Abstract: We will discuss some recent results on the $L^2$-theory of the $\overline{\partial}$-equation on the domain $\{\vert z_1 \vert < \vert z_2\vert <1\}$ in $\mathbb{C}^2$. This is joint work with Mei-Chi Shaw.

Abstract: The rational numbers are dense in the reals, so we can always approximate a real as closely as we'd like to by a rational number. But how close can the rational get if we restrict the size of the denominator? If only a few lucky ones can ever get close, the number will be mighty lonely. Many great mathematicians have worked on this problem, and in this talk we'll give a historical survey of their results, along with examples of transcendental and Liouville numbers - the least lonely of them all.