Logarithmic Geometry was invented (or discovered) in the 1980's, with crucial ideas contributed by Deligne, Faltings, Fontaine, Illusie, and especially K. Kato. Its roots go back much further, at least to the work by Mumford and others on toroidal embeddings, as well as the theory of semistable reduction. The goal of all these theories is to understand, in a systematic way, the related phenomena of compactification and degeneration in algebraic and arithmetic geometry, with applications to number theory. I will attempt to explain the main ideas and foundations of Kato's version of log geometry, with an emphasis on the geometric and topological aspects and applications. The course will consist of five ninety-minute lectures, for which the tenative plan is as follows.

### Course Plan

1. Motivation, background, and basic concepts

2. Geometry of monoids and monoid actions

3. Homomorphisms of monoids: exact, integral, and saturated

4. Log schemes

5. Topology and cohomology of log schemes

Homework