**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.