Regular Solids and Isolated Singularities: Advanced Lectures in Mathematics

I: Regular Solids and Finite Rotation Groups.- §1. The Platonic Solids.- §2. Convex Polytopes.- §3. Regular Solids.- §4. Enumeration and Realization of Regular Solids.- §5. The Rotation Groups of the Platonic Solids.- §6. Finite Subgroups of the Rotation Group SO(3).- §7. Normal Subgroups.- §8. Generators and Relations for the Finite Subgroups of SO(3).- II: Finite Subgroups of SL(2,G) and Invariant Polynomials.- §1. Finite Subgroups of SL(2,C).- §2. Quaternions and Rotations.- §3. Four-Dimensional Regular Solids.- §4. The Orbit Spaces S3/G of the Finite Subgroups G of SU(2).- §5. Generators and Relations for the Finite Subgroups of SL(2,C).- §6. Invariant Divisors and Semi-Invariant Forms.- §7. The Characters of the Invariant Divisors.- §8. Generators and Relations for the Algebra of Invariant Polynomials.- §9. The Affine Orbit Variety.- III: Local Theory of Several Complex Variables.- §1. Germs of Holomorphic Functions.- §2. Germs of Analytic Sets.- §3. Germs of Holomorphic Maps.- §4. The Embedding Dimension.- §5. The Preparation Theorem.- §6. Finite Maps.- §7. Finite and Strict Maps.- §8. The Nullstellensatz.- §9. The Dimension.- §10. Annihilators.- §11. Regular Sequences.- §12. Complete Intersections.- §13. Complex Spaces.- IV: Quotient Singularities and Their Resolutions.- §1. Germs of Invariant Holomorphic Functions.- §2. Complex Orbit Spaces.- §3. Quotient Singularities.- §4. Modifications. Line Bundles.- §5. Cyclic Quotient Singularities.- §6. The Resolution of Cyclic Quotient Singularities.- §7. The Cotangent Action.- §8. Line Bundles with Singularities.- §9. The Resolution of Non-Cyclic Quotient Singularities.- §10. Plumbed Surfaces.- §11. Intersection Numbers.- §12. The Homology of Plumbed Surfaces.- §13. TheFundamental Group of a Plumbed Surface Minus its Core.- §14. Groups Determined by a Weighted Tree.- §15. Topological Invariants.- V: The Hierarchy of Simple Singularities.- §1. Basic Concepts.- §2. The Milnor Number.- §3. Transformation Groups.- §4. Families of Germs.- §5. Finitely Determined Germs.- §6. Unfoldings.- §7. The Multiplicity.- §8. Weighted Homogeneous Polynomials.- §9. The Classification of Holomorphic Germs.- §10. Three Series of Holomorphic Germs.- §11. Simple Singularities.- §12. Adjacency.- §13. Conclusion and Outlook.- References.