoriously difficult because small edits to the prime numbers can destroy the conjecture's truth, indicating a delicate underlying structure that eludes current methods. He contrasts this with the Green-Tao theorem, which proves the existence of arithmetic progressions of arbitrary length in primes, demonstrating the robustness of such patterns. Tao then delves into the Collatz conjecture, a seemingly simple problem of iterating a function (if even, divide by two; if odd, multiply by three and add one) and conjecturing that all numbers eventually reach one. He explains that while statistically numbers tend to decrease, the probabilistic arguments don't rule out rare 'outlier' numbers that might grow infinitely, as suggested by analogies to Conway's Fractran system.

Terence Tao identifies the Riemann Hypothesis and P versus NP as some of the hardest unsolved problems in mathematics. He explains that the Riemann Hypothesis, concerning the distribution of primes viewed multiplicatively, implies that primes behave as randomly as possible. Its disproof could have significant implications for cryptography. He notes that while evidence leans towards P not equaling NP, the problem's complexity lies in the vast number of approaches that have been ruled out. The conversation shifts to Grigori Perelman's famous refusal of the Fields Medal and Millennium Prize for solving the Poincare Conjecture. Tao reflects on the differing motivations of mathematicians and the complexities of peer recognition. He also offers advice to young people for pursuing careers, emphasizing the importance of transferable skills, adaptability, and problem-solving in an uncertain world. He encourages students struggling with math to seek out diverse learning resources outside traditional classrooms, and embraces the potential for tools like Lean to democratize mathematical research by enabling broader public participation and collaboration.

Tao discusses the inherent limitations of individual human cognition in mastering the vastness of modern mathematics. He suggests that while evolution hasn't endowed humans with a specific 'math center' in the brain, other cognitive centers (visual, linguistic, problem-solving) can be repurposed for mathematical thinking, leading to diverse learning styles. He advocates for educational approaches that cater to these different styles. He expresses optimism about the future of human civilization, highlighting the creative and enthusiastic nature of younger generations and the transformative power of scientific progress. He concludes by emphasizing that humanity, as a collective and augmented intelligence, demonstrates far greater capabilities than individuals alone, citing examples like Math Overflow where complex problems are swiftly tackled by the community. He reflects on the rapid advancements in AI and technology, acknowledging that what seems impossible today might become trivial homework in the future.