Moduli theory of algebraic varieties and classification theory of compact complex spaces -- Moduli spaces for polarized algebraic varieties -- Group quotients in the category of analytic spaces and the category of algebraic spaces -- Applications of the quotient theorems to moduli of algebraic varieties -- Quotients for affine schemes by reductive algebraic groups -- Quotients in the category of schemes -- Mumfordโs construction of the moduli variety for curves and polarized abelian varieties. Other applications of Mumfordโs quotient theory -- Other methods of treating moduli problems. Artinโs method of algebraic stacks. Griffithsโs method of period maps -- Compactification of moduli spaces -- Fine moduli spaces. The universal families for stable curves with level n-structure -- Applications of moduli theory to fibre spaces and the additivity formula for the Kodaira dimension of fibre spaces. Open problems

1. Mathematics
2. Algebra
3. Mathematics
4. Algebra