Perturbations, Deformations, and Variations (and “near-misses”) in Geometry, Physics, and Number Theory

Contents 1. Introduction 307 2. Universal unfoldings 309 3. Special, general, very general, generic... 313 4. Moduli: Parameters describing the deformations of a mathematical structure 314 (1). Classifying ellipses in the plane up to Euclidean congruence. 314 (2). Deforming Riemann surfaces. 315 5. Moduli and lattices 317 6. Lattices in real vector spaces 317 (1). Lattices in the plane. 318 (2). From lattices in the plane to elliptic curves over C. 319 7. The modular curves X 0 (N) play a central role in three different classification problems 320 8. Deformations of group representations 321 9. Mirror symmetry – and back to deforming lattices in the plane 325 10. Quantization in physics, and noncommutative geometry 327 11. Near-misses in algebra and number theory (approximation of irrationalities) 329 References 334