Avery St. Dizier

#### Brief biography

I was born right here in Baton Rouge, and I have lived here my whole life. I went to St. Aloysius for elementary school, where I joined the Boy Scouts and later became an Eagle Scout. I continued on to Redemptorist High School, where I joined Mu Alpha Theta and discovered that math was actually fun and was the thing for me. I am now a junior mathematics major here at LSU, and I am enjoying every day here. I live on campus with three other math majors, and I love the dynamic that we have when we work together and help each other out. I enjoy regular trips to the UREC with my friends for rock climbing and racquetball.

#### Mathematical and Educational Activities

In an effort to better understand the intricacies and difficulties of com- municating mathematics, I regularly tutor in various subjects. I enjoy the challenge of working to convey what I know in a cohesive form to someone else. Communicating the ideas I have been taught exposes the gaps in my own understanding and helps me to consolidate my knowledge, keeping me up to date on information from classes that I might otherwise forget from disuse. Tutoring also allows me to give back to the mathematical community at LSU, which often supports me, particularly in providing opportunities for undergraduate mathematical research.

This semester I am enrolled in an independent reading course under Professor Achar. I intend to complete the groundwork for an undergraduate honors thesis project this semester and to write the thesis next semester or over the summer, time permitting. We will be studying reflection groups.

Reflections are mappings from a Euclidean space to itself that are isome- tries whose set of fixed points is a hyperplane. Concretely, in two dimensions, a reflection would be like folding a piece of paper. Points located along the fold line are fixed, they do not move. Points not along the fold line are sent to their mirror image across the fold. It is also easy to visualize a reflection in three dimensions: the only difference is that we reflect across a plane instead of a line. Reflections can be defined mathematically in any finite number of dimensions.

When we take a set of reflections and combine them in all possible ways, what we get is called a reflection group. It is a collection of transformations of space, each of which can be described as a series of reflections. Reflec- tion groups have remarkable geometric and group theoretic properties. We will be exploring various topics in reflection groups and their more general counterparts, Coxeter groups.