# Week 1

## Session 1

This is the first session, which will just be an introduction to the course. No pre-class material. Details are summarised here:

## Session 2

### Pre-class videos 58.38

Most sessions will only have 20-40 minutes of pre-class videos. This "feature-length" session 1 is there:

• to get us quickly to the point where we can answer questions and get started on in-depth projects;

• to give us more time to focus on the deeper ideas later in the course.

• ### Pre-class exercises

Exercise:

Compute $\exp\begin{pmatrix}0&x\\ 0&0\end{pmatrix}$ and $\exp\begin{pmatrix}0&a&c\\ 0&0&b\\ 0&0&0\end{pmatrix}$ .

Exercise:

Verify the Baker-Campbell-Hausdorff formula for $\exp\begin{pmatrix}0&a_{1}&c_{1}\\ 0&0&b_{1}\\ 0&0&0\end{pmatrix}\exp\begin{pmatrix}0&a_{2}&c_{2}\\ 0&0&b_{2}\\ 0&0&0\end{pmatrix}$ . [Hint: In this case, the formula is $\exp\left(X+Y+\frac{1}{2}[X,Y]\right)$ with no further correction terms. Can you see why?]