Well, in at least some parts of Afghanistan, girls trying to be educated caused disasters where they were brutalized, raped, or it was considered some kind of horrible religious offense that they should even be allowed to be educated. In Maryam Mirzakhani's case, though, she was educated in schools and undergraduate college in Iran and apparently was quite pro-active and non-shy about it, requesting and getting special education. Apparently she had a sympathetic headmaster in her school who wanted to help her. ------------ This Guardian piece S.Anderson pointed me to re the field medalists seems quite excellent... but that is since they just copied it from the IMU: http://www.theguardian.com/science/alexs-adventures-in-numberland/2014/aug/1... "Another outstanding result of Avila is his work, with Giovanni Forni, on weak mixing. If one attempts to shuffle a deck of cards by only cutting the deck - that is, taking a small stack off the top of the deck and putting the stack on the bottom - then the deck will not be truly mixed. ... but if a closed interval is cut into several subintervals... e.g. cut into four pieces, ABCD, and then one defines a map on the interval by exchanging the positions of the subintervals so that, say, ABCD goes to DCBA. (Always apply a "derangement" permutation.) By iterating the map, one obtains a dynamical system called an 'interval exchange transformation'. Avila and Forni showed that almost every interval exchange transformation is weakly mixing; in other words, if one chooses at random an interval exchange transformation, the overwhelmingly likelihood is that, when iterated, it will produce a dynamical system that is weakly mixing." --that could be used to make a pseudo-random number generator, by iterating a 4 piece bijective map, each piece of the form x --> x+a. However, this would be fairly poor since it will require a large number of iterations (at least comparable to the number of bits in a machine word) before it becomes likely that adjacent integer x are no longer. Also, it seems to need "if" statements, which are somewhat slow. Still, it might be a useful ingredient in a combination generator... http://w3.impa.br/~avila/qmath.pdf considers a certain linear "Schrodinger" operator arising in a 1-dimensional physics model, the "wave function" being defined only at integer locations. This is just an infinite matrix. Avila & Jitomirskaya prove the eigenvalue spectrum of this matrix, is a Cantor set. My mind is somewhat boggled. I mean, the matrix is countably-infinite-dimensional. A Cantor set, although 'discrete', has the same cardinality as the real numbers. How can the number of eigenvalues exceed the number of dimensions of the matrix??? http://en.wikipedia.org/wiki/Cantor_set#Cardinality "Bhargava asked, what can be said about the rational points on a typical curve? In joint work with Arul Shankar and also with Christopher Skinner, Bhargava came to the surprising conclusion that a positive proportion of elliptic curves have only one rational point and a positive proportion have infinitely many. Analogously, in the case of hyperelliptic curves of degree 4, Bhargava showed that a positive proportion of such curves have no rational points and a positive proportion have infinitely many rational points. For higher degree hyperelliptic curves... the typical hyperelliptic curve has no rational points at all." --this is useful for generating conjectures... Martin Hairer: his work sounds potentially very important for trying to rigorize physics. "Mirzakhani... closed geodesics on a hyperbolic surface. These are closed curves whose length cannot be shortened by deforming them. A now-classic theorem proved more than 50 years ago gives a precise way of estimating the number of closed geodesics whose length is less than some bound L. The number of closed geodesics grows exponentially with L; specifically, it is asymptotic to exp(L)/L for large L... Mirzakhani looked at what happens to this 'prime number theorem for geodesics' when one considers only the closed geodesics that are simple, meaning that they do not intersect themselves. The behavior is very different in this case: the growth of the number of geodesics of length at most L is no longer exponential in L but asymptotic to c*L^(6g - 6) for large L (going to infinity), where the constant c depends on the hyperbolic structure and g is the genus of the surface." She apparently knows all about "moduli space", whatever that is. "Mirzakhani, showed the system known as the 'earthquake flow,' which was introduced by William Thurston (a 1982 Fields Medalist), is chaotic." --what the hell did that mean? I think, not sure, that the "earthquake map" means this. If you have a hyperbolic surface (i.e. closed 2-manifoldl with constant negative curvature) and it has some simple closed geodesics then cut it along the geodesic then glue it back together shifted circularly. One could also do this for, say, a sphere (cut along a great circle and glue back one half rotated), or a flat torus. Is that right? What is the meaning of what it is claimed she proved? -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)