Plus Blog

August 20, 2014
secret garden

As we attempt to close our overstuffed suitcases for the flight home, we are very pleased that Plus too is overflowing with goodies from this years International Congress of Mathematicians in Seoul.

From the excitement of the announcement of the Fields medallists to meeting mathematicians from around the world, we've had a brilliant time. We've still got a few articles and podcast yet to publish, but for now, you can find all our coverage from ICM 2014 here.

Thank you to all mathematicians for their talks and to those who generously gave their time for interviews and explanations. A huge thank you to the local organisers of the conference, the staff at the media office and the many enthusiastic and helpful volunteers who made the event such a success. Thank you!

August 20, 2014

"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 Rubik's cube

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 way.

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 (see here for some articles we have produced in this context). 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.

August 13, 2014

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.

This is not to say, however, that there are not brilliant women mathematicians. Quite the contrary. We have been lucky enough to with and reported on the work of many – Corinna Ulcigrai, Julia Gog, Helen Mason and Nathalie Vriend to name just a few.

MM

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.

Guests at the opening ceremony of ICM 2014

The prizewinners and guests on stage at the opening ceremony of ICM 2014

You can read more about Maryam Mirzakhani and the other Fields medallists, and about the ICM 2014 on Plus.

August 11, 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!

Binary

Sunset over Seoul.

August 7, 2014
Surface

Image © Oliver Labs.

Our image of the week shows a beautiful surface with many singularities, constructed using computer algebra. The image was created by Oliver Labs, using the software Singular and visualised using Surf. © Oliver Labs.

The picture is one of the images that appears in the book 50 visions of mathematics, which celebrates the 50th anniversary of the Institute of Mathematics and its Applications.

You can see previous images of the week here.

August 7, 2014
Binary

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 for

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.

Easy!

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.

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