Triangulation Conjecture Disproved

This is a story from Quanta Magazine about a Romanian-American mathematician by the name of Ciprian Manolescu (Wikipedia link) who was born on December 24th 1978. He is now approaching his 39th birthday. The article appeared in January of 2015 but I only read about today via Flipboard. He managed to disprove the triangulation conjecture, which posits that all manifolds (abstract spaces) can be triangulated (one of the most famous problems in topology). He has "the sole distinction of writing three perfect papers at the International Mathematical Olympiad: Toronto, Canada (1995); Bombay, India (1996); Mar del Plata, Argentina (1997)" and "he was elected as a member of the 2017 class of Fellows of the American Mathematical Society for contributions to Floer homology and the topology of manifolds" (quoting from the Wikipedia article).

The article mentions invariants and uses the well known Euler Formula for polyhedra stating that the number of vertices V, faces F, and edges E in a convex 3-dimensional polyhedron, satisfies V + F - E = 2.

An “invariant” is a tool mathematicians use to compare spaces, or manifolds. One famous example is the Euler characteristic, shown here. To calculate it for any two-dimensional manifold, first carve the manifold into polygons (here we use triangles). Next, add the number of faces to the number of vertices and subtract the number of edges. Every sphere will have an Euler characteristic of 2, no matter how the manifold is carved up.

To quote from the article:

He created a new invariant, which he named “beta,” and used it to create a proof by contradiction. Here’s how it works: As we have seen, the triangulation conjecture is equivalent to asking whether there exists a homology 3-sphere with certain characteristics. One characteristic is that the sphere has to have a certain property — a Rokhlin invariant of 1. Manolescu showed that when a homology 3-sphere has a Rokhlin invariant of 1, the value of beta has to be odd. At the same time, other necessary characteristics of these homology 3-spheres require beta to be even. Since beta cannot be both even and odd at the same time, these particular homology 3-spheres do not exist. Thus, the triangulation conjecture is false.

Here is a link to Ciprian Manolescu's UCLA page.