When I read the news that Maryam Mirzakhani had sadly passed away with breast cancer aged 40 I was honestly shocked. I remember finding out that she was the first women to win the Fields Medal in 2014 and feeling a huge sense of pride that we had achieved such a big milestone in mathematics – a mostly male dominated subject.

Professor at Stanford Universtiy, Mirzakhani was awarded the notorious Field’s Medal for her work on complex geometry and dynamic systems. She specialised in areas of theoretical mathematics that “*read like a foreign language by those outside of mathematics”* such as moduli spaces, Teichmüller thoery, hyperbolic geometry, Ergodic theory and symplectic geometry. By mastering these fields, Mirzakhani could describe the geometric and dynamic complexities of curved surfaces, spheres, donut shapes and even amoebas in a huge amount of detail. Furthermore, her work had implications in a vast amount of fields, ranging from cryptography to the physics of how the universe was created.

#### Moduli Spaces

Moduli Spaces can be thought of as geometric solutions to geometric classification problems. In broad terms, a moduli problem consists of three main categories:

**Objects:**which geometric objects do we want to parametrise?**Equivalences:**when do we identify two objects as being isomorphic?**Families:**how do we allow our objects to vary?

Read more here.

#### Teichmüller Theory

Teichmüller theory, which brings together an array of fundamental ideas from different mathematical fields (including complex analysis, hyperbolic geometry, differential geometry, etc), is concerned with the Teichmüller space.

To get an short introduction to Teichmüller theory, click here.

#### Hyperbolic Geometry

Hyperbolic geometry is a non-Euclidean geometry, where the parallel postulate of Euclidean geometry is replaced with:

“

For any given line R and point P not on R, in the plane containing both line R and point P there are at least two distinct lines through P that do not intersect R.” – Wikipedia

#### Ergodic Theory

Ergodic theory was initially developed to solve problems in statistical physics and is a branch of mathematics that studies “dynamical systems with an invariant measure”. An invariant measure is a measure that is preserved by some function.

#### Symplectic Geometry

Symplectic Geometry is a branch of differential geometry and differential topology that studies symplectic manifolds. These are differentiable manifolds that have a closed, non-degenerate 2-form.

“Mirzakhani once described her work as ‘

like being lost in a jungle and trying to use all the knowledge that you can gather to come up with some new tricks, and with some luck you might find a way out’.”– Guardian

Mirzakhani will be remembered not only for her extraordinary work, but also as being an inspiration to thousands of women to pursue maths and science.

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