turkmath.org

Logarithmic structures were introduced into algebraic geometry by Kato, Fontaine, and Illusie as a generalization of the toric embeddings studied earlier by Mumford and others. Roughly speaking, the theory furnishes a category where the nicest object is a smooth variety equipped with a normal crossings divisor. We will survey some applications of this machinery in the construction and study of various moduli spaces and in the context of degeneration formulas in Gromov-Witten and Donaldson-Thomas theory. We will also discuss a version of this theory in differential geometry---where manifolds with corners are the nicest objects---and explain how this is related to the algebro-geometric side of the theory.