Mirzakhani and Meanders
As part of the panel event which took place as part of this year’s HLF on the life and influence of Mayam Mirzakhani, we heard from several people who have found her inspirational. Two of the young researchers shared their experiences in following in her lead, to try to ‘make a footprint in the field’; we also heard from Andrea Vera Gajardo, organiser of the May 12 Initiative which celebrates women in mathematics every year on Maryam Mirzakhani’s birthday; and Hélène Barcelo, from the Mathematical Science Research Institute, spoke about the chair named in Mirzakhani’s honour. It was an impressive collection of stories showing her wide-reaching impact as a role model.
But the session also included an insight into Mirzakhani’s impact as a mathematician – Vincent Delecroix, a researcher at LaBRI in Bordeaux, studies dynamical systems and curves on surfaces, and described a recent problem he and his colleagues worked on which brought them into Maryam Mirzakhani’s mathematical world.
The work concerned studying a particular type of curve called a ‘meander’. Given a horizontal line dividing a space in two, a meander is a single closed curve which goes above and below the line, and is made up of semicircular arcs of different sizes.
The number of times the meander crosses the line can be any even number from two upwards, and one problem Delecroix has been working on is enumeration of these curves – for a given number of crossings, how many different possible meanders are there?
With two crossings, there is only one possible meander (a circle), and for four there are two, shown above left. For 6 crossings there are 8 possible meanders (shown above right), and for 8 crossings there are 42. The number of possible meanders of each order (an order N meander has 2N crossings) has been counted for smaller numbers of crossings (it’s known for up to 84 crossings), but in general there’s no currently known simple formula for calculating these numbers. The number of meanders of order N is denoted MN, and researchers are trying to understand the behaviour of this sequence – how fast does it grow?
These curves can be considered in their literal interpretation – if you’re building a road across land with a river on it, the river might cross the road, and depending on the shape of the river and the placement of the road, it will cross the road a different number of times. But it’s also a more theoretical question – in mathematics, the surface you’re building a road on might not be two-dimensional, and might use a different type of geometry. Topologists study the number of possible curves on a shape as a way of understanding its structure, and enumerative geometry considers these kinds of questions.
We can vary the definition of a meander in a few different ways – for example, we’re considering closed curves here, but an ‘open meander’ is one where the two ends of the curve don’t join up, and these can have odd numbers of crossings. You can also consider the line you’re crossing to be on the surface of an object like a sphere – in which case the question becomes more like, how many rivers cross the equator 2N times? In this case, we consider the left-hand crossing to be directly next to the right-hand crossing, as the line wraps around and connects back to itself.
Some of the arches making up the meander are small ones which connect a crossing to the one directly next to it (this includes arches connecting the two ends, if you’re wrapped around a sphere). Such an arch is called a minimal arch, and this is another aspect of meanders which can be studied – how many meanders of order N (with 2N crossings) have K minimal arches? The answer is denoted M(N,K), and this is a particular subject of study for Delecroix and his collaborators.
In their paper, ‘Enumeration of Meanders and Masur-Veech Volumes’, they give a formula for counting the number of meanders with less than a given number of crossings, which gives an indication of how this number will grow. The formula is given here, although Delecroix admits ‘it’s not so nice’:
This is where the connection to Maryam Mirzakhani becomes apparent: she also worked on counting the number of possible curves on surfaces – although in her case she was studying hyperbolic spaces. These involve a different definition of what a straight line is, and involve ‘cusps’ – points in the space where the geometry is distorted and behaves strangely. Mirzakhani studied simple closed curves – which join back up without crossing themselves – and multi-curves (collections of curves).
Mirzakhani proved a theorem counting the number of possible curves, and determining how it changes as you increase the length of the curves. As above, the formulae were generalisations of how quickly the numbers grow as the length of the curve increases to a limit – but if you look at the formula, you might notice some common elements, highlighted in colour here:
“Counting curves and counting meanders are actually very similar,” Delecroix explains – “[you can] see a meander as a pair of simple closed curves on a surface; the minimal arches are very much related to the cusps. You can use Mirzakhani’s theorem to prove our result – basically the two statements are equivalent.”
Mathematics often involves using theorems and results proved by other mathematicians – it’s one of the great strengths of a subject in which truth can be considered universal. In this case the connection between the two different areas means that Delecroix and his collaborators can now use tools Mirzakhani developed in continuing this research – it’s a part of her legacy, as for anyone who adds to our collective knowledge by research in maths.