Moduli of Curves and Abelian Varieties

Moduli of Abelian Varieties: A Short Introduction and Survey.- 1 Introduction.- 2 Elliptic Curves.- 3 Abelian Varieties.- 4 The Torelli Morphism.- 5 Cycles on the Moduli Space of Abelian Varieties.- 6 The Moduli Space Ag in Positive Characteristic.- 7 The Moduli Space in Mixed Characteristic.- Remarks on Moduli of Curves.- 1 Introduction.- 2 Mapping Class Groups.- 3 The Torelli Group.- 4 Moduli Spaces of Curves.- 5 Deligne-Mumford-Knudsen Completion.- 6 Covers of Moduli Stacks.- 7 Tautological Classes.- 8 Stability.- 9 A Proarithmetic Hull of the Mapping Class Group.- 10 The Witten Conjecture.- 11 Complete Subvarieties of Moduli Spaces.- 12 Intersection Theory.- 13 Stable Maps and the Virasoro Conjecture.- A Stratification of a Moduli Space of Polarized Abelian Varieties in Positive Characteristic.- 1 Introduction.- 2 Elementary Sequences and Final Filtrations.- 3 Explicit Description of Some of the Strata.- 4 Standard Types.- 5 Moving in a Stratum.- 6 The Raynaud Trick.- 7 Results.- 8 Some Questions.- Cycles on the Moduli Space of Abelian Varieties.- 1 Introduction.- 2 The Tautological Subring of Ag.- 3 The Tautological Ring of A? g.- 4 Cycle Relations from Grothendieck-Riemann-Roch.- 5 On the Torsion of the Class ? g.- 6 The Ekedahl-Oort Stratification in Positive Characteristic.- 7 The Ekedahl-Oort Strata as Degeneracy Loci.- 8 The Degeneracy Locus Uø and ?0(p).- 9 The Cycle Classes of the Strata.- 10 Effectivity of Tautological Classes.- 11 Some Additional Results.- Locally Symmetric Families of Curves and Jacobians.- 1 Introduction.- 2 Background and Definitions.- 3 Homomorphisms to Mapping Class Groups.- 4 Maps of Lattices to Mapping Class Groups.- 5 Locally Symmetric Hypersurfaces in Locally Symmetric Varieties.- 6 Geometry of the Jacobian Locus.- 7 LocallySymmetric Families of Curves.- 8 Locally Symmetric Families of Jacobians.- 9 Appendix: An Example.- A Conjectural Description of the Tautological Ring of the Moduli Space of Curves.- 1 Introduction.- 2 Known Results.- 3 The Conjectures and the Evidence.- 4 Calculations.- 5 Other Relations.- Correspondences between Moduli Spaces of Curves.- 1 Introduction.- 2 Correspondences for Mapping Class Groups: Closed Surfaces.- 3 Action of the Correspondences on the Tautological Classes.- 4 Hecke Operators Attached to Finite Abelian Covers.- 5 Correspondences via Ramified Covers.- 6 Correspondences Acting on a Lie Algebra.- Fields, Strings, Matrices and Symmetric Products.- 1 Introduction.- 2 Particles, Symmetric Products and Fields.- 3 Second-quantized Strings.- 4 Light-Cone Quantization of Quantum Field Theories.- 5 Light-Cone Quantization of String Theories.- 6 Matrix Strings and Interactions.- 7 String Theories in Six Dimensions.- Addresses.