In this thesis, we study the properties of tropical geometry and survey the recent development of tropical geometry, especially Mikhalkin's method of counting curves. We also briefly study tropical geometry in three-dimensional case.
1.Introduction-----------------------------------p.1
2.Motivation-------------------------------------p.3
3.Properties of Tropical Geometry----------------p.27
4.The Application: Enumerative Geometry----------p.36
5.Tropical Geometry in R3------------------------p.49
1.Statistics of frameworks and motions of panel structures,a projective geometry introduction.
2.Tropical algebraic geometry.(Gathmann)
3.Welschinger invariant and enumeration of real rational curves.
4.Tropical algebraic geometry.(Itenberg)
5.First steps in tropical geometry.
6.Counting plane curves of any genus.
7.Non-archimedean amoebas and tropical varieties.
8.Gromov-witten classes, quantum cohomology and enumerative geometry.
9.Rational tropical curves in Rn.
10.Amoebas of algebraic varieties and tropical geometry.
11.Enumerative tropical algebraic geometry in R2.
12.Patchworking singular algebraic curves, non-archimedean amoebas, and enumerative geometry.
13.Dequantization of real algebraic geometry on logarithmic paper.
14.Spinor states of real rational curves in real algebraic convex 3-manifolds and enumerative invariants.
15.Invariant of real rational symplectic 4-manifolds and lower bounds in real enumerative geometry.