Moduli space and shortest paths on curved surfaces<\/h2>\n
Mirzakhani's primary research lay at the intersection of hyperbolic geometry, topology, and dynamical systems. She focused on Riemann surfaces\u2014curved surfaces possessing hyperbolic geometry\u2014and studied moduli spaces, which are spaces where each point represents a specific hyperbolic surface.<\/p>\n
Her renowned work involves calculating the Weil\u2013Petersson volume of moduli spaces, expressed as a polynomial function of boundary lengths:<\/p>\n
$$ V_{g,n}(L_1, \\dots, L_n) = \\sum_{|\\alpha| \\le 3g-3+n} c_\\alpha , L^{2\\alpha} $$<\/p>\n
Where $g$ is the genus of the surface and $n$ is the number of boundary components, she employed a technique involving the counting of simple closed geodesics to systematically calculate these volumes, thereby solving a problem that had remained open for decades.<\/p>\n
Another equally renowned achievement is the collaboration with Alex Eskin on the dynamics of billiards on polygonal tables; rather than studying a single billiard table, they investigated the \u201cuniverse of all possible billiard tables\u201d and proved the \u201cmagic wand theorem,\u201d which reveals a surprising regularity in the trajectories within these complex spaces.<\/p>\n
A historic Fields Medal<\/h2>\n
In 2014, in Seoul, Mirzakhani was awarded the Fields Medal for her \"outstanding contributions to the dynamics and geometry of Riemann surfaces and their moduli spaces.\" Upon receiving the award, she expressed her deep honor and her hope that she would inspire the next generation of female scientists and mathematicians.<\/p>\n
Tragically, just three years later, she discovered that the breast cancer she had previously overcome had spread to her bone marrow; she passed away on July 14, 2017, leaving behind an intellectual legacy that continues to inspire future generations of mathematicians worldwide\u2014particularly young women in science and mathematics who see that there is no ceiling on what is possible.<\/p>\n
Thought-provoking activities<\/h2>\n\n- Why is the moduli space of Riemann surfaces viewed as a \u201cspace of spaces,\u201d and how does this concept differ from Euclidean geometry?<\/li>\n
- Explain the difference between Euclidean geometry and hyperbolic geometry in terms of the sum of the angles of a triangle.<\/li>\n
- How does Mirzakhani's research on billiard dynamics link geometry with dynamics? Let's consider a simple dynamical system that students are familiar with.<\/li>\n
- Discuss Mirzakhani's role as an inspiration for gender equality in STEM.<\/li>\n<\/ol>\n
\nReferences<\/p>\n
Britannica. (2026, July). Maryam Mirzakhani. Encyclop\u00e6dia Britannica<\/em>. https:\/\/www.britannica.com\/biography\/Maryam-Mirzakhani<\/p>\nKlarreich, E. (2014, August 12). A tenacious explorer of abstract surfaces. Quanta Magazine<\/em>.<\/p>\nMyers, A., & Carey, B. (2014, August 12). Stanford’s Maryam Mirzakhani wins Fields Medal. Stanford Report<\/em>. https:\/\/news.stanford.edu\/stories\/2014\/08\/stanfords-maryam-mirzakhani-wins-fields-medal<\/p>\n
\nThis article is part of the \u201cThis Day in Mathematical History\u201d series\u2014retracing the stories of the mathematicians who shaped the world of mathematics, day by day throughout the year.<\/em><\/p>\n![\"\"](\"https:\/\/mathed.bru.ac.th\/wp-content\/uploads\/2026\/07\/ChatGPT-Image-14-\u0e01.\u0e04.-2569-12_54_12-300x200.jpg\")
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References<\/p>\n
Britannica. (2026, July). Maryam Mirzakhani. Encyclop\u00e6dia Britannica<\/em>. https:\/\/www.britannica.com\/biography\/Maryam-Mirzakhani<\/p>\n Klarreich, E. (2014, August 12). A tenacious explorer of abstract surfaces. Quanta Magazine<\/em>.<\/p>\n Myers, A., & Carey, B. (2014, August 12). Stanford’s Maryam Mirzakhani wins Fields Medal. Stanford Report<\/em>. https:\/\/news.stanford.edu\/stories\/2014\/08\/stanfords-maryam-mirzakhani-wins-fields-medal<\/p>\n This article is part of the \u201cThis Day in Mathematical History\u201d series\u2014retracing the stories of the mathematicians who shaped the world of mathematics, day by day throughout the year.<\/em><\/p>\n
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