#### UM Math Graduate Students Seminar

Dr. Drew Armstrong
University of Miami

will present

#### What is... a Root System?

Friday, September 23, 2011, 4:00pm
Ungar Building Room 402

Abstract: Fact 1: Every isometry $$\varphi:\mathbb{R}^n\to\mathbb{R}^n$$ is affine. That is, we can write $$\varphi(x)=Ax+b$$ for some matrix $$A$$ and vector $$b$$. Fact 2: (Cartan-Dieudonné) Every othogonal matrix $$A\in O(n)$$ is the product of at most $$n$$ reflections. Conclusion 1: The study of euclidean geometry is the study of the orthogonal group $$O(n)$$. Conclusion 2: The orthogonal group is generated by (infinitely many) reflections. Question 1: What can be said about groups generated by finitely many reflections? Answer 1: Quite a lot, actually.

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