The four winners of the Fields Medal for 2018 have been announced. The Fields Medal is awarded every 4 years at an international gathering of mathematicians and is considered the Nobel Prize for Mathematics. However, there is one key difference: recipients must be 40 years old or younger.

This years recipients were announced on Wednesday 1st August at the International Congress of Mathematicians in Rio de Janeiro. They are:

Caucher Birker, 40, of the University of Cambridge in England: “for his proof of the boundedness of Fano varieties and for contributions to the minimal model program.”

Allesio Figalli, 34, of the Swiss Federal Institute of Technology in Zurich: “for his contributions to the theory of optimal transport, and its application to partial differential equations, metric geometry, and probability.”

Akshay Venkatesh, 36, of the Institute for Advanced Study in Princeton and Stanford University in California: “for his synthesis of analytic number theory, homogeneous dynamics, topology, and representation theory, which has resolved long-standing problems in areas such as the equidistribution of arithmetic objects.” Click here for more information.

Peter Scholze, 30 (one of the youngest recipients), of the University of Bonn in Germany: “for transforming arithmetic algebraic geometry over p-adic fields through his introduction of perfectoid spaces, with application to galois representations and for the development of new cohomology theories.” Click here for more information.

Mathematicians celebrate the new year with the recent discovery of the largest prime. It has 23,249,425 digits, which is 910,807 digits more than than the previously known largest prime.

“If every second you were to write five digits to an inch then 54 days later you’d have a number stretching over 73 miles (118 km) — almost 3 miles (5 km) longer than the previous record prime.”

On December 26th 2017, Jonathan Pace, a GIMPS volunteer for 14 years, discovered the 50th Mersenne prime, which is also the largest prime known to mankind:

“Today we take it for granted that the concept of zero is used across the globe and our whole digital world is based on nothing or something. But there was a moment when there wasn’t this number.”

-Marcus du Sautoy

So when was the number zero conceptualised?

Recently, scientists have traced the origins of zero to an ancient Indian text, called the Bakhshali manuscript, which has been housed in the UK since 1902. Radiocarbon dating has revealed that this script originated in the third or fourth century. This is about 500 years earlier than scholars previously believed.

The Bakhshali manuscript, which was first found by a farmer in 1881 in what is now Pakistan, is inscribed on 70 pieces of birch bark and contains hundreds of zeroes.

Several ancient cultures independently came up with symbols for the concept of ‘nothing’, however the dot symbol in the Bakhshali script is the one that evolved into the hollow-centred version of the symbol that we use today.

“It also sowed the seed for zero as a number, which is first described in a text called Brahmasphutasiddhanta, written by the Indian astronomer and mathematician Brahmagupta in 628AD.”

A generalised Fermat Prime is a prime number of the form for a >0. It is called ‘generalised’ as a Fermat Prime is a number of this form with a = 0.

The discovery was made by Sylvanus A. Zimmerman of the United States.

“Until now only 392 generalised Fermat primes had been found: this new discovery makes 393. At 6,253,210 digits long, it’s now the 12th largest of all known primes, and the second-largest known non-Mersenne prime.”

Polya Prize: This was awarded to Professor Alex Wilkie FRS from the University of Oxford due to his contributions to model theory and its connections to real analytic geometry.

Senior Whitehead Prize: Professor Peter Cameron, from the University of St Andrews, was awarded this prize for his research on combinatorics and group theory.

Senior Anne Bennett Prize: Awarded to Professor Alison Etheridge FRS from the University of Oxford for her “research on measure-valued stochastic processes and applications to population biology” as well as outstanding leadership.

Naylor Prize and Lectureship: This was given to Professor John Robert King from the University of Nottingham due to profound contributions to the theory of non-linear PDEs and applied mathematical modelling.

Berwick Prize: This was awarded to Kevin Costello of the Perimeter Institute in Canada for his paper entitled The partition function of a topological field theory (published in the Journal of Topology in 2009). In this paper Costello “characterises the function as the unique solution of a master equation in a Fock space.”

Whitehead Prize:

Dr Julia Gog (University of Cambridge) for her research on the mathematical understanding of disease dynamics, in particular influenza.

Dr András Máthé (Univeristy of Warwick) due to his insights into problems in the fields of geometric measure theory, combinatorics and real analysis.

Ashley Montanaro (University of Bristol) for her contributions to quantum computation and quantum information theory.

Dr Oscar Randal-Williams (University of Cambridge) due to his contributions to algebraic topology, in particular the study of moduli spaces of manifolds.

Dr Jack Thorne (University of Cambridge) for research in number theory, in particular the Langlands program.

Professor Michael Wemyss (University of Glasgow) for his “applications of algebraic and homological techniques to algebraic geometry.”

When I read the news that Maryam Mirzakhani had sadly passed away with breast cancer aged 40 I was honestly shocked. I remember finding out that she was the first women to win the Fields Medal in 2014 and feeling a huge sense of pride that we had achieved such a big milestone in mathematics – a mostly male dominated subject.

Professor at Stanford Universtiy, Mirzakhani was awarded the notorious Field’s Medal for her work on complex geometry and dynamic systems. She specialised in areas of theoretical mathematics that “read like a foreign language by those outside of mathematics” such as moduli spaces, Teichmüller thoery, hyperbolic geometry, Ergodic theory and symplectic geometry. By mastering these fields, Mirzakhani could describe the geometric and dynamic complexities of curved surfaces, spheres, donut shapes and even amoebas in a huge amount of detail. Furthermore, her work had implications in a vast amount of fields, ranging from cryptography to the physics of how the universe was created.

Moduli Spaces

Moduli Spaces can be thought of as geometric solutions to geometric classification problems. In broad terms, a moduli problem consists of three main categories:

Objects: which geometric objects do we want to parametrise?

Equivalences: when do we identify two objects as being isomorphic?

Teichmüller theory, which brings together an array of fundamental ideas from different mathematical fields (including complex analysis, hyperbolic geometry, differential geometry, etc), is concerned with the Teichmüller space.

To get an short introduction to Teichmüller theory, click here.

Hyperbolic Geometry

Hyperbolic geometry is a non-Euclidean geometry, where the parallel postulate of Euclidean geometry is replaced with:

“For any given line R and point P not on R, in the plane containing both line R and point P there are at least two distinct lines through P that do not intersect R.” – Wikipedia

Ergodic Theory

Ergodic theory was initially developed to solve problems in statistical physics and is a branch of mathematics that studies “dynamical systems with an invariant measure”. An invariant measure is a measure that is preserved by some function.

Symplectic Geometry

Symplectic Geometry is a branch of differential geometry and differential topology that studies symplectic manifolds. These are differentiable manifolds that have a closed, non-degenerate 2-form.

“Mirzakhani once described her work as ‘like being lost in a jungle and trying to use all the knowledge that you can gather to come up with some new tricks, and with some luck you might find a way out’.”

– Guardian

Mirzakhani will be remembered not only for her extraordinary work, but also as being an inspiration to thousands of women to pursue maths and science.

Recently, James Davis found a counterexample to John H. Conway’s ‘Climb to a Prime’ conjecture, for which Conway was offering $1,000 for a solution.

The conjecture states the following:

“Let n be a positive integer. Write the prime factorisation in the usual way, where the primes are written in ascending order and exponents of 1 are omitted. Then bring the exponents down to the line, omit the multiplication signs, giving a number f(n). Now repeat.”

For example, f(60) = f(2^2 x 3 x 5) = 2235. As 2235 = 3 x 5 x 149, f(2235) = 35149. Since 35149 is prime, we stop there.

Davis had a feeling that the counterexample would be of the form

where p is the largest prime factor of n. This motivated him to look for x of the form

The number Davis found was 13532385396179 = 13 x 53^2 x 3853 x 96179, which maps to itself under f (i.e. its a fixed point). So, f will never map this composite number to a prime, hence disproving the conjecture.

Researchers have recently discovered that solutions to the Riemann zeta function correspond to the solutions of another function that may make it easier to solve the Riemann hypothesis.

Dorje Brody, a mathematical physicist at Brunel University London, says that “to our knowledge, this is the first time that an explicit—and perhaps surprisingly relatively simple—operator has been identified whose eigenvalues [‘solutions’ in matrix terminology] correspond exactly to the nontrivial zeros of the Riemann zeta function“.

Now what remains to be proven is that all of the eigenvalues are real numbers rather than imaginary ones.

This newly discovered operator has close ties with quantum physics. In 1999, Michael Berry and Jonathan Keating made the conjecture (now called the Berry-Keating conjecture) that if such an operator exists, then it should correspond to a theoretical quantum system with particular properties. However, no one has ever found such a system before now.

In general, mathematicians are optimistic that the eigenvalues are actually real, and there is a strong argument for this based on PT symmetry (a concept from quantum physics which says that you can change the signs of all four components of space-time and the result will look the same as the original).

The Abel Prize 2017 has been awarded to Yves Meyer of the École normale supérieure Paris-Saclay in France due to his “pivotal role in the development of the mathematical theory of wavelets”, which has applications in data compression, medical imagery and the detection of gravitational waves.

Meyer, aged 77, will receive 6 million Norwegian krone (around £600,000) for the prize, which aims to recognise outstanding contributions to mathematics. It is often called the ‘Nobel Prize’ of mathematics.

The Abel Prize was previously won by Andrew Wiles in 2016, who solved Fermat’s Last Theorem.

Biography

Yves Meyer, born on the 19th July 1939, grew up in Tunis in the North of Africa. After graduating from École normale supérieure de la rue d’Ulm in Paris and completing a PhD in 1966 at the University of Strasbourg, he became a professor of mathematics at the Université Paris-Sud, then the École Polytechnique and then Université Paris-Dauphine. He then moved to École normale supérieure Paris-Saclay in 1995, until formally retiring in 2008, although he still remains an associate member of the research centre.

Vera Rubin died on 25th December 2016, aged 88. Rubin was an American astronomer who pioneered work on galaxy rotation rates.

In the 1960s and 70s, Rubin and her collogue Kent Ford noted a discrepancy between the predicted angular motion of galaxies and their observed motion, whilst studying galactic rotation curves.

This led Rubin to conclude that some unseen mass must be influencing the rotation of galaxies. As a result, in an attempt to explain the galaxy rotation problem, the theory of dark matter was created. The existence of this ‘invisible mass’ was first theorised by Fritz Zwicky in the 1930s but until Rubin and Ford’s work it had not been proven to exist.

Although initially it was met with skepticism from the scientific community, Rubin’s results have been confirmed over the subsequent decades.

Emily Levesque from the University of Washington said in an interview with Astronomy magazine:

This discovery “utterly revolutionised our concept of the universe and our entire field.”

It is considered one of the most significant results of the 20th century.

However, Rubin never received the Nobel Prize for Physics, although she was frequently mentioned as a candidate for it. It has been 53 years since a women has won a Nobel Prize in Physics, and now that Vera Rubin has passed away, she is no longer eligible. But, we can take some consolation in the fact that Rubin was indifferent to not being nominated for the Nobel Prize.

“Fame is fleeting,” Rubin said in 1990 to Discover magazine. “My numbers mean more to me than my name. If astronomers are still using my data years from now, that’s my greatest compliment.”