Maths has always been about pushing the boundaries of knowledge, but the 21st century has also looked inward, forging a synthesis of previously disparate subjects. For example, Andrew Wiles’ proof of Fermat’s last Theorem was a triumph of algebraic geometry and number theory working side-by-side, and many ideas from string theory and quantum mechanics in physics have inspired new constructions in differential geometry. Nowhere was this philosophy more apparent than in the 2018 Fields Medal winners, several of which have drawn from many different fields in their most revolutionary works. In this article, we look at the maths that won the medals.

#### Caucher Birkar

Anyone who has plotted a graph of a quadratic equation has dabbled in *algebraic geometry* to some extent – for the subject is about geometric objects that come from polynomial expressions. These are known as *algebraic curves*, and simple examples in 2 variables include the parabola ($y=x^2$) and the circle ($x^2 + y^2 = 1$).

Unlike school students however, professional algebraic geometers such as Fields medal winner Caucher Birkar typically work over the complex numbers, as the resulting theory is stronger, and more beautiful. For example, if we use real numbers, a line and a quadratic curve will meet 0, 1, or 2 times. But over the complex numbers (if we count correctly), we will always have precisely 2 intersections! More generally in fact, if you have two polynomial curves of degree $n$ and $m$ respectively, then they will meet $n \times m$ times – for example a cubic curve and a quadratic curve should meet in 6 places. This powerful result is known as **Bezout’s Theorem**.

A major problem of algebraic geometry is classifying algebraic curves. If we use only two variables (say $x$ and $y$) to form a polynomial equation, then the resulting curve in complex space will look like a 2-dimensional surface (as opposed to the 1-dimensional curve we get over the real numbers). The great mathematician Bernhard Riemann discovered that these surfaces will always look like donuts, but perhaps with more than 1 hole, or no holes at all. The number of holes is called the genus of the surface; knowing just the genus (which mathematicians can calculate using the powerful **Riemann-Roch theorem**) gives you a good idea of what the whole object looks like. Though the case of these ‘Riemann surfaces’ has long been dealt with, classifying higher-dimensional complex algebraic curves is a much harder problem. Over the 20th Century, mathematicians have attempted to simplify the situation by breaking down complicated curves into easier-to-understand pieces; this is known as the `Minimal Model program’ – but although this has worked in some low-dimensional settings, it was unsuccessful in general.

The work of Caucher Birkar has furthered this program immensely. A collaboratively-written paper (known now as BCHM after the authors Birkar, Cascini, Hacon and McKernan) manages to prove the existence of minimal models for a swathe of algebraic curves, cracking the field wide open, and providing geometers with new tools to work with. Since this paper he has managed to cover some of the gaps he left, proving in 2016 a long-standing conjecture about a different class of curves known as `Fano varieties’.

#### Peter Scholze

The German fields medal winner Peter Scholze, though also known for his work in algebraic geometry, focuses on a different aspect of the subject – its application to the field of number theory. This is the branch of maths dedicate to studying the*integers*(1, 2, 3 …). At school, finding prime factors or lowest common multiples of numbers is a simple exercise of number theory.

If geometry and number theory sound incompatible to you, consider the classic problem of finding integers that satisfy $x^2 + y^2 = z^2$ – such solutions are known as `Pythagorean triples’. Though as stated this is a number theory problem, it is converted into geometry very easily. Dividing by $z^2$, we get the equation $p^2 + q^2 = 1$, where $p=\tfrac{x}{z}$ and $q= \tfrac{y}{z}$. This is just the equation of a circle, of radius 1! So the problem is reduced to finding *rational* points on the circle of radius 1 – which for example can be done by drawing slopes with a rational gradient from the top of the circle. Considering just the rational solutions to an algebraic equation is known as *arithmetic* or *Diophantine* geometry, and it’s the same kind of geometry that led to the solution of ‘**Fermat’s Last Theorem**‘ – by the above strategy, the problem of finding integer solutions to $x^n + y^n = z^n$ is equivalent to finding rational points on the curve $p^n + q^n = 1$.

The work of Peter Scholze builds on this legacy by continuing to provide a geometric framework to solve number-theoretic problems. Though *rational* algebraic curves have been well-understood for a while, another more mysterious type of algebraic curve is the `$p$-adic’ type – these are the curves that Scholze works with. His work has built on the shoulders of the 20th century mathematical giants Grothendieck and Fontaine by creating an entirely new type of mathematical object, the fabulously named *Perfectoid spaces*, which have helped Algebraic Geometers study $p$-adic curves in unforeseen ways.

#### Akshay Venkatesh

The work of Akshay Venkatesh stands out as particularly cross-discipline: his most famous papers stitch together the theories of dynamical systems, topology and algebra to prove remarkable facts of number theory.

For example, one of his most famous results regards polynomial expressions of degree 2 (examples are $z^2$, or $x^2 + 2xy + y^2$; note each component of these expressions is a power of two, or two variables multiplied together). These are known as *quadratic forms*, and much study has gone into them for hundreds of years. For example, **Lagrange’s Four-Square Theorem** (proved in 1770) states that every positive integer can be written as the sum of four square numbers, and another way of saying this is that the quadratic form $a^2 + b^2+ c^2 + d^2$ generates every positive integer, through different choices of $a$, $b$, $c$ and $d$.

Quadratic forms are the source of a long-standing open problem in mathematics – how can we tell whether a change of variables will transform one quadratic form into another? For example, the two examples given above are in fact equivalent, as if we replace $z$ with $x+y$, we get the same expression – we say that $x^2 + 2xy + y^2$ *represents* $z^2$. This procedure can simplify a quadratic form; in this case it changes a form of two variables into a form with one! It was proven in 1978 that if $P$ is a quadratic form of $p$ variables, and $Q$ is one of $q$ variables, then $Q$ represents $P$ if $q > 2p + 2$, no matter what $P$ and $Q$ are. This has since been beaten – Akshay Venkatesh’s work exploits dynamical systems theory to prove that in fact, the same is true if $q > p+4$ – an astonishing and unexpected improvement.

#### Alessio Figalli

Alessio Figalli is an Italian mathematician, whose main interests lie in the theory of optimal transport; though his impressive portfolio of results extends to geometry, probability and the Schrödinger equation in physics. The theory of optimal transport begins simply enough – it asks the question `what is the most efficient way to move a mass from point A to point B’? For example, if there are several factories producing a product, and a number of locations where that product is to be delivered, what is the most efficient way (in terms of total distance travelled) to have the items delivered? Clearly this is a very applicable subject – indeed advances in optimal transport theory have brought improvements in city design, plumbing systems, image processing as well as a better understanding of many biological processes.One of Figalli’s most impressive achievements is the application of this theory to understand the changing shape of crystals. Crystals try to maintain the most efficient shape possible with their current energy level – this is analogous to the way that a soap bubble attempts to minimise its surface area while enclosing a fixed amount of air, and so forms a sphere. If a crystal is heated, it will try to optimise its structure to match the new energy level, by changing shape.

Figalli answered the question of how drastically this can happen by modelling the change in shape of the crystal as a transport problem – and showed that the molecules of the crystal move proportionally to the square root of the energy increase. This demonstrated that crystal shapes are stable, i.e. a huge change in shape could only result from a similarly huge change in energy.