Tropical varieties, maps and gossip

Preface......Page 5
List of notations......Page 11
Quick algebraic definitions......Page 13
Foundations......Page 14
Tropical geometry of hypersurfaces......Page 21
Tropical geometry in higher codimension......Page 23
Tropical geometry and valuation theory......Page 30
Tropical unirational varieties......Page 33
Some classes of tropically unirational varieties......Page 35
Combining reparameterizations......Page 38
Birational projections......Page 41
Very local reparameterizations......Page 45
Concluding remarks......Page 48
Modules over semirings and homomorphisms......Page 49
Linear functionals on idempotent modules......Page 60
Ranks and vanishing conditions......Page 64
Modules over linearly ordered semirings......Page 70
Conclusion......Page 72
A category of tropical linear spaces......Page 73
Valuated matroids and tropical linear spaces......Page 75
Foundations......Page 76
Basic constructions and relations......Page 83
Parameter spaces......Page 92
Functions and their graphs......Page 94
Linear functions......Page 95
Extensions of matroids......Page 104
A category of tropical linear spaces......Page 125
The tropical linear monoid......Page 133
The gossip monoid......Page 139
Foundations......Page 140
Relation to the gossip problem......Page 144
Polyhedral structure of the gossip monoid......Page 146
Graphs with detours......Page 147
n 3......Page 151
n = 4......Page 153
Bibliography......Page 157
Index......Page 162
Summary......Page 165
Curriculum Vitae......Page 167