Title: A century (or more) of entropy 

Abstract: One of the great triumphs of 19th -century science was the emergence of thermodynamics. This is a subject of great power and generality, setting down the rules for what is possible and, even more crucially, what is not possible: there can be no perpetual motion machines, heat flows from hot bodies to cold bodies, and any effort to convert energy from one form to another always involves a bit of waste. A central, if slightly mysterious concept in thermodynamics is the entropy, which is introduced first as a bookkeeping device but then becomes fundamental. In the formulation of statistical mechanics, the bridge connecting our microscopic description of atoms and molecules to the macroscopic phenomena of our everyday experience, entropy reappears as a measure of the number of states that are accessible to all the atoms and molecules. In the mid-twentieth century, entropy makes yet another appearance, first as a quantitative measure of information, and then as a limit on the amount of space that we need to record that information. It is astonishing that the same concept reaches from steam engines to the internet, and from molecules to language. In this lecture I will try to give a sense for these four different notions of entropy, and their connections with one another, hoping to give a sense for the unifying power of mathematics.