🤓of the day(ish): Michael Atiyah

Atiyah is one of the main forces at the intersection of differential geometry and topology of the 20th century. Along with people like Raoul Bott and Nigel Hitchin, he found innovative ways to use smooth manifolds to explore discrete topology https://i.imgur.com/FK3d0HA.jpg

ヽ(⌐■_■)ノ♪♬ @ @gauntlet/🤖Ventures/tldr/Aera \\ ¯\_(ツ)_/¯ Struggling with the nature of randomness \\ main: @gaa \\ /red-bull /defi-research maxi

But Atiyah did much more than tie functional analysis to topology. He extended classical cohomological theories (de Rham, Dolbeault) to classes of topological (e.g. equivalence classes of Fibre and Vector bundles over a base space) and used spectral sequences to map classical results to the abstract realm

His work in K-Theory (which I mainly ended up reading via Hatcher's excellent introduction to topological K-Theory) is extremely beautiful — one can use K-Theory to construct geometric objects to represent classical algebraic problems such as division algebras and diophantine equations

Finally, I'd be remiss to not mention the Atiyah-Donaldson-Hitchin-Manin construction of instantons — physicists had posited that instantons (gauge connection forms) existed from quantum tunneling, but it took ADHM to formalize how to compute instantons for generic spacetimes and manifolds

One of his works that impacted me the most when I was in undergrad was the Atiyah-Singer index theorem. It was so counterintuitive (at first) that a smooth, infinite dimensional object (differential operators on a manifold) could somehow give you info about discrete, combinatorial structure in a smooth manifold

did I see this here? I came across this recently and enjoyed this series: https://youtu.be/yS6XQ7MElVI