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An extended abstract of Lie theory after Sophus Lie2 Sep 2013
- Published on Amazon.com
Not that Armand doesn't mention that Sophus Lie's initial effort was to make a unified closed form solution to differential equations, but if you want to know about that mathematics, you have to go somewhere else - such as Thomas Hawkins; a book I'm hoping to get to soon enough next.
Armand Borel's book treats of Elie Cartan's differential geometry in terms of Sophus's Lie's work, then Hermann Weyl's Lie algebra work; the interplay between Lie algebra's and Lie groups; global and local aspects. Then there's Chevelley's algebraic groups; algebraic groups are varieties and groups combined. Varieties are a vast generalization of the fundamental theorem of algebra due to David Hilbert. Chevelley also found the algebraic groups of E8 - the last Lie group not solved by Elie Cartan; it was recently solved by computer around 2001. Armand Borel's book also doesn't get into Hilber's fifth problem about Lie groups.
Coming from another direction, in the eighteen hundreds, mathematicians found invariance theory and abstract algebra through Galois's work. In the nineteen hundreds, Emmy Noether unified both; but quickly, Lie theory in general put even Emmy Noether's great work in (non)commutative algebra in its place in much the same way Newton/Liebniz's calculus put algebra, trigonometry, and geometry in their places, and could solve their problems with comparable ease.
I don't know if it would be correct to consider Lie theory the ultimate mathematics of today; topology probably would stand up and be heard; but, then again they get combined with one another anyways. Overall, this book is a thrill ride and guide to much of the frontier's of modern mathematics.