"Ingenuity", "far reaching vision", "unerring sense", "deep
curiosity" — and best of all,
"extraordinary creativity". These are some of the words that have been
used to praise this year's Fields medallists at the International
Congress of Mathematicians (ICM). These words aren't specific to maths. They
could be used to describe anyone whose work is about discovery and
beauty; writers, poets, or musicians for example. If there has
been one overarching theme at this ICM, it's just how creative a
subject mathematics is. To imagine what has never been imagined
before, spot new connections, look at things in a new light and find hidden patterns lies at the
heart of doing maths.
A great example is Fields medallist Manjul Bhargava, who in a flash
of inspiration discovered that a centuries' old maths problem
could be solved using a Rubik's cube, reducing long and tedious
calculations, quite literally, to child's play. But singling out
specific examples gives the wrong impression:
discovering the undiscovered and creating new ideas is central to the
work of every mathematician, whether they are presenting their work here at the ICM or working away at home.
What guides these creative minds is often a sense of beauty.
"There is an aesthetic side," says Phillip Griffiths,
winner of this year's Chern Medal. "You find out what the most harmonious
properties of a [mathematical] structure are, and then
you let those guide you." You might be wrong, of course,
things might be more complex, or perhaps even simpler, than you think,
but still, it's often aesthetic considerations that lead the
The creative aspect of maths is one of mathematics' best kept
secrets. If you didn't have an inspiring teacher
at school, then you probably think that maths is all about repetition
with no room for imagination and exploration. "Many teachers still
teach that way; here is the problem, here is how you solve the
problem," says Martin Grötschel, Secretary of the International
Mathematical Union. "So what you do is learn recipes, how to solve a
quadratic equation, etc. But you are not taught why and how you get to this equation. The really successful countries, where
maths education has improved a lot, have taken the approach of teaching
students to figure out for themselves why things go this way or the
other way." (You can listen to our interview with Grötschel here.)
Fields medallist Bhargava suggested something similar to us: perhaps maths
students, rather than learning how to apply "finished" mathematical tools, should
be allowed to play with maths problems that have inspired past
masters, to develop their mathematical intuition and get a sense of
the excitement and adventure of doing mathematics (see our interview
with Bhargava to find out more) — the sense of discovering, and
shaping, something new.
A lot of research is currently going into finding out how to best
teach maths at school, part of it supported by the International
Mathematical Union. Luckily we know lots of teachers who
are passionate about creativity in maths. And we are currently involved in a project that
aims to foster this creativity, an effort that has resulted in the website Wild Maths. Equally luckily, we know lots of mathematicians who are
passionate about communicating the essence of maths to a wider
audience (Bhargava is one of them). So perhaps it won't be long until
mathematics' best kept — and most exciting — secret becomes public knowledge.
One of the reasons we love maths is that at its core it doesn't care about who you are, where you are from, the language you speak, the colour of your skin or your gender. Mathematics is a universal language. We feel this as people doing mathematics. But as we attend conferences, read journals and interview the many wonderful mathematicians we meet, we are reminded that as women we are in a minority.
Maryam Mirzkhani, the first female Fields medallist
But currently in the UK only 6% of maths professors are women, despite women making up over 44% of maths undergraduates (see this paper from the London Mathematical Society). The reasons for this are not well understood but the problem seems to be exacerbated by the small number of women itself: fewer women in the community mean they are overlooked when names are sought for speakers or prizes and the relatively few women are disproportionally asked to sit on committees and participate in other non-research activities at the detriment of their research time. And of course this can be compounded by the broken career patterns and other challenges that stem from child-rearing and family responsibilities.
The under-representation of women at the highest levels of maths has been recognised by learned societies, universities and government resulting in many initiatives, such as the Athena Swan Charter in the UK. Thankfully we now rarely hear of experiences of direct sexism from female mathematicians. Instead for many women, particularly young female mathematicians, there remains an uneasiness about being one of a very few women in a maths department. They have spoken of being "the odd one out" or feeling like "the other" in their work places. More role models – high profile women in the mathematical community – are needed both for younger and established female mathematicians.
So we are thrilled that the brilliant mathematician Maryam Mirzakhani has been awarded the Fields Medal at the ICM 2014 in Seoul, Korea – the first women to be so recognised. Mirzakhani's work is on investigating mathematical surfaces and the geometric structures they can have. She has been recognised for her "rare combination of superb technical ability, bold ambition, far-reaching vision, and deep curiosity" which led to "striking and highly original contributions to geometry and dynamical systems".
Now that the IMU has finally recognised a woman for her mathematical achievements with a Fields medal, we hope that more female mathematicians will be nominated and recognised in the future. It seems fitting that Mirzakhani received her prize from a female head of state, President Park Geun-hye of South Korea, in a ceremony led by Ingrid Debauchies, the president of the International Mathematics Union, and mc-ed by Seon-Hee Lin, a professor from Seoul National University. These four women made up a third of the guests on the stage at the opening ceremony – we hope the recognition of women in mathematics stays at at least this level in the future.
The prizewinners and guests on stage at the opening ceremony of ICM 2014
We have set up our temporary headquarters in Seoul, South Korea! We are here to attend the International Congress of Mathematicians, a huge maths conference that happens every four years. We are particularly excited because we'll witness the the award of the Fields Medal, one of the highest honours in maths. The Fields medal is awarded every four years by the International Mathematical Union to up to four mathematicians for "outstanding mathematical achievement for existing work and for the promise of future achievement." (There is an age limit of 40 though, so one half of Plus is sadly already out of the running.)
We will be reporting live on the award of the medals, as well as a range of other prizes that will be handed out, and no doubt meet many fascinating mathematicians to interview. Stay tuned!
Most people are aware of the fact that computers work using strings of
0s and 1s. But how do you write numbers using only those two symbols?
To see how, let's first remind ourselves of how the ordinary decimal way
of writing numbers works. Let's take the number 4302 as an example. The
digit 4 in this number doesn't stand for the number 4, rather it stands
for 4000, or 4 x 1000. Similarly, 3 doesn't stand for 3 but for 300 = 3
x 100, 0 stands for 0 x 10, and 2 stands for 2 x 1. So 4302 means
4 x 1000 + 3 x 1000 + 0 x 10 + 2 x 1.
Similarly, 7396 stands for
7 x 1000 + 3 x 100 + 9 x 10 + 6 x 1.
What do the numbers 1000, 100, 10 and 1, which appear in these
expressions, have in common? They are all powers of 10:
1000 = 103
100 = 102
10 = 101
1 = 100.
To write a number in decimal notation, you first write it as a sum of
consecutive powers of 10 (with the largest power on the left) and then
pull out the coefficients of these powers. We can do the same with
powers of 2 rather than 10. For example, the binary number 110 stands
1 x 22 + 1 x 21 + 0 x 20 = 4 + 2 +0 = 6
(written in decimal).
And the binary number 10001 stands for
1 x 24 + 0 x 23 + 0 x 22 + 0 x 21
+ 1 x 20
= 16 + 0 + 0 + 0 + 1 = 17 (written in decimal).
You can convince yourself that a binary number only consists of the
digits 0 or 1: when you write a number as a sum of consecutive powers of
2, no other coefficients are necessary.
This sorts out the natural numbers, but what about numbers that have a
fractional part? To write a number between 0 and 1 in binary, you play
the same game using powers of 1/2 instead of powers of 2. For example,
0.75 = 1/2 + 1/4 = 1 x 1/21+ 1 x 1/22.
In binary, 0.75 is written as 0.11. The binary number 0.1001 stands for
the decimal number
1 x 1/21+ 0 x 1/22 + 0 x 1/23+ 1 x 1/24
= 1/2 + 1/16 =0.5625.
You can find out more about the positional way of writing numbers here and about the use of 0s and 1s in logical operations here.