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Atiyah-Singer Index Theorem and Poincaré Conjecture

Posted by geoffreyzheng under

Science and Technology
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I have a very interesting Chinese colleague, let’s call him LJ. I’ll probably write about him another day. He’s a Columbia math Ph.D. and a very good math teacher, especially when it comes to anecdotes about mathematicians, particularly every Fields Medalist.

The other day he mentioned that a partner of our company worked on index theorem before going into finance. “What is index theorem?” I asked, innocently.

LJ spent the whole lunch time explaining a grossly simplified version of the Atiyah-Singer theorem in terms of an n-dimensional Gauss-Bonnet theorem: the integral of Gaussian curvature equals to Euler characteristics (Vertices – Edges + Faces). Curvature is a geometrical invariant, and EC a topological invariant. The theorem relates the two apparently distinct values, as they are both intrinsic properties of a manifold.

Atiyah won Fields Medal for it in 1966, and he’s got some interesting co-winners. One is Paul Cohen and another is Alexander Grothendieck. LJ told us some bizarre stories of them before (I’ll retell Cohen’s in another piece), and he thinks Grothendieck is the greatest mathematician in the last 100 years.

The 4th and final 1966 Fields Medalists, Stephen Smale, is the first in a string of FMists mostly for their work on the Poincaré Conjecture. Smale proved it for n >= 5, Michael Freedman did it for n = 4, and Grigori Perelman nailed the coffin with a lot of fascinating controversies.

There’s another Fields Medalist, William Thurston, whose work is related to PC. It was his geometrization conjecture, which subsumes PC, that Perelman actually proved.

Another mathematician central to the PC circus, Richard Hamilton, probably deserves FM for Ricci flow, which he introduced at the age of 39. LJ took his class in Columbia, and said that Shing-Tung Yau recognized Hamilton’s work early on.

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