Diophantine equations and local-global principles: into the wild

研究多项式方程的整数(整数)解是数学中最古老的领域，包含了数千年来一直没有解决的问题。此外，它在密码学和安全方面的应用使其成为纯数学中影响最大的领域之一。密码系统依靠数学问题的计算难度来保护我们的数据。多项式方程的整数解是支撑现代密码系统的困难问题的自然来源。例如，它可以声称椭圆曲线密码术(ECC)的发展是功劳。这是一种公钥加密系统，十多年来一直被美国国家安全局和微软等大公司广泛使用。例如，当我们在互联网上购物时，ECC被用来保护我们的信用卡详细信息。网络安全在保护个人、公司和国家层面的数据方面具有至关重要的国家重要性，随着越来越多的经济、行政和社会互动在网上发生，网络安全在日常生活中的作用越来越大。发展椭圆曲线所需的深刻知识是通过在数学中进行蓝天研究而获得的，其中最著名的例子是安德鲁·怀尔斯1995年的费马大定理证明。这涉及到一族特殊的多项式方程，即x^n+y^n=z^n。当n=2时，这是毕达哥拉斯关于直角三角形边长的方程。这个方程有无穷多个整数解(例如x=3，y=4，z=5)，我们甚至有一个公式。然而，当n大于2时，这种行为是非常不同的。费马在1637年猜想，当n大于2时，方程x^n+y^n=z^n不存在正整数解。这一事实的证明花了350多年的时间，需要非常先进的数学技术的发展。2019年9月，谷歌宣布他们已经实现了量子霸主地位，开发出了一台量子计算机，可以在200秒内完成一项任务，而顶级超级计算机需要1万年才能完成一项任务。这一惊人的成就给保护我们数据的密码系统带来了迫在眉睫的危机。一台量子计算机可以在几秒钟而不是几千年内解决当前密码系统背后的数学问题，它将能够解密加密的数据，并危及其安全性。安全机构和科技公司正在紧急寻找新的、更困难的数学问题来支撑后量子密码系统，他们热衷于与数学家合作来实现这一点。我的建议是研究比椭圆曲线更大、更复杂的多项式方程的整数解，使用数论、代数、几何和分析的各种技术。现代的方法首先寻找所谓的局部解，然后研究它们的集合是否可以拼凑在一起形成全局(即整数)解。然而，这种局部-全局方法并不总是成功的。我将研究其失败的原因，并在方程式家族中对这些失败的频率进行统计分析。我将开拓进取地处理那些由于其复杂性而迄今未被触及的案例：我书名中的“狂野”是数学家用来描述其行为特别难以处理的数学对象的形容词。数论最近的突破意味着，解决这些狂野问题的时机已经成熟。我将与领先的密码学家合作，探索我对新的复杂数学问题的研究产生的可能性，这些问题可以用来支撑可以抵抗量子计算机攻击的密码系统。

Studying integer (whole number) solutions to polynomial equations is the oldest field in mathematics, containing problems that have remained unsolved for millennia. Furthermore, its applications to cryptography and security make it one of the most high-impact areas of pure mathematics. Cryptosystems rely on the computational hardness of mathematical problems to protect our data. The realm of integer solutions to polynomial equations is a natural source of hard problems to underpin modern cryptosystems. For example, it can claim credit for the development of elliptic curve cryptography (ECC). This is a public key cryptographic system that has been widely used for over a decade by big players such as the USA National Security Agency and Microsoft. For instance, ECC is used to protect our credit card details when we make purchases over the internet. Cybersecurity is of crucial national importance in protecting data at the individual, corporate and state level and its role in daily life is increasing as more of our economic, administrative and social interactions take place online.The deep knowledge of elliptic curves needed for the development of ECC was gained by pursuing blue sky research in mathematics, of which the most famous recent example is Andrew Wiles' 1995 proof of Fermat's Last Theorem. This concerns one particular family of polynomial equations, namely x^n+y^n = z^n. When n=2, this is Pythagoras' equation relating the side lengths of a right-angled triangle. There are infinitely many integer solutions to this equation (e.g. x = 3, y = 4, z = 5) and we even have a formula for them. However, when n is greater than 2, the behaviour is very different. Fermat conjectured in 1637 that there were no positive integer solutions to the equation x^n+y^n = z^n for n greater than 2. The proof of this fact took more than 350 years and required the development of very advanced mathematical techniques. In September 2019, Google announced that they had achieved 'quantum supremacy', having developed a quantum computer that performed a task in 200 seconds where a top-range supercomputer would take 10,000 years. This stunning achievement presents a looming crisis for the cryptosystems protecting our data. A quantum computer that can solve the mathematical problems underlying current cryptosystems in seconds rather than millennia would be able to decrypt encrypted data and compromise its security. Security agencies and technology companies are urgently seeking new, and harder, mathematical problems to underlie post-quantum cryptographic systems and they are keen to collaborate with mathematicians to achieve this.My proposal is to study integer solutions to a much larger and more complex class of polynomial equations than elliptic curves, using a wide variety of techniques from number theory, algebra, geometry and analysis. The modern approach looks first for so-called local solutions and then investigates whether a collection of them can be patched together to form a global (meaning integer) solution. However, this local-global method is not always successful. I will study the reasons for its failure and conduct a statistical analysis of the frequency of these failures within families of equations. I will break new ground by tackling cases that have so far been untouched due to their complexity: the 'wild' in my title is an adjective used by mathematicians to describe mathematical objects whose behaviour is particularly difficult to handle. Recent breakthroughs in number theory mean the time is ripe to grapple with these wild problems. I will collaborate with leading cryptographers to explore possibilities arising from my research for new hard mathematical problems that can be used to underpin cryptosystems that can resist attacks by quantum computers.