Problems of Unlikely Intersections: The Zilber-Pink Conjecture for Shimura varieties

One of the oldest topics in mathematics is the study of Diophantine equations (named after the 3rd century Hellenistic mathematician Diophantus of Alexandria).

A Diophantine equation is simply a polynomial equation, usually with two or more unknowns, whose coefficients are whole numbers (integers) or fractions (rationals). Consider, for example, 5X+7Y=1, or Y=3X^2.

The aim, given a Diophantine equation, is to find integer or rational values for the unknowns (X and Y in the above) so that the equation holds. Such a combination of values is referred to as an integral or rational solution to the equation. In general, the problem of finding integral or rational solutions to a Diophantine equation is extremely difficult and only the simplest cases can be handled explicitly.

To take a more conceptual approach, one can think of a Diophantine equation as a geometric object (consider, for example, the parabola Y=X^2). In more modern times, it has become clear that this perspective has a key role to play, and some of the greatest mathematical advances of the 20th century have led to the striking discovery that geometry plays a governing role in arithmetic problems such as these. Indeed, this was profoundly demonstrated in 1983, when Faltings proved Mordell's conjecture, which states that a Diophantine equation in two variables satisfying certain geometric conditions has only finitely many rational solutions. In 1986, Faltings was awarded a Fields Medal for his proof.

Mordell's conjecture gave rise to a number of other finiteness conjectures, due to Andre, Lang, Manin, Mumford, Oort, and others. However, although these conjectures shared obvious similarities, it was unclear how they related to one another. This was until the Zilber-Pink Conjecture came along; in a vast new conjecture, it simultaneously generalised all of the aforementioned conjectures. It achieved this in part by working within the rich mathematical objects known as (mixed) Shimura varieties.

The Zilber-Pink Conjecture is a problem of Unlikely Intersections, which is a name derived from the simple principal that, in a space of dimension d, two geometric objects of dimensions n and m, respectively, are highly unlikely to meet if the sum of n and m is less than d. Consider, for example, two lasers fired from opposite corners of a laboratory; we expect them to miss each other because the lasers are lines (and, hence, of dimension 1) being fired in 3-dimensional space, and 1+1=2 is less than 3.

Problems of Unlikely Intersections have produced a flurry of activity in recent years, in large part due to new tools coming from mathematical logic. These were first applied by Pila and Zannier to the so-called Manin-Mumford Conjecture, and their approach has inspired a general strategy, which has already had profound effects in the subject.

The proposed research seeks to obtain new arithmetic results for Shimura varieties that, due to previous work of the author and his collaborator, are known to yield significant progress towards the Zilber-Pink Conjecture via extensions of the Pila-Zannier strategy. It also seeks to obtain effective results (in other words, results with numeric outputs that are in principal computable) in the setting of the Zilber-Pink Conjecture. It will achieve this latter aim using new tools from the geometry of differential equations that have already produced results in simpler settings.

This project explores problems at the heart of pure mathematics. As such, it deliberately aims to answer questions so natural that we find ourselves powerless but to ask them, not because they pertain necessarily to real world problems, but because they are fundamental. Time and again, answers to these sorts of questions have proven useful beyond imagination.

The work sits within a wider programme of research known as problems in Unlikely Intersections. Ostensibly concerned with questions from arithmetic geometry, this programme distinguishes itself through the manner in which it combines tools from other areas of mathematics to solve its problems. As such, I speculate that interactions between my work and these other areas may eventually lead to impact, in addition to impact arising from the arithmetic aspects.

Number theory itself is now at the centre of a revolution. The world is creating digital information at an unprecedented rate. Indeed, we have created more data in the last few years than in the rest of human history. It is imperative, therefore, that we have the means to securely and efficiently store, manipulate, and transmit this new kind of information. Indeed, the functioning of our monetary system, the security of our online data, and our ability to prevent cyber attacks all depend on our development of these capabilities. Number theory gave rise to one of the early crypto-systems, known as RSA, and has continued to produce newer, more efficient systems, such as those based on elliptic curves, and even higher dimensional objects. Shimura varieties are the parameter spaces for such objects, and it is plausible that properties of Shimura varieties will eventually become relevant to cryptography. Furthermore, as quantum computers become more powerful, we will be forced to develop more secure crypto-systems. Lattice cryptography is a potential candidate, and it is possible that our work in reduction theory might one day have something to add here.

Impact arising through interactions with model theory and the geometry of differential equations is more speculative. However, model theory is evidently an important source for innovation in computer science, data handling, and machine learning, whereas eventual applications coming from the geometry of differential equations are almost limitless; many disciplines, including engineering, meteorology, and medicine, all depend on our understanding of differential equations.

Of course, it is almost impossible to predict the eventual uses of pure mathematics. It is likely that my research will have to pass through the hands of many other academics before finding applications. What I am able to do is disseminate my results widely, feeding forward and back into the many areas of research related with this project.

There will be, however, almost immediate economic and societal benefits of this project. If my research is funded, my department has agreed to provide me with funding for a PhD student. This will give me the opportunity to train a new researcher in this subject, who will then be in a strong position to either enter research, and further develop the results of this project, or one of many high-tech, in-demand industries, such as data science, cryptography, or finance.
Similarly, in allowing me to diversify my research programme, this project will put me in the position to provide opportunities for postdoctoral research assistants, as well as further PhD students. I believe I am perfectly suited to take on these responsibilities, given that I am a passionate teacher working in a vibrant and diverse field of research.

This project will serve to strengthen the UK's reputation in an internationally important area of research, which will feed into the country's ability to attract students and professionals, as its universities and industries both compete for mathematical minds.