Information and interactive communication

Notions of entropy and information, pioneered by Shannon, have been very powerful tools in coding theory. Coding theory aims to solve the problem of one-way communication: sending a message from Alice to Bob using as little communication as possible, sometimes over a noisy channel.
We will discuss several extensions of information-theoretic notions to the two-way communication setting. We use them to prove a direct sum theorem for randomized communication complexity, showing that implementing $k$ copies of a functionality requires substantially more communication than just one copy.
More generally, we will show that information cost $I(f)$ can be defined as a natural fundamental property of a functionality $f$. We will describe several new tight connections between $I(f)$, direct sum theorems, interactive compression schemes, and amortized communication complexity.