Rong Zhou is one of the Lecturers who started their role at the Faculty in the midst of the pandemic. We catch up with him to talk about equations that can’t be solved, surprising connections in number theory, and the Faculty coming back to life.
"One thing that's really coming out now that restrictions are ending is the energy of the place," he says. "It's not just the Faculty, the students and postdocs contribute to this too. It's a real pleasure, and very invigorating, to teach students who are so motivated, so enthusiastic and capable."
"It's amazing to work with all these people who are experts in their fields. Any kind of question I might have about any area of maths, there is someone here I can ask." Rong Zhou
Zhou started his lectureship at the Department of Pure Mathematics and Mathematical Statistics (DPMMS) in the summer of 2020 and is part of the number theory group. His love for number theory goes back to his own days as an enthusiastic Cambridge undergraduate. What fascinated him back then was the idea that some equations, unlike the ones you usually meet at school, don't come with a neat formula for their solutions. It's not just that nobody has been able to find a general formula yet, there simply isn't one — and it's possible to prove this mathematically.
"When I heard about this, it completely blew my mind," says Zhou. "How can you prove that a solution doesn't exist? Then I learnt about the proof and thought, 'this is incredibly beautiful, I want to keep working on this'. That's how I got my start in this area."
Angels, devils, and bathrooms
These days Zhou is still interested in equations, but the overarching theme of his research is one of surprising connections. In the 1960s the Canadian mathematician Robert Langlands discovered an astonishing link between geometry and number theory. Langlands formulated a web of conjectures around this link, collectively known as the Langlands programme. The task of proving them all is perhaps the biggest single project in mathematics.
The geometric side of the Langlands programme can be traced back to the 19th century. As many had done before them, mathematicians at the time were attempting to prove that any geometry you could possibly think of must satisfy all the axioms of geometry that Euclid had laid down around 2000 years earlier. Ironically, the quest for this proof led them to discover geometries that didn't satisfy all of these axioms. One of them is now known as hyperbolic geometry and is significantly different from Euclid's geometry: for example, the angles in a hyperbolic triangle don't add up to 180 degrees, but to less than that.
Perhaps the most famous representations of hyperbolic geometry occur in the works of the artist MC Escher. A famous wood engraving of Escher's shows copies of angels and devils filling up a disc. Even though to our eyes the angels and devils get smaller and smaller towards the edge of the disc, in the hyperbolic way of measuring area, they are all the same size. Escher's work represents a tiling of the hyperbolic plane. It's a bit like a tiling of a bathroom wall by squares, only that Escher's tiles are more complex and the underlying geometry is hyperbolic.
Tracking a spider
The bathroom analogy is useful, though, to understand what is to come. Imagine you wanted to trace the journey of a spider crawling around a bathroom wall extending infinitely in all directions. Because the square tiles are all identical, you could mark the spider's path on a single tile. When the spider leaves the starting tile by crossing, say, its right edge, you simply imagine it re-entering the same tile via the corresponding point on the left edge and continue along as, in reality, it did on the adjacent tile. Similarly, if the spider leaves the original tile via its bottom edge, you can imagine it re-entering via the top edge. And so on — eventually, the spider's entire path is marked as bits of path on the original tile.
Since you're treating the right edge of the original tile as the same as the left, you might as well glue the edges together. This gives you a cylinder. And since you imagine the top edge to be identical to the bottom edge, you might as well glue them together too, turning the cylinder into a donut. The surface of this donut now represents the entire infinite tiling— the spider's journey can be represented by a path winding round and round this surface in different directions.
In a roughly similar way, when you are given certain types of tilings of the hyperbolic plane, you can form surfaces that represent the entire tiling. A special class of these surfaces are called modular curves. (Technically, a modular curve is a quotient of the complex upper half-plane by the action of a congruence subgroup of the modular group.)
Ancient equations
Modular curves are interesting to geometers studying hyperbolic geometry, but never in a million years would you connect them to number theory, which is interested in questions such as 'which numbers, if any, satisfy a given equation?' Of particular interest to number theorists are polynomial equations (which are made up of powers of the variables involved) whose coefficients are all integers. An example is
x2+y2=1.
One of the first people known to have extensively studied these equations was Diophantus of Alexandria nearly 2000 years ago. Diophantus was interested in whether the equations he was looking at have integer solutions, but if you allow the solutions to come from the larger family of real numbers, then the equation above defines a circle: the points in the plane whose coordinates (x, y) satisfy the equation describe a circle of radius 1 around the point (0,0). If you allow the variables to come from the even larger world of complex numbers, then the solutions describe, not just a one-dimensional curve like the circle, but a two-dimensional surface.
A miracle
And here comes the miracle: it turns out that the modular curves born deep in the heart of geometry can also double up as solution surfaces of polynomial equations with integer coefficients — more complicated polynomial equations than our example, but polynomial equations nevertheless. And it also turns out that mathematical functions called modular forms, defined on the modular surfaces, can give you deep insights into the kind of questions that already fascinated Diophantus, such as 'what are the integer solutions', or 'how do any integer solutions relate to the prime numbers'?
What we have just described is one of the astonishing connections involved in the Langlands programme. It has been leveraged to much success in number theory already. Zhou is taking this connection further: his work focuses on a generalisation of those modular curves that form the meeting point between number theory and geometry, called Shimura varieties. "Shimura varieties are higher-dimensional analogues of modular curves," he explains. "The idea is that by studying the geometry of Shimura varieties you can prove cases of the Langlands programme."
Apart from making headway in the Langlands programme, Zhou also hopes to further explore another important role that Shimura varieties play in number theory by using them to study elliptic curves and their counterparts in higher dimensions. These objects played a crucial role in Andrew Wiles' proof of Fermat's famous last theorem, and holds great promise for future developments.
Awesome people
In returning to DPMMS for a Lectureship, Zhou is coming full circle. "I am now colleagues with people I used to look up to, who mentored me and taught me when I was an undergraduate," he says. "The first time I encountered Shimura varieties was in a course taught by Tony Scholl, who is now my office neighbour. I still learn a lot from Tony these days and it's really cool to be able to talk to him."
"Generally, it's amazing to work with all these people who are experts in their fields. Any kind of question I might have about any area of maths, there is someone here I can ask. But my colleagues are not just a great resource, they are also very awesome people. When I arrived here I felt completely embraced and welcomed by the department, not just by the academics, but also by the administrative staff who take care of all the non-mathematical aspects of my research life."
"I feel very lucky, it's a brilliant place to work in."
