[fermenflo]

Mirzakhani will always have a special place in my heart.

I was an undergraduate mathematics student when I discovered her work. It was a paper on closed geodesics and there was something special about her writing. Her approaches were simple an elegant -the kind that made you, as a reader, feel accomplished for understanding such a complex subject. It wasn't long until she was placed among other grand mathematicians that I looked up to.

A year later she died. I wasn't even aware of her health. It sucked to see an idol go so young. But it's incredible what she accomplished within her lifetime. She'll always be one of the greatest.

[benrbray]

For the curious, do you have a link to the paper?

[fermenflo]

I don't remember if this is the exact paper but as united893 already posted, the paper was probably:

https://annals.math.princeton.edu/wp-content/uploads/annals-...

Long story short: Imagine you have an object, place an ant on the surface of the object, and then instruct the ant to walk in a straight line forever. Will the ant ever end up in the same place it started (with the same initial direction)? If so, then it has formed a closed geodesic. For some objects, the answer is obvious. For a perfect sphere, the answer is always yes. In fact, any sphere-like object (imagine warping/contorting a sphere without tearing or poking holes in it) will always have at least 3 such closed geodesics: https://en.wikipedia.org/wiki/Theorem_of_the_three_geodesics.

Miriam managed to construct an amazing formula that, when given the number of holes in an object, can give you the probability of forming a closed geodesic when starting from a random point in a random direction.

This: https://www.youtube.com/watch?v=Sx-kAlEpiZk is a great video that goes over what I explained and a couple other great achievements of hers. Worth a watch.

[united893]

Here's her two most cited papers on Geodesics. First one is her most cited work.

https://www.math.stonybrook.edu/~mlyubich/Archive/Geometry/T...

https://annals.math.princeton.edu/wp-content/uploads/annals-...