## Congratulations to two Cambridge mathematicians who have been elected fellows of the Royal Society. Professor Hugh Osborn has been recognised for his comprehensive body of work on quantum field theory, and Professor Jack Thorne for his multiple breakthroughs in diverse areas of algebraic number theory.

### Jack Thorne

Much of my work has been done in collaboration with others and I am very grateful to have had the opportunity to work with and learn from so many excellent mathematicians. Jack Thorne

Jack Thorne works in the field of number theory, famous for results that are simple to state but fiendishly difficult to prove — so much so that tools and methods from many other areas of maths have become integral parts of the field . "Part of the attraction of number theory is that it draws on ideas from so many parts of mathematics, including algebraic geometry, representation theory, and analysis," says Thorne. "This is particularly true of the *Langlands programme*, which blends these themes in an amazingly subtle and harmonious way." The Langlands program is a program of research set out in the 1970s by mathematician Robert Langland that bridges previously unconnected areas of mathematics.

Andrew Wiles used some these new connections for his proof of *Fermat's last theorem*. In 1637 Fermat stated, without giving a proof, that the equation

*a ^{n}*+

*b*=

^{n}*c*

^{n}has no integer solutions for *a*, *b* and *c*, for any value of *n* greater than 2. This deceptively simple statement was only proved more than 350 years later, by Wiles, with help from Richard Taylor. In finally proving the result Wiles and Taylor brought together sophisticated mathematical results from algebraic geometry and analysis to prove a more general result, called the *Shimura-Taniyama conjecture* (you can find out more about Fermat's last theorem on *Plus* magazine).

Thorne has taken as a starting point Wiles' and Taylor's work on the interaction of number theory with other areas of mathematics which exploit a symmetry that is inherent in certain mathematical equations. "Wiles' achievement was proving Fermat's last theorem was true for integers," says Thorne. "But number theorists care about all number systems, not just the integers." Thorne's work includes looking for solutions to Fermat's equation within these other number systems, rather than just for the integers. Examples of these number systems are number fields that extend the integers - for example, starting off with the integers but also including some algebraic numbers, such as the square root of two (find out more about symmetries and number fields on *Plus* magazine).

Thorne has been part of a major project to extend the Shimura-Taniyama conjecture to new number fields and has been recognised as a central force in extending the methods used by Wiles and Taylor to dramatically increase their mathematical scope.

"I am thrilled to be recognised in this way by the Royal Society," he says. "Much of my work has been done in collaboration with others and I am very grateful to have had the opportunity to work with and learn from so many excellent mathematicians." With his election, Thorne becomes the youngest living Fellow of the Royal Society.

### Hugh Osborn

While Thorne’s work concerns numbers, the fundamental building blocks of mathematics, Hugh Osborn's work concerns the fundamental building blocks of matter: the elementary particles that make up the physical world.

You don't have to be young to do potentially interesting work. I was elected FRS for work I did past the age of 50 or even 60. There is some hope for late developers! Hugh Osborn

"Elementary particle physics made rapid progress in the 1950s and later and I was always excited by this area of fundamental physics," he says. "The subject has undergone many twists and turns but there has been genuine progress, although there are still many unanswered questions."

In physics, the theory used to describe a situation can change depending on the scale we are interested in. This isn't new. Isaac Newton's theories work well for understanding and predicting the world we experience at our ordinary human scale. But when things are very large, such as in cosmology, or moving at speeds close to the speed of light, Einstein's theories of relativity are the theories to use. Einstein's theories are effective theories for describing cosmological events, Newton's theories are effective theories for mechanics on our human scale. And both Einstein and Newton's theories are unhelpful at the very smallest scales, where instead we need quantum physics.

The mathematical framework used to describe physical theories at the quantum scale is called *quantum field theory*. Osborn has been interested in how these mathematical descriptions of a quantum system change with changing energy scales. In quantum physics energy is equivalent to the length – you can think of changing the energy scale as zooming in or out on a mathematical microscope that you are using to describe the system, giving you a quantum field theory for each magnification level of your microscope. The technical name for this zooming between quantum field theories as you change scale is the *renormalisation flow*. (You can find out more about quantum field theory and the renormalisation flow on *Plus* magazine.)

Osborn has been particularly interested in conformal field theories: such theories remain the same as you change scale by zooming in or out – the picture you see in your mathematical microscope remains the same. If you reach a conformal field theory in your zooming process, you have a fixed point of the renormalisation flow, something that can give great insight into the underlying physics. (For a more familiar analogy, this sort of fixed point could give information comparable to that of the Curie temperature for a magnet, the temperature above which a metal loses its inherent magnetism.)

Calculations in quantum field theories are often not straightforward and making predictions about realistic physical systems may be difficult. Osborn's work on conformal field theories have led to results that now play a crucial role in the resurgence of a mathematical tool that is known as the *bootstrap*. Using conformal symmetry and the bootstrap can provide a way of determining some quantities to a much greater accuracy than was previously possible.

"I am glad some of my work has been part of significant developments," Osborn says, recognising that "many of my contemporaries, in Cambridge and elsewhere, also made important contributions". And while it's always a joy to celebrate those who revolutionise their field at a very young age, Osborn hopes that the recognition he received will help to also spur on researchers at a later stage of their careers. "You don't have to be young to do potentially interesting work," he says. "The reason I was elected to FRS was for work I did past the age of 50 or even 60. There is some hope for late developers!"

*The image above shows the Royal Society building in London. Photo reproduced under CC BY 3.0*.