Beautiful Minds

Today is the one-year anniversary of Beautiful Minds of Math! As I progress into a second year of writing posts on this blog, I wanted to cover and celebrate an influential yet humble mathematician who inspired many to explore math: Maryam Mirzakhani, an Iranian mathematician and the first woman to receive the Fields Medal in 2014 (she was also an IMO gold medal winner!). 

When we study geometry, we usually start with the idea that parallel lines never meet and that the angles of a triangle always add up to 180 degrees. However, Maryam Mirzakhani built her career on exploring spaces where those rules do not hold: her work focused on hyperbolic geometry and Riemann surfaces, areas that look very different from the flat geometry of Euclidean space, the kind that most high school students in general see in class. 

In hyperbolic geometry, space curves inward; this means many lines can pass through a point without ever meeting another line, and many other Euclidean truths that we take for granted, such as the sum of all the angles in the triangle, no longer hold true. Mirzakhani studied these surfaces by looking at a phenomenon that describes how one surface can change into another while keeping certain key features: Moduli spaces. This observation of moduli spaces allowed her to topologically understand how shapes behave when they are stretched or bent in different ways. 

A major part of her research involved studying geodesics, the shortest paths between two points in a curved space: on a sphere, for example, a geodesic is a great circle like the equator. Mirzakhani developed ways to count and describe closed geodesics that loop around without crossing themselves. Her results gave mathematicians new tools to predict how these paths behave and how complex surfaces can be organized.

Her work has connections to many areas of science despite its heavy emphasis on theoretical ideas. Concepts from hyperbolic geometry appear in interdisciplinary fields beyond math, such as physics. This is especially important in understanding the shape of the universe and the paths of particles. They also appear in network science and computer modeling, where the relationships between nodes often resemble the structure of curved spaces. Her insights into how surfaces evolve over time help researchers describe systems that change continuously. 

Mirzakhani’s discoveries have also had a lasting influence on interdisciplinary fields such as computer science. Her study of curved surfaces helps researchers understand how information moves through complex networks. In computer science, hyperbolic geometry offers a way to represent data that grows quickly, like the structure of the internet or the connections in artificial intelligence models. These ideas make it easier to map and search through huge amounts of information without losing efficiency, and they also improve how computers learn from patterns by showing new ways to group and compare data. Her work provides the mathematical foundation for displaying high-dimensional information in simpler forms that humans can interpret in areas such as data visualization and computer graphics. Mirzakhani’s mathematics shows that even the most abstract ideas can become practical tools for building technology and solving real problems.

What I admire most about Mirzakhani is how she approached mathematics. Her visualization of complex problems made them more approachable and reminded me of the way I try to draw diagrams to plot certain information given in competition math problems (especially those that are graph or geometry-based). That made me see that thinking through pictures and sketches is not a distraction but a valid way to reason about math. Her example encouraged me to keep exploring concepts that seem too abstract at first and to see mathematics as something that can grow from curiosity.

Maryam Mirzakhani showed that imagination and persistence can open new directions in mathematics. Her research continues to influence how mathematicians think about geometry and motion, and her story continues to inspire students like me to keep going when the problems look too complicated to solve.

White, Martin L. “Maryam Mirzakhani | Biography & Facts.” Encyclopaedia Britannica, Encyclopaedia Britannica, Inc., www.britannica.com/biography/Maryam-Mirzakhani. Accessed 10 Nov. 2025. Encyclopedia Britannica

Kalaydzhieva, Nikoleta. “The Mathematics of Maryam Mirzakhani.” Chalkdust Magazine, 18 Oct. 2017, chalkdustmagazine.com/features/mathematics-maryam-mirzakhani/. Accessed 10 Nov. 2025. chalkdustmagazine.com

“Maryam Mirzakhani: 1977–2017.” Notices of the American Mathematical Society, vol. 65, no. 10, Nov. 2018, pp. 1221-1239, American Mathematical Society, www.ams.org/journals/notices/201810/rnoti-p1221.pdf. Accessed 10 Nov. 2025.