Riemann surface or Babylon 5 monster? Only a genius can tell …

Today Maryam Mirzakhani, aged 37, became the first woman ever awarded the Fields prize for mathematics, a prize that is sometimes described as the “Nobel prize of maths.” She was awarded the prize for her work on “Riemann Surfaces and their moduli spaces,” which you can look up in wikipedia but good luck with that. Riemann surfaces are a kind of manifold, which is a space that globally has a complex structure that cannot be easily described mathematically but that reduces locally to a Euclidean space. A good way to think about manifolds is as the problem of ironing your shirt. Globally, your shirt has a twisted and contorted structure which means you can’t conceive of it as a flat surface suitable for ironing; but you can fold out small sections of it into a simple plane, and iron those sections. Manifold theory is essential for higher work in physics, since quantum mechanical topology is not straightforward. The wikipedia page has some nice examples of Riemann surfaces for basic functions plotted in the complex plane (that is, a plane with complex numbers). The example for the square root function shows an application of the theory of Riemann surfaces (I think): you can plot the real part of the square root on the vertical axis, and then obtain the surface for the complex part by a simple 180 degree rotation. For the average mortal, obtaining a result like that will probably make your eyes bleed. For Dr. Mirzakhani I guess it’s breakfast reading.

Dr. Mirzakhani first came to love mathematics in Iran, where she completed high school and undergraduate studies. I find it very interesting that the first woman to win the Field’s prize was educated in a nation that we westerners consider to be very sexist, and furthermore that she comes from a middle-income country. There are nearly a billion people living in high-income, supposedly comparatively gender-equal nations, but the first female Fields prize winner comes from a middle-income country with a bad record on women’s rights. I think this is indicative of two things: first of all, Iran’s strong support of science; and secondly, the west’s overbearingly sexist attitude towards maths and science. While we in the west like to pride ourselves on the equality of the sexes, it is my opinion that attitudes towards femininity and science in the west are still very backward, and there are major cultural and institutional factors that push women away from fields that they are perfectly capable of performing well in. We also see this in the world of gaming and nerd pursuits, where women are vastly under-represented. This problem does not exist in Asia, where women are encouraged to take up scientific and nerdy pursuits. Certainly in Japan, there is no question about whether a woman could or should do mathematics – it is to be encouraged and admired, and many forms of mathematics that we in the west would consider to be “advanced” or “optional” parts of education (and therefore, through institutional and cultural pressure, tend to select men to learn) are considered an essential and basic part of a woman’s education in Japan. I see this as an Asia-wide phenomenon, and I suspect that it is true of Iran as well that women are considered capable of mathematical achievement. In this aspect of gender equality, I think the west has a long way to go.

Dr. Mirzakhani is also a sterling example of another aspect of maths education that I consider important, and that I have written about before on this blog: it depends very strongly on the attitude of your teachers, and especially on their ability to get students engaged in mathematics and to keep them trained. Dr. Mirzakhani was not originally interested in mathematics, but had her interest fired by a brother’s stories and a teacher’s encouragement. She also was not initially very good at mathematics, but stuck at it, saying:

I can see that without being excited mathematics can look pointless and cold. The beauty of mathematics only shows itself to more patient followers

It takes time and encouragement to develop mathematical skills, and teachers who ignore the slower students because they assume they lack “talent,” or who discourage certain groups or people from taking up this field, are both denying their society the chance to deepen and broaden the level of cultural knowledge of an essential discipline, and also are denying the possibility of access to a beautiful and inspirational world of thought, simply on the basis of their own prejudices. Dr. Mirzakhani obviously benefited from a series of teachers who like to inspire interest and support effort, and don’t judge their students’ potential on the basis of poor early development or gender. The world needs more teachers like those who encouraged Dr. Mirzakhani. Dr. Mirzakhani herself commented on barriers to entering and staying in mathematics earlier this year, suggesting that they are not being lowered:

The social barriers for girls who are interested in mathematical sciences might not be lower now than they were when I grew up. And balancing career and family remains a big challenge. It makes most women face difficult decisions which usually compromise their work

Hopefully this award will be another small step to breaking down some of those social barriers, and encouraging more women into mathematics.

The Guardian article on Dr. Mirzakhani also contains a very nice and powerful quote from another Fields prize winner, Manjul Bhargava:

The mathematics that has been the most applicable and important to society over the years has been the mathematics that scientists found while searching for beauty; and eventually all beautiful and elegant mathematics tends to find applications

I think the importance of beauty and aesthetic sense in driving discoveries in mathematics and physics is often understated, but when you listen to mathematicians and physicists talk it is clear that it is a really important part of how they conceive of problems and solutions. There is an unexpected and deep relationship between our sense of symmetry and beauty, and the deep truths of the natural world. This is also the reason that people who understand mathematics find it so compelling and almost mystical in its beauty, and why I think it is not just an issue of shrunken talent pools when some groups of people are prevented from fully enjoying this field – they are being held back from being part of something truly profound. It’s good to see that whatever barriers still exist for women entering mathematics in Iran or the west, Dr. Mirzakhani was able to overcome them and join this small group of people peering into the deep mysteries of our universe.