# Maryam Mirzakhani and Multidimensional Manifolds

##### “The beauty of mathematics only shows itself to more patient followers.”– Maryam Mirzakhani (1977-2017)

Throughout history, women have played a pivotal role in changing our world with their discoveries and research in STEM. Therefore, it is essential to shed light upon their astounding achievements in order to encourage girls and women in our current society. One such pioneering mathematician was Maryam Mirzakhani, an Iranian mathematician who became the first and only woman to date to be awarded the prestigious Fields Medal for her work on geodesics.

##### Understanding Riemann Surfaces

Riemann surfaces are a key structure in topology. A Riemann surface is a two-dimensional manifold formed by an analytic function. A manifold is a topological area in which there is a resemblance to our classical, Euclidean geometric shapes. Manifolds can occur on numerous dimensions; whilst one-dimensional manifolds encompass geometric areas like lines, two-dimensional manifolds are deemed as ‘surfaces’ (this is where the word ‘surface’ in ‘Riemann surfaces’ arises from). Meanwhile, analytic functions are functions relating to complex algebra. In order to understand, this we need to envision numbers as a number line: The numbers along this number line are referred to as real numbers. However, in mathematics, the variety of numbers extends beyond this number line, resulting in complex numbers. Typically, they are depicted with the symbol i, representing ‘imaginary numbers’. Complex numbers can be explained through the following equation:

i2 = -1

Using real numbers, this equation is not possible, as the square of a real number is always positive. Therefore, complex numbers can be envisioned along a Cartesian plane: Complex numbers in themselves form a set that encompasses all numbers. The formula that represents this set is as follows:

z = x  + yi

In which:

• x represents the real numbers
• y represents the coefficient of the imaginary number (e.g., 3i); therefore, this is the imaginary part of z

The function of f(z), which plays a pivotal role in understanding complex functions, can be defined as

f(z) = u+vi

In which u is the equivalent of x and v is the equivalent of y in the previous equation.

Therefore, a function is described as being analytic if a complex derivative of f’(z) is obtained. These analytic functions are the functions out of which Riemann surfaces are generated. Riemann surfaces contain complex structures: these are manifolds in which the atlases that form the shapes are holomorphic in nature (the term holomorphic simply refers to a function that has a complex variable in it).

##### Maryam Mirzakhani’s Research

Maryam Mirzkhani was awarded the Fields Medal for her “outstanding contributions to the dynamics and geometry of Riemann surfaces and their moduli spaces”. This primarily focused upon her work on geodesics and other works regarding Riemann surface dynamics, including her research relating to Teichmüller spaces. One of her initial mathematical breakthroughs was counting closed geodesics on specific Riemann surfaces (hyperbolic Riemann surfaces, which differs from Euclidean geometrical Riemann surfaces). This, according to topologist Benson Farb, would be the “key object to unlocking the structure and geometry of the whole surface”. She then focused upon Teichmüller spaces. This resulted in her proving a conjecture made by William Thurston that theorised that in Teichmüller spaces the earthquake flow (in which one hyperbolic manifold changes into another) is ergodic- this simply means that it moves throughout all the points in the Riemann surface.

Born on the 12th of May in 1977 in Iran, Maryam Mirzakhani always had an aptitude in mathematics since she was a child. She won two gold medals in the International Mathematical Olympiad, making her the first Iranian to do so. After earning her degree in mathematics from the Sharif University of Technology in Iran, she earned her PhD from Harvard University in 2004, where her PhD thesis was centralised around counting geodesics (one of her monumental achievements). After becoming a research fellow at the Clay Mathematics Institute, she was a professor at Princeton University and subsequently at Stanford University. In 2014, she earned the Fields Medal award, making her the first woman in history and to date (as of July 2021) to win this prestigious award. Unfortunately, Mirzakhani died at the age of 40 due to cancer. Despite her death, her legacy lives on. The 12th of May is now deemed as International Women in Mathematics Day in her honour. Her achievements and contributions to mathematics and society continue to inspire young girls and women today. As Mirzakhani once said: “I like crossing the imaginary boundaries people set up between different fields.”

##### References

• https://media.newyorker.com/photos/596d3b7a25778a3e4d191cd7/4:3/w_999,h_749,c_limit/Roberts-Maryam-Mirzakhanis-Pioneering-Mathematical-Legacy.jpg
• http://groups.csail.mit.edu/mac/users/kkylin/riemann.html
• https://www.math.toronto.edu/~drorbn/classes/0405/Topology/BlownTorus.html
• https://www.ziptiedomes.com/images/sphere/2vsphere.jpg

Informational Sources

• Mirzakhani, M., 2006. Simple geodesics and Weil-Petersson volumes of moduli spaces of bordered Riemann surfaces. Inventiones mathematicae, 167(1), pp.179-222.
• https://math.mit.edu/~jorloff/18.04/
• https://web.stanford.edu/~aaronlan/notes.html
• https://www.mathunion.org/imu-awards/fields-medal/fields-medals-2014
• https://www.quantamagazine.org/maryam-mirzakhani-is-first-woman-fields-medalist-20140812/

Image Sources

• Torus: https://mathworld.wolfram.com/Torus.html
• Riemann Surface (f(z) =  z1/3): https://upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Riemann_surface_cube_root.svg/178px-Riemann_surface_cube_root.svg.png
• Riemann Surface (f(z) = log z): https://upload.wikimedia.org/wikipedia/commons/thumb/a/ab/Riemann_surface_log.svg/155px-Riemann_surface_log.svg.png ### Ishtanetra Siva

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