(T) My high-school math teacher liked to draw perfect planes and spheres in a Euclidian space. With those perfect drawings, he expected that we will better understand geometry in a three dimensional spaces.

When thinking about a difficult math problem, Professor Maryam Mirzakhani said “you don’t want to write down all the details. But the process of drawing something helps you somehow to stay connected.” Maryam said that her 3-year-old daughter, Anahita, often exclaims, “Oh, Mommy is painting again!” when she sees the mathematician drawing. “Maybe she thinks I’m a painter,” Maryam said.

I have been trying to get an intuition of the work of Professor Mirzakhani, who was the first woman to receive the field medal in 2014. This is an extremely intimidating task but the more you start to articulate the beginning of an intuition about her work, the more you are motivated to learn more.

Here are a selection of materials to start the journey of understanding her work.

My suggestion would be:

1. Reading: “Field medal news release 2014: The Work of Maryam Mirzakhani“
2. Watching Alex Wright’s video: “Mirzakhani’s universe of abstract surfaces“
3. Get familiar on Wikipedia of the key mathematical concepts, the illumination problem and mathematical billards
4. Read Terrence Tao’s blog posts: “Avila, Bhargava, Hairer, Mirzakhani” and “Maryam Mirzakhani“
5. Read Alex Wright’s paper: “ A tour through Mirzakhani’s work on moduli spaces of Riemann surfaces“
6. Deeper dive again on Wikipedia on the key mathematical concepts
7. Start exploring Maryam’s lectures and other materials from other mathematicians
8. Start exploring Maryam’s papers on Google Scholar and other papers in similar fields
9. Deeper dive again on the key mathematical concepts from lectures and papers
10. Continue exploring Maryam’s papers, lectures, and other materials from other mathematicians

Professor Mirzakhani’s research topics and mathematical concepts

Her research areas focused on:

• Techmuller theory, hyberbolic geometry, ergodic theory, simplectic geometry and dynamical systems theory

Key mathematical concepts needed to understand her work includes:

• the Teichmuller-type moduli spaces, Riemman surface, the moduli spaces of Riemman surfaces, closed geodesics, the special linear group SL₂(R), the Witten conjoncture, the illumination problem, mathematical billards

Professor Mirzakhani’s papers and talks

• Google Scholar

• 2012 Marston Morse Lectures on Dynamics on the Moduli Spaces of Curves at IAS
• Maryam’s Lecture: Dynamics on the moduli spaces, I

• Maryam’s Lecture: Dynamics on the moduli spaces, II

• Maryam’s Lecture: Dynamics on the moduli spaces, III

• Harvard University, Current Developments in Mathematics 2014
• Maryam’s Lecture: Dynamics Moduli Spaces of Curves I

• Maryam’s Lecture: Dynamics Moduli Spaces of Curves II

Professor Alex Wright on the work of Professor Mirzakhani

Alex Wright worked with Professor Mirzakhani at Stanford, and had for PhD advisor Alex Eskin who collaborated with Maryam on key papers.

• Mirzakhani’s universe of abstract surfaces
• A tour through Mirzakhani’s work on moduli spaces of Riemann surfaces

Professor Terence Tao on the work of Professor Mirzakhani

• Blog post: Avila, Bhargava, Hairer, Mirzakhani
• Blog post: Maryam Mirzakhani

Stanford University’s conference on the work of Professor Mirzakhani

• Conference to honor The Mathematical Legacy of Maryam Mirzakhani

Other interesting materials:

• Quanta Magazine: A Tenacious Explorer of Abstract Surfaces
• Field medal news release 2014: The Work of Maryam Mirzakhani
• Professor Curtis T. McMullen (who was Maryam’s PhD advisor): The Work of Maryam Mirzakhani
• Paper: “Everything is illuminated” from Samuel Lelievre, Thierry Monteil, and Barak Weiss
• Seminaire Bourbaki – Jean-Francois Quint – Rigidite des SL2(R) – Orbites dans les espaces de modules de surfaces plates

• Pierre Dehornoy, Courbes et surfaces: Le monde de Maryam Mirzakhani

Note: The picture above is from the Jardin du Thabor in Rennes.

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