preface
 Part I Moduli Spaces of Riemann Surfaces
  1 Mapping class groups and Dehn twists
   Mapping class groups
   Dehn twists
   Fundamental groups and mapping class groups
  2 Conformal structures and a rough classification
   Conformal structures
   Spherical cases
   Flat cases
   Hyperbolic cases
  3 Geometry of the upper half plane
  4 Hyperbolic surfaces
   Closed geodesics on a hyperbolic surface
   Geodesic shear
   Geodesic representation
   Disjunction of geodesics
   Pair of pants
   Pants decompositions
   Fenchel-Nielsen coordinates
   Hyperbolic surfaces with cusps
   Pants decomposition in the presence of cusps
  5 Quadratic differentials
   Local normal form
   Structure defined by a holomorphic quadratic differential
   Quadratic differentials and the Teichm¨uller flow
  6 Ribbon graphs and ideal triangulation of Teichm¨uller space
   Jenkins-Strebel differentials
   Combinatorial description of a graph
   Ribbon graphs
   The dual ribbon graph and the arc complex
   Metrized ribbon graphs
   Ideal triangulation of thickened Teichm¨uller space
  7 The homotopy type of the moduli spacesMg,n
   The homotopy type ofMg,1
   The moduli space of curves as a virtual classifying space
  8 A brief review of Dolbault cohomology and Serre duality for Riemann surfaces
   Dolbault cohomology of a coherent sheaf
   Riemann-Roch and Serre duality for a Riemann surface
  9 Deformation theory of Riemann surfaces
   Orbifolds
   Towards the complex tangent bundle of Teichm¨uller space
   Deformation theory
   The Kodaira Spencermap
   TheWeil-Petersson metric
   The quasi-projective structure on the universal family
  10 Harvey bordification and Deligne-Mumford compactification
   The curve complex
   Harvey’s bordification
   Stable pointed curves
   Deformations of nodes
   Deformations of nodal curves
   The Deligne-Mumford compactification
   The idea behindGeometric Invariant Theory
   Projectivity of the moduli space of stable pointed curves
   Getting Harvey’s bordification from the D-M compactification
  11 Cohomological properties ofMg,P
   Harer’s stability theorem
   Hopf algebra structure on stable cohomology
  12 Tautological algebras
   Duality on orbifolds
   Review of the Gysinmap
   The moduli space of stable pointed curves as a category
   Vector bundles on moduli spaces of curves
   Weighted graphs
   The notion of a tautological algebra
   A theoremof Kontsevich
   Faber’s conjectures
   Primitivity of the kappa classes
  References
 Part II Introduction to Tropical Curves, Tropical Moduli and Teichm¨ uller Spaces
  1 Introduction
   1.1 A general overview
   1.2 A specific motivation fromgeometric group theory
   1.3 Interaction between moduli spaces of tropical curves and outer space ofmetric graphs
  2 Definitions and basic facts on Out(Fn) and the outer space Xn
   2.1 Combinatorial group theory and motivations from mapping class groups
   2.2 Geometric group theory, metric graphs and outer space Xn
   2.3 Action of Out(Fn) on outer space Xn and applications
   2.4 A natural approach to construct complete geodesic metrics on Xn
  3 Tropical semifields and tropical polynomials
   3.1 The tropical semifield T
   3.2 Tropical polynomials
  4 Tropical curves
   4.1 Plane tropical curves Γp
   4.2 Newton polygons and tropical curves
   4.3 Balancing condition at vertices
   4.4 Abstract smooth tropical curves
   4.5 Identification between smooth tropical curves and metric graphs
  5 Definitions of tropical varieties andmotivations
   5.1 Tropical hypersurfaces and higher codimension tropical subvarieties of Rn
   5.2 Tropicalization of polynomials and tropical varieties as non-Archimedean amoebas
   5.3 Tropical varieties as logarithmic limit sets
   5.4 Tropical varieties as limits of Archimedean amoebas
   5.5 Affine manifolds and tropical affinemanifolds
  6 Some alternative descriptions of tropical varieties and related spaces
   6.1 Fans and tropical fans
   6.2 Stacky fans
   6.3 Tropical hypersurfaces as super currents
   6.4 Tropical varieties versus Berkovich spaces
  7 Applications of tropical geometry
   7.1 Logarithmic compactification of algebraic varieties
   7.2 Topology of real algebraic curves
   7.3 Geometric and combinatorial group theory
   7.4 Three-dimensional topology
   7.5 Boundary of Teichm¨uller space Tg
   7.6 Enumerative algebraic geometry
   7.7 Mirror symmetry of Calabi-Yaumanifolds
   7.8 Arithmetic algebraic geometry
  8 Moduli spaces of tropical curves and tropical Teichm¨uller spaces
   8.1 Moduli space Mtr
   n of tropical curves and a general philosophy on moduli spaces
   8.2 Gromov-Hausdorff distance and topology onMtr n
   8.3 The moduli spaceMtr
   n and tropical Teichm¨uller space T tr g as stacky fans
   8.4 The tropical Teichm¨uller space T tr g as a tropical space
   8.5 The moduli spaceMtr n as a tropical orbifold
   8.6 Compactifications of the moduli spaceMtr n
  9 Jacobian variety of a Riemann surface and the Siegel upper-half space
   9.1 Polarized algebraic varieties
   9.2 Abelian varieties
   9.3 Jacobian varieties of Riemann surfaces
   9.4 Moduli space of principally polarized abelian varieties Ag
  10 Tropical abelian varieties, their moduli spaces, and Jacobians of tropical curves
   10.1 Jacobian variety of tropical curves andmetric graphs
   10.2 Tropical abelian varieties
   10.3 Principally polarized tropical abelian varieties and their moduli space Atr n
  11 The Torelli Theorem for tropical curves
   11.1 Period of tropical curves and contractions of graphs
   11.2 Tropical Torelli theoremand its failure
  12 Complete invariant metrics on outer space Xn
   12.1 Tropical Jacobian map and invariant complete geodesic metrics on Xn
   12.2 Tropical Jacobian map and a complete pesudo-Riemannian metric on Xn
   12.3 Complete pseudo-Riemannian metric on Xn via lengths of pinching loops
   12.4 Finite Riemannian volume of the quotient Out(Fn)\Xn
  References
 Index