Rognes preprints in progress (as of Sep 2009), including: Complex oriented logarithmic structures, Logarithmic topological cyclic homology
Logarithmic structures of Fontaine-Illusie.
See also various log cohomology theories
[arXiv:0909.0288] Geography of log models:theory and applications from arXiv Front: math.AG by Sung Rak Choi, Vyacheslav Shokurov An introduction to geography of log models with applications to positive cones of FT varieties and to geometry of minimal models and Mori fibrations.
arXiv:1006.5870 Logarithmic Geometry and Moduli from arXiv Front: math.AG by Dan Abramovich, Qile Chen, Danny Gillam, Yuhao Huang, Martin Olsson, Matthew Satriano, Shenghao Sun We discuss the role played by logarithmic structures in the theory of moduli.
http://mathoverflow.net/questions/70186/more-questions-about-log-structures
arXiv:1001.0466 Parabolic sheaves on logarithmic schemes fra arXiv Front: math.NT av Niels Borne, Angelo Vistoli We show how the natural context for the definition of parabolic sheaves on a scheme is that of logarithmic geometry. The key point is a reformulation of the concept of logarithmic structure in the language of symmetric monoidal categories, which might be of independent interest. Our main result states that parabolic sheaves can be interpreted as quasi-coherent sheaves on certain stacks of roots.
Oslo 2009 The aim of the conference is to develop the current interplay between arithmetic algebraic geometry and stable homotopy theory. Using structured ring spectra one can form topological structure sheaves for moduli stacks of elliptic curves or other abelian varieties, whose ring spectra of global sections define powerful new homology theories. Using sheaves of infinity-categories one can form topological crystalline cohomology, with associated de Rham-Witt complexes closely related to topological cyclic homology and p-adic algebraic K-theory. For complete local rings of mixed characteristic, or for structured ring spectra of mixed chromatic types, there are log (= logarithmic) geometric extensions of these theories, leading to de Rham-Witt complexes with log poles, log topological cyclic homology and log K-theory. The hope is that both algebraic geometers and homotopy theorists will have something to learn from the modern developments in these adjacent fields.
nLab page on Logarithmic structures