In this talk, I'll start with definition of Carlitz polynomial and discuss about some analogy of that with polynomial $X^m -1$ over rational numbers . After that, we will see the module structure of $\bar{F_p(T)}$ as $F_p(T)$-module using Carlitz polynomial. So, that leads us to the definition of Carlitz extension which is analogue of cyclotomic extension over rational numbers. At the end, we will see the analogue of Quadratic Reciprocity over finite fields.

The classifying space $BG$ of a group $G$ can be filtered by a sequence of subspaces $B(q,G)$, using the descending central series of free groups. The smallest subspace in this filtration is $B(2,G)$ which is obtained from commuting elements in the group. When $G$ is finite describing these subspaces as homotopy colimits is convenient to study the cohomology, and also generalized cohomology theories. I will describe the complex $K$-theory of $B(2,G)$ modulo torsion, and discuss examples where non-trivial torsion part appears.

Let $\mathcal{H} \subset \mbox{Hol}(\mathbb{D})$ be a Hilbert space of analytic functions. A finite positive Borel measure $\mu$ on $\mathbb{D}$ is a Carleson measure for $\mathcal{H}$ if \[ \|f\|_{L^2(\mu)} \leq C \|f\|_{\mathcal{H}}, \qquad f \in \mathcal{H}. \] Equivalently, we can say that $\mathcal{H}$ embeds in $L^2(\mu)$. In 1962, Carleson solved the corona problem. But, besides solving this difficult problem, he opened many other venues of research. For example, he characterized such measures (now called Carleson measures) for the Hardy-Hilbert space $H^2$. However, the same question perfectly makes sense for any other Hilbert space of functions. We will discuss Carleson measures for the classical Dirichlet space $\mathcal{D}$.

We resume next week.