# Spaces

I read about the sad passing of Maryam Mirzakhani in July, and the extraordinary trajectory of her career in mathematics. But I did not know much about what she was actually doing. A recent post in Quanta Magazine, with the title: Why Mathematicians Like to Classify Things, caught my attention because the title suggested that the post was about one of the most important ways that mathematics succeeds – namely by finding sameness among diversity. I found that the work discussed in this post addresses the mathematical world to which Mirzakhani has made significant contributions. Looking further into the content of the post and Mirzakhani’s experience invigorated both my emotional and my intellectual responses to mathematics.

A Quanta article by Erica Klarreich was written in 2014, when Mirzakhani won the Fields Medal. There Klarreich tells us that when Mirzakhani began her graduate career at Harvard, she became fascinated with hyperbolic surfaces and, it seems, that this fascination lit the road she would journey. These are surfaces with a hyperbolic geometry rather than a Euclidean one.  They can only be explored in the abstract. They cannot be constructed in ordinary space.

I find it worth noting that the ancestry of these objects can be traced back to the 19th century when, while investigating the necessity of Euclid’s postulate about parallel lines, mathematicians brought forth a new world, a new geometry, known today as hyperbolic geometry.  This new geometry is sometimes identified with the names of mathematicians János Bolyai and Nikolai Ivanovich Lobachevsky.  Bolyai and Lobachevsky independently confirmed its existence when they allowed Euclid’s postulate about parallel lines to be replaced by another.  In hyperbolic geometry, given a line and a point not on it, there are many lines going through the given point that are parallel to the given line.  In Euclidean geometry there is only one.  With this change, Bolyai and Lobachevsky developed a consistent and meaningful non-Euclidean geometry axiomatically.  Extensive work on the ideas is also attributed to Carl Friedrich Gauss. One of the consequences of the change is that the sum of the angles of a hyperbolic triangle is strictly less than 180 degrees.  The depth of this newly discovered world was ultimately investigated analytically.  And Riemann’s famous lecture in 1854 brought definitive clarity to the notion of geometry itself.

With her doctoral thesis in 2004, Mirzakhani was able to answer some fundamental questions about hyperbolic surfaces and, at the same time, build a connection to another major research effort concerning what is called moduli space. The value of moduli space is the other thing that captured my attention in these articles.

In his more extended piece for Quanta, Kevin Hartnett provides a very accessible description of moduli space that is reproduced here:

In mathematics, it’s often beneficial to study classes of objects rather than specific objects — to make statements about all squares rather than individual squares, or to corral an infinitude of curves into one single object that represents them all.

“This is one of the key ideas of the last 50 years, that it is very convenient to not study objects individually, but to try to see them as a member of some continuous family of objects,” said Anton Zorich, a mathematician at the Institute of Mathematics of Jussieu in Paris and a leading figure in dynamics.

Moduli space is a tidy way of doing just this, of tallying all objects of a given kind, so that all objects can be studied in relation to one another.

Imagine, for instance, that you wanted to study the family of lines on a plane that pass through a single point. That’s a lot of lines to keep track of, but you might realize that each line pierces a circle drawn around that point in two opposite places. The points on the circle serve as a kind of catalog of all possible lines passing through the original point. Instead of trying to work with more lines than you can hold in your hands, you can instead study points on a ring that fits around your finger.

“It’s often not so complicated to see this family as a geometric object, which has its own existence and own geometry. It’s not so abstract,” Zorich said.

This way of collapsing one world into another is particularly interesting.  And one of the results in Mirzakhani’s doctoral thesis concerned a formula for the volume of the moduli space created by the set of all possible hyperbolic structures on a given surface.  Mirzakhani’s research has roots in all of these – hyperbolic geometry, Riemann’s manifold, and moduli space.

Her work, and the work of her colleagues, is often characterized as an analysis of the paths of imagined billiard balls inside a polygon. This is not for the sake of understanding the game of pool better, it’s just one of the ways to see the task at hand.  Their strategies are interesting and, I might say, provocative . With this in mind, Hartnett provides a simple statement of process:

Start with billiards in a polygon, reflect that polygon to create a translation surface, and encode that translation surface as a point in the moduli space of all translation surfaces. The miracle of the whole operation is that the point in moduli space remembers something of the original polygon — so by studying how that point fits among all the other points in moduli space, mathematicians can deduce properties of billiard paths in the original polygon.   (emphasis added)

The ‘translation surface’ is just a series of reflections of the original polygon over its edges.

These are beautiful conceptual leaps and they have answered many questions that inevitably concern both mathematics and physics.  In 2014, Klarreich’s article captured some of Mirzakhani’s thoughtfulness:

In a way, she said, mathematics research feels like writing a novel. “There are different characters, and you are getting to know them better,” she said. “Things evolve, and then you look back at a character, and it’s completely different from your first impression.”

The Iranian mathematician follows her characters wherever they take her, along story lines that often take years to unfold.

In the article she was described as someone with a daring imagination.  Reading about how she experienced mathematics made the nature of these efforts even more striking.   There is a mysterious reality in these abstract worlds that grow out of measuring the earth.  The two and three dimensional worlds of our experience become represented by ideals which then, almost like an Alice-in-Wonderland rabbit hole, lead the way to unimaginable depths.   We find purely abstract spaces that have volume.  We get there by looking further and looking longer.  I feel a happy and eager inquisitiveness when I ask myself the question: “What are we looking at?”  And I would like to find a new way to begin an answer. It seems to me that Mirzakhani loved looking.  A last little bit from Klarreich:

Unlike some mathematicians who solve problems with quicksilver brilliance, she gravitates toward deep problems that she can chew on for years. “Months or years later, you see very different aspects” of a problem, she said. There are problems she has been thinking about for more than a decade. “And still there’s not much I can do about them,” she said.