Diophantine equations and local-global principles: into the wild

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.

Cybersecurity is of crucial national importance and its role is increasing as more of our economy moves online. Cryptosystems rely on the computational hardness of mathematical problems to protect our data. The realm of Diophantine equations, containing problems that remain unsolved after thousands of years, is a natural source of hard problems to underpin modern cryptosystems. For example, equations called elliptic curves are the basis for ECC, a cryptosystem in use for over a decade by big players such as the US National Security Agency and Microsoft.

Google's 2019 quantum supremacy announcement presents a looming crisis for data protection: 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. Technology companies and security agencies are urgently seeking harder mathematical problems to underlie post-quantum cryptographic systems and are keen to collaborate with mathematicians to achieve this.

I will study a much larger and more complex class of equations than elliptic curves. I will work with leading cryptographers at Microsoft to investigate possibilities arising from my research for new hard mathematical problems that can be used to underpin cryptosystems that can resist attacks by quantum computers. This work is exploratory: it may have an immediate impact on data security but, even if not, a deeper understanding of these equations may lead to far-reaching future applications. The point is that it is hard to predict the impact of research in pure mathematics: 19th century researchers studying elliptic curves would be amazed at how much we rely on them today for internet transactions and national security. By wide dissemination of my results through publications in leading journals, open access repositories, and talks for broad audiences as well as specialists, I will maximise the chances of finding applications for my research. I will maintain pages on arXiv, Research Gate and Google Scholar, as well as my own website, to increase visibility of my work.

The Coalition for African Research and Innovation states that the future of the continent depends on rapid improvement in conditions for research and development. To this end, the African Mathematics Millennium Science Initiative have partnered with the London Mathematical Society (LMS) and the International Mathematics Union to develop the Mentoring African Research in Mathematics programme, which aims to address the brain drain of mathematical talent from Africa by improving the local research environment and establishing mentoring relationships between African mathematicians and their European counterparts. My involvement with this and other grassroots initiatives such as the Nesin Mathematics Village in Turkey will support mathematical talent in developing and emerging economies. The ensuing knowledge exchange will be mutually beneficial: I will learn new approaches to research and teaching, improve my communication and engagement skills, and develop best practice to be shared with the Reading outreach team.

A 2018 Public Accounts Committee report identified the STEM skills gap as one of the UK's key economic problems. At the same time, a Microsoft study found that having an inspiring role model almost doubles the number of girls interested in STEM. Through outreach and engagement, I will encourage the next generation to pursue careers in STEM and will become a role model for women in maths. I will mentor younger women and support their careers by leading a Women in Numbers collaborative research project. I will work with influential organisations such as EPSRC, the LMS and Women in Numbers to tackle structural and cultural barriers for women and minorities in mathematics. The benefits will be both immediate on an individual level and long term for the UK economy in terms of addressing the STEM skills gap.