User:ProboscideaRubber15/sandbox/mirz

https://watermark.silverchair.com/rnm116.pdf?token=AQECAHi208BE49Ooan9kkhW_Ercy7Dm3ZL_9Cf3qfKAc485ysgAAAicwggIjBgkqhkiG9w0BBwagggIUMIICEAIBADCCAgkGCSqGSIb3DQEHATAeBglghkgBZQMEAS4wEQQMRLvgODApMPUTdUo4AgEQgIIB2uSWzoxgMo71_f2-mhFTK4ubS3IDKqfHdV6Obo_R_lcZmwWDhdaM7isQ1k6i7RQ_dIll2d8-fDwj_Q981fonIUgxOE5SuADCLMlnYOvImuxKt4xgQSRQ6nILnRtJOBSdfeggxQnl9fRKC6QAKWvbc63Zh3ewSe_MSzI1fwmLgSa48BEuI5iR1js9G-XtwEBV99hNOTCu_ZHI1UM0V2R3j6bsTdqkErmPnQKIuQpXe2zD-A4P9iLz64Ai1TbfVelYNkKy9Fkm0H86_DLc1yzqsfr88c5N-GwRDzIwILWdDE93MKX7J6oV5G2KICGpnsbTEgVdvLphhYxgdKEYGv2IMuEMtzTXJFZuCFdyPjDhANVYk_q8gb8xIDz5qlYa8r4NQKsZrFyQkTcxCta3KqHbLLqU6jTts4beiUH3nNIgv0GU7PYj5JKx5tYulQ70skF-geoq1JQHxur0X1JttqtyDxOjoBJXwEvy8xhNuNu0qXIFJyRcgQCaf2Pxl-gnmvSHXOJViHqrGZDcNYP4lFE3fwWBe31B7pTvTQszixNlIDEa4-rYyva3wDqIgslXTgSJey0mxhNrzAv14sxgUNvdz9InQ4WxnJJutLqiYMBrnz_71Ayhz29TDRzmqg

https://www.ams.org/notices/201304/rnoti-p490.pdf

http://www.ams.org/notices/201409/rnoti-p1074.pdf

http://www.ams.org/journals/notices/201803/201803FullIssue.pdf

Previous Version
Mirzakhani made several contributions to the theory of moduli spaces of Riemann surfaces. In her early work, Mirzakhani discovered a formula expressing the volume of the moduli space of surfaces of type (g,n) with given boundary lengths as a polynomial in those lengths. This led her to obtain a new proof for the formula discovered by Edward Witten and Maxim Kontsevich on the intersection numbers of tautological classes on moduli space, as well as an asymptotic formula for the growth of the number of simple closed geodesics on a compact hyperbolic surface, generalizing the theorem of the three geodesics for spherical surfaces. Her subsequent work focused on Teichmüller dynamics of moduli space. In particular, she was able to prove the long-standing conjecture that William Thurston's earthquake flow on Teichmüller space is ergodic.

In 2014, with Alex Eskin and with input from Amir Mohammadi, Mirzakhani proved that complex geodesics and their closures in moduli space are surprisingly regular, rather than irregular or fractal. The closures of complex geodesics are algebraic objects defined in terms of polynomials and therefore they have certain rigidity properties, which is analogous to a celebrated result that Marina Ratner arrived at during the 1990s. The International Mathematical Union said in its press release that "It is astounding to find that the rigidity in homogeneous spaces has an echo in the inhomogeneous world of moduli space."

Rewrite
Mirzakhani made several contributions to the theory of moduli spaces of Riemann surfaces. Mirzakhani’s early work solved the problem of counting simple closed geodesics on hyperbolic Riemann surfaces by finding a relationship to volume calculations on moduli space. Geodesics are the natural generalization of the idea of a "straight line" to "curved spaces". Slightly more formally, a curve is a geodesic if no slight deformation can make it shorter. Closed geodesics are geodesics which are also closed curves—that is, they are curves that close up into loops. A closed geodesic is  simple if it does not cross itself.

A previous result, known as the “prime number theorem for geodesics”, established that the number of closed geodesics of length less than L grows exponentially with L—it is asymptotic to $$ e^L/L$$. However, the analogous counting problem for simple closed geodesics remained open, despite being “the key object to unlocking the structure and geometry of the whole surface,” according to University of Chicago topologist Benson Farb. Mirzakhani’s 2004 PhD thesis solved this problem, showing that the number of simple closed geodesics of length less than L is polynomial in L. Explicitly, it is asymptotic to $$ cL^{6g-6}$$, where g is the genus (roughly, the number of “holes”) and c is a a constant depending on the hyperbolic structure. This result can be seen as a generalization of the theorem of the three geodesics for spherical surfaces.

Mirzakhani solved this counting problem by relating it to the problem of computing volumes in moduli space—a space whose points correspond to different complex structures on a surface genus g. In her thesis, Mirzakhani found a volume formula for the moduli space of bordered Riemann surfaces of genus g with n geodesic boundary components. From this formula followed the counting for simple closed geodesics mentioned above, as well as a number of other results. This led her to obtain a new proof for the formula discovered by Edward Witten and Maxim Kontsevich on the intersection numbers of tautological classes on moduli space.

Her subsequent work focused on Teichmüller dynamics of moduli space. In particular, she was able to prove the long-standing conjecture that William Thurston's earthquake flow on Teichmüller space is ergodic. One can construct a simple earthquake map by cutting a surface along a finite number of disjoint simple closed geodesics, sliding the edges of each of these cut past each other by some amount, and closing the surface back up. One can imagine the surface being cut by strike-slip faults. An earthquake is a sort of limit of simple earthquakes, where one has an infinite number of geodesics, and instead of attaching a positive real number to each geodesic one puts a measure on them.

In 2014, with Alex Eskin and with input from Amir Mohammadi, Mirzakhani proved that complex geodesics and their closures in moduli space are surprisingly regular, rather than irregular or fractal. The closures of complex geodesics are algebraic objects defined in terms of polynomials and therefore they have certain rigidity properties, which is analogous to a celebrated result that Marina Ratner arrived at during the 1990's. The International Mathematical Union said in its press release that "It is astounding to find that the rigidity in homogeneous spaces has an echo in the inhomogeneous world of moduli space."