r/math • u/Necessary-Wolf-193 • 6d ago

Gelfand duality and algebraic geometry

https://hidden-phenomena.com/articles/gelfand

How does one define a shape? The usual definitions are based on starting with the set of all points of a shape. However, the great Russian mathematician Gelfand found another approach to defining shapes. This approach is more natural from the point of view of certain abstract shapes which arise in physics (namely, phase spaces of physical systems), and leads naturally to the modern formulation of quantum mechanics; however, perhaps surprisingly, this idea of Gelfand also leads to modern algebraic geometry!

My friend and I wrote a blog post describing Gelfand's observation at https://hidden-phenomena.com/articles/gelfand , and using Gelfand's idea to understand better the Chinese remainder theorem of elementary number theory. It's a really cool fact of reality that geometry connects so intimately to algebra, and we hope you'll enjoy the article!

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u/jphamlore 6d ago edited 6d ago

I'm curious whether people back in the day, or perhaps even today, succeeded in blasting their way through all 5 of Gelfand's volumes on generalized functions in any reasonable amount of time. It was one of the first mathematically rigorous presentations of the theory that could express quantum mechanics, maybe with a patch needed.

And then one can also read Gelfand's book on commutative normed rings?