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Log geometry was developed to study compactification and degeneration phenomena in algebraic and analytic geometry. It is fair to say that log schemes play the role of “algebraic manifolds with boundary.” I will attempt to review the main concepts of log geometry and then to illustrate how they help us understand the topology of degenerations in the complex setting. For example, a proper semistable family over a disc gives rise to a smooth proper and saturated morphism X/S of log analytic spaces over the log disc. It turns out that the underlying germ of the map of topological spaces X_top/S_top can be recovered from the log fiber X_0/S_0 of X/S over the log point. Furthermore, there are simple formulas for the d_2 differentials and the action of the monodromy on the E_2 terms of the “nearby cycles” spectral sequence, in terms of the combinatorics of the log structure on X_0/S_0.

This is joint work with Piotr Achinger.