Regular Solids and Isolated Singularities

The last book XIII of Euclid's Elements deals with the regular solids which therefore are sometimes considered as crown of classical geometry. More than two thousand years later around 1850 Schl~fli extended the classification of regular solids to four and more dimensions. A few decades later, thanks to the invention of group and invariant theory the old three dimensional regular solid were involved in the development of new mathematical ideas: F. Klein (Lectures on the Icosa hedron and the Resolution of Equations of Degree Five, 1884) emphasized the relation of the regular solids to the finite rotation groups. He introduced complex coordinates and by means of invariant theory associated polynomial equations with these groups. These equations in turn describe isolated singularities of complex surfaces. The structure of the singularities is investigated by methods of commutative algebra, algebraic and complex analytic geometry, differential and algebraic topology. A paper by DuVal from 1934 (see the References), in which resolutions play an important rele, marked an early stage of these investigations. Around 1970 Klein's polynomials were again related to new mathematical ideas: V. I. Arnold established a hierarchy of critical points of functions in several variables according to growing com plexity. In this hierarchy Kleinls polynomials describe the "simple" critical points.

Inhaltsverzeichnis:

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 Analyti c 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. The Fundamental 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.