# The Fields Medal Winners 2022

## July 12th 2022 by Anand M

Every 4 years a great event takes place in the world of Mathematics, the presentation of the Fields Medal. Many liken it to the Nobel Prize for Mathematics, but in reality, it is not. The Fields Medal is awarded to mathematicians under the age of 40 who have excelled in their field. The Nobel Prizes (science-based ones) are presented to those that have made a significant achievement in their field and are usually presented to those near the end of their careers.

## The 2022 Fields Medals

This year's Fields Medalists are explained below. We’ve tried to explain the work of each person in an approachable way.

## Hugo Duminil-Copin

*“For solving long-standing problems in the probabilistic theory of phase transitions in statistical physics, **especially in dimensions three and four.”*

*“For solving long-standing problems in the probabilistic theory of phase transitions in statistical physics,*

*especially in dimensions three and four.”*

Duminil-Copin has won the award for his work in looking at the mathematical understanding of phase transitions. A phase transition occurs when a substance changes its state of matter, say liquid to gas (boiling) or solid to liquid (melting).

In our Preliminary Physics course, we look at the process of phase transition in the topic **Waves and Thermodynamics*** .* We learn that when an object changes state there needs to be energy added or removed (latent heat). This energy is then transferred to and from the individual particles (atoms or molecules) to allow them to break apart (e.g. in boiling) or come together (e.g. freezing).

The work of Duminil-Copin has given us a better understanding of the microscopic processes that are involved. Any process in thermodynamics can be considered a statistical process as we look at the behaviour of the group of particles rather than trying to look at each particle. For instance, the temperature is a measure of the average kinetic energy of a group of all particles. In this way statistical mechanics allows us to better understand the behaviour of a system of particles, just like looking at census data can allow you to infer information about a population. Duminil-Copin’s work provides us with new techniques for understanding the nature of these particles and provides us with mathematical techniques to understand how a system evolves in 2, 3 and 4 dimensions (the fourth dimension being science).

International Mathematical Union. (2021). [Hugo Duminil-Copin]. Fields Medals 2022. https://www.mathunion.org/imu-awards/fields-medal/fields-medals-2022

## June Huh

*“For bringing the ideas of Hodge theory to combinatorics, the proof of the Dowling–Wilson conjecture for geometric lattices, the proof of the Heron–Rota–Welsh conjecture for matroids, the development of the theory of Lorentzian polynomials, and the proof of the strong Mason conjecture.”*

*“For bringing the ideas of Hodge theory to combinatorics, the proof of the Dowling–Wilson conjecture for geometric lattices, the proof of the Heron–Rota–Welsh conjecture for matroids, the development of the theory of Lorentzian polynomials, and the proof of the strong Mason conjecture.”*

The citation above has some big words in it! Huh has been able to prove many conjectures. A conjecture in mathematics is an open statement that has not been proven. In our Extension 2 Mathematics course we learn about the * Nature of Proof*, and how a statement that we think is true is called a conjecture, but when we prove that it is true it is a theorem. Huh’s work allowed several key conjectures in the area of geometric combinatorics to be proven and hence are now theorems.

Geometric combinatorics is a field of mathematics in which we are concerned with counting properties of geometric shapes. A simple version of this is the number of faces that can be generated by looking at n-points in some 3D lattice (or even at higher dimensions). At first, it seems quite abstract, but in today's digital world representing the world in geometric space has led to things such as GPS, Google Maps, Internet routing and many more. One interesting application of some of their work of Huh includes the way in which mega satellite constellations such as Space X’s Starlink are able to encircle the globe and provide coverage.

One of the conjectures that June Huh along with his collaborators proved was the Dowling-Wilson conjecture.

International Mathematical Union. (2021). [June Huh]. Fields Medals 2022. https://www.mathunion.org/imu-awards/fields-medal/fields-medals-2022

## James Maynard

**“For contributions to analytic number theory, which have led to major advances in the understanding of the structure of prime numbers and in Diophantine approximation”**

**“For contributions to analytic number theory, which have led to major advances in the understanding of the structure of prime numbers and in Diophantine approximation”**

Maynard has done a significant amount of work in the area of prime numbers and their behaviour. This also extends into the area of Diophantine equations (and their approximations).

Maynard’s work looks at the distribution of prime numbers, and in particular, looking at prime gaps. Maynard’s innovative approaches have reduced the so-called “gap” between prime numbers. The goal of mathematicians is to bring the gap down to two (The Twin Prime Conjecture - that there are infinitely many primes that are separated by a gap of 2). Often people ask why we even care about prime numbers (and more general problems in number theory), and the answer is quite simple. Prime numbers are the basis of all other numbers, something that we get from the Fundamental Theorem of Arithmetic, which states that every positive integer can uniquely be expressed as the product of primes. Thus the more we know about primes, the more we know about numbers and hence mathematics in general.

Maynard has also worked extensively on the Diophantine equations, which are polynomial equations with integer solutions. Maynard’s work focuses on methods to approximate solutions to such equations.

Maynard is also a great maths communicator with several videos on Numberphile explaining his work, they come highly recommended.

International Mathematical Union. (2021). [James Maynard]. Fields Medals 2022. https://www.mathunion.org/imu-awards/fields-medal/fields-medals-2022

#### Maryna Viazovska

**“For the proof that the E8 lattice provides the densest packing of identical spheres in 8 dimensions, and further contributions to related extremal problems and interpolation problems in Fourier analysis.”**

**“For the proof that the E8 lattice provides the densest packing of identical spheres in 8 dimensions, and further contributions to related extremal problems and interpolation problems in Fourier analysis.”**

I’ll be honest this is the hardest to really understand and explain. The way to pack most efficiently circles (in 2D space), spheres (in 3D space) or hyper-spheres (in 4D or higher space) is a long-standing problem in mathematics. For instance, in 2D it was proven by Lagrange in 1770 that hexagonal packing was the most efficient packing method. Viazoska’s work shows how we can efficiently pack hyperspheres in 8-dimensional space. She was able to show using new approaches to the problem how the E8 lattice (which is a topological lattice in 8-dimensional space).

Again a strange problem to solve, but often the techniques used to solve these problems are more important than the problem itself and this is the case in this area. Higher-dimensional spaces are used in areas of Physics and Chemistry to explain the universe we live in.

International Mathematical Union. (2021). [Maryna Viazovska]. Fields Medals 2022. https://www.mathunion.org/imu-awards/fields-medal/fields-medals-2022

## The Lone Aussie

Of the 64 people that have received the medal since it was first awarded in 1936, only one was an Australian. In 2006 South Australian-born and educated Terance Tao was awarded the Fields Medal “For his contributions to partial differential equations, combinatorics, harmonic analysis and additive number theory". Which is a long way of saying that he’s made some great advancements in the field of Mathematics.

Tao was a child prodigy and these days works at the University of California Los Angles, (UCLA). His current work is mainly in the area of Number Theory and in particular Prime Numbers.