2023. Mathematics for physicists - 2
Syllabus, very preliminary (as decided by the physics teacher):
++ 1. Topology on a space, categories; topology as a category
++ 2. Presheaf as a functor from the category of open sets into the category of vector spaces (rings, etc)
++ 3. Sheaves: gluing axiom
4. Affine schemes, Zariski topology, sheaf of regular functions 𝒪_X. Example: plane affine curve
5. Complex analytic manifolds.
++ 6. Tangent sheaf, examples. (Analytic and algebraic categories)
7. Deformation of a complex structure and sheaf deformations. Case of affine curves (analytic category)
8. ++ Čech cohomology.
9. Deformations and 1-Čech cohomology of the tangent sheaf (algebraic ana analytic categories).
10. Transcendental approach – operator of almost complex structure, integrability of almost complex structures.
11-12. Generalized Beltrami differentials, and Kodaira–Spencer map.
13. Dolbeault cohomology, comparing with Čech cohomology.
Homework
Homework 1. Products-and-intersections-of-ideals; version of 2023.10.24; (also an interesting example of primary decomposition)
Homework 2023.11.16, Vector fields on projective space, and Lie algebra sl_n
Whiteboards
1. Class 1: 2023.10.17: Topology; some commutative algebra; Zariski topology (PDF)
2. additional file: Ideals and rings - 2 (PDF)
3. Classes 2-3: 2023.10.18: Categories; Commutative Algebra; and Geometry
all the material on the "equivalence of categories" (pages 6 - 9 1/2) you can safely skip
Some references
Algebraic geometry:
Miles Reid, Undergraduate algebraic geometry, paragraph 1; Appendix to Chapter 1; paragraphs 3-4, paragraph 6
Complex analytic geometry:
Herbert Clemens, A scrapbook of complex curve theory, 2.8 (differential on a cubic curve), 2.9 (beginning) (periods of a cubic curve), 2;10* (Remarks on Serre duality) , 4.1*-4.3* (exponential sequence and Jacobian; first Chern class)
("star" means "additional material, somewhat more difficult"
De Rham cohomology:
Bott and Tu, "Differential forms in algebraic topology", I.1 : de Rham complex (ignore the "compact support case", 1.2* (starred) Mayer-Vietoris sequence,
Sheaves and Čech cohomology:
Bott and Tu, II.10 pages 108-113
or
Wells, Differential Analysis on Complex Manifolds, (3rd edition), pages 36-41 (sheaves on complex analytic manifolds), pages 63-64: Čech cohomology