On Root Systems -- Ben Elias
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Root systems embody some of the most beautiful symmetry available in mathematics, while being very easy to visualize and quite accessible.  Put briefly, a root system is a finite set of vectors that fall into some sort of integral lattice, and are closed under their own reflections. I intend to describe root systems and some of their most basic properties, with the goal of proving the classification of root systems by Dynkin Diagrams.  Dynkin Diagrams are ubiquitous structures in mathematics that show up almost magically in diverse fields, and are best introduced with root systems.