以下是亚历克斯·弗里德曼和陶哲轩（一位备受推崇的数学家，被誉为“数学界的莫扎特”）之间的一次对话。他们讨论了广泛的议题，包括高难度的数学问题、纳维-斯托克斯方程、计算机辅助证明、人工智能在数学中的作用以及素数的本质。 陶哲轩首先讨论了Kakeya针问题，解释了它如何与偏微分方程、数论和波传播等多个领域相关联。他解释说，对它的研究与波的行为有关，特别是关于奇异点和纳维-斯托克斯方程（控制流体动力学）。他深入探讨了他发表的关于“平均三维纳维-斯托克斯方程”的论文，以及它如何试图理解流体中的奇异点是否能够形成。 陶哲轩详细阐述了为什么难以证明纳维-斯托克斯方程的一般性质，他提到了麦克斯韦妖的概念，即即使在统计上不太可能，理论上也可能出现极其不可能的配置。他用这个概念来说明，即使存在会使流体趋于平静的粘滞力，流体的能量也可能集中到越来越小的尺度上。为了解决这个问题，陶哲轩讨论了他如何通过改变平均方程中的物理定律来设计一个爆破，一个能量达到奇点的强制情况。他提到这项工作提供了一种障碍，并排除了某些证明方式。 他描述了解决纳维-斯托克斯问题的另一种方法，认为它可以通过构建一个液态图灵机来解决。他阐述说，水可以用来制造逻辑门，从而创建一个可以自我复制的机器。液态图灵机可能会导致爆破。陶哲轩指出，康威的生命游戏对这个想法产生了影响，系统的复制是这个构造的灵感来源。他还深入探讨了结构和随机性之间的二分法。 陶哲轩随后将话题转向了数学、物理和工程之间的区别。物理学收集观察结果并提出模型，而数学则接受这些模型并描述由此产生的后果。数学从公理开始，并询问会得出什么结论，而科学则从一个问题开始。他说我们需要实验和理论两方面。 讨论过渡到人工智能在数学中的作用，从计算机辅助证明和Lean形式证明编程语言开始。Lean能够逐行形式化代码，如果信任编译器，在数学上，就能 100% 保证论证的正确性。但人类的因素和模式识别可能会丢失。陶哲轩还提到了Alpha Proof，DeepMind 的 AI，它接受了对成功和失败证明进行强化学习的训练。 进入数论领域，陶哲轩深入研究了素数的性质，素数通常被称为“数学的原子”。他解释了如何生成自然数，即加 1。素数，2 3 5 7 是乘法的。讨论还涵盖了孪生素数猜想的挑战。正如 Paul Radar 所说，虽然这是一个难题，但陶哲轩取得了一些进展，他解释说大约 60% 的统计输入会拖回来。 后来，陶哲轩解释了庞加莱猜想，这是一个千禧年难题。他解释了这个挑战：如果任何三维物体都可以把它变成一个三维球体？如果可以降低维度，它就可以解决大量问题。 对话还涉及了作为数学家的个人方面，包括情感投入的挑战以及在工作中找到正确平衡的重要性。陶哲轩最后分享了他对数学未来的希望。

This is a conversation between Alex Friedman and Terence Tao, a highly regarded mathematician known as the "Mozart of Math." They discuss a wide range of topics, including difficult math problems, Navier-Stokes equations, computer-assisted proofs, the role of AI in mathematics, and the nature of prime numbers. Tao begins by discussing the Kakeya needle problem, explaining how it connects to various areas like partial differential equations, number theory, and wave propagation. He explains how the study of it relates to the behavior of waves, specifically concerning singularities and the Navier-Stokes equations, which govern fluid dynamics. He dives into a paper he published concerning the "average three-dimensional Navier-Stokes equation," and how it tries to understand whether singularities in fluid can form. Tao elaborates on why it's difficult to prove general properties about the Navier-Stokes equations, invoking the concept of Maxwell's demon, which is the idea that extremely improbable configurations can theoretically emerge even though they are statistically unlikely. He uses this concept to illustrate how the energy of a fluid can potentially be concentrated into a smaller and smaller scale despite the presence of viscous forces, which would calm things down. To address this, Tao discusses how he engineered a blow-up, a forced situation where energy reaches a singularity, by altering the laws of physics in an average equation. He mentions that this work provides an obstruction and rules out certain ways to prove things. He describes another approach to the Navier-Stokes problem, suggesting that it might be resolvable by constructing a liquid Turing machine. He elaborates that water could be used to make logic gates, creating a machine that would replicate itself. The liquid touring machine could potentially lead to a blowup. Tao notes the influence of Conway's Game of Life on this idea, where the system replicates, as inspiration for this construct. He also dives into the dichotomy between structure and randomness. Tao then pivots to the difference between mathematics and physics, with engineering also in there. Physics gathers observations and suggests models, and mathematics takes the models and describes what consequences come from them. Mathematics starts with axioms and asks what conclusions come, while the sciences start with a question. He said one needs both experimental and theoretical. The discussion transitions to the role of AI in mathematics, starting with computer-assisted proofs and the Lean formal proof programming language. Lean enables formalizing a line by line code that mathematically, if you trust the compiler, gives a 100% guarantee that the arguments are correct. But the human element and pattern recognition can be lost. Tao also touches on Alpha Proof, the DeepMind AI that's trained on reinforcement learning on successful and failed proofs. Moving into the realm of number theory, Tao delves into the properties of prime numbers, often called the "atoms of mathematics." He explains how to generate the natural numbers, you can add 1. Primes, 2 3 5 7 are multiplicative. The challenges of the twin prime conjecture are also covered. While it's a struggle, as said by Paul Radar, Tao has made some progress by explaining that about 60% of the inputs in statistics will drag back. Later, Tao explains the Poincaré Conjecture, a Millennium Prize Problem. He explains the challenge: what if any three-dimensional object could you turn it into a three-dimensional sphere? If you can reduce dimensions, it solves a huge number of problems. The conversation also touches on the personal aspects of being a mathematician, including the challenges of emotional investment and the importance of finding the right balance in one's work. Tao concludes by sharing his hope for the future of mathematics.