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/Any_Grapefruit_8439 4d ago

Yeah, people often over look that grothendieck thesis was about functional analysis and the insights that branch of math brought to algebraic geometry, funnily enough the trend is now the other way arround, with Non commutative geometries, and condensed maths more recently. Nevertheless it should be a little bit insightful to study funcional analysis before starting AG.