Self-intersection local time $\beta_t$ is a measure of how often
a Brownian motion (or other process) crosses itself. Since Brownian
motion in the plane intersects itself so often, a renormalization
is needed in order to get something finite. LeGall proved that
$E e^{\gamma \beta_1}$ is finite for small $\gamma$ and infinite
for large $\gamma$. It turns out that the critical value is related
to the best constant in a Gagliardo-Nirenberg inequality. I will discuss
this result (joint work with Xia Chen) as well as large deviations
for $\beta_1$ and $-\beta_1$ and LILs for $\beta_t$ and $-\beta_t$.
The range of random walks is closely related to self-intersection
local times, and I will also discuss joint work with Jay Rosen
making this idea precise.