The most recent version of the lecture notes is lecturenotes.pdf.
Old versions
Version 2018-03-12 new: Chapter 1: Why tropical scheme theory?
Version 2018-03-14 new: Chapter 2: Semirings
Version 2018-03-17 new: Chapter 3: Monoids with zero
Version 2018-04-05 new: Chapter 4: Blueprints; small additions to Chapters 2 and 3
Version 2018-05-09 new: Chapter 5: Ordered blueprints; minor changes in Chapters 2, 3 and 4
Version 2018-05-21 new: Chapter 6: Valuations; minor changes in Chapters 2, 3, 4 and 5
Personal note: Due to time restraints, I have to suspend writing for the time being.