Maximal degenerations, log structures, wall crossing and theta functions

A maximal degeneration of an algebraic variety is, in a
sense, a decomposition of the variety into primitive pieces. Log
geometry provides a framework to investigate such degenerations.
There is an underlying discrete geometry building the bridge to
tropical geometry. Corrections to the naive complex geometry can be
organized by a wall structure in tropical geometry. In favourable
situations one obtains canonical degenerations, and a basis of
canonical sections of an ample line bundle generalizing classical
theta functions.

In the talk I will survey this circle of ideas at the example of K3
surfaces. The theta functions along with ideas from mirror symmetry
suggest a canonical toroidal compactification of moduli spaces of
polarized K3 surfaces. This is joint work with Mark Gross and partly
with Paul Hacking and Sean Keel.