This is a contributed entry in progress

Index of Algebraic Geometry

Algebraic Geometry (AG), and non-commutative geometry/. On the other hand, there are also close ties between algebraic geometry and number theory.

Outline

Disciplines in algebraic geometry

1. Birational geometry, Dedekind domains and Riemann-Roch theorem
2. Homology and cohomology theories
3. Algebraic groups: Lie groups, matrix group schemes,group machines, linear groups, generalizing Lie groups, representation theory
4. Abelian varieties
5. Arithmetic algebraic geometry
6. duality
7. category theory applications in algebraic geometry
8. indexes of category, functors and natural transformations
9. Grothendieck's Descent theory
10. `Anabelian geometry'
11. Categorical Galois theory
12. higher dimensional algebra (HDA)
13. Quantum Algebraic Topology (QAT)
14. Quantum Geometry
15. computer algebra systems; an example is: explicit projective resolutions for finitely-generated modules over suitable rings

Cohomology

1. cohomology groups are defined and then cohomology functors associate Abelian groups to sheaves on a scheme; one may view such Abelian groups them as cohomology with coefficients in a scheme.
2. Cohomology functors
3. fundamental cohomology theorems
4. A basic type of cohomology for schemes is the sheaf cohomology
5. Whitehead groups, torsion and towers
6. xyz

Seminars on Algebraic Geometry and Topos Theory (SGA)

1. SGA1
2. SGA2
3. SGA3
4. SGA4
5. SGA5
6. SGA6
7. SGA7

Algebraic varieties and the GAGA principle

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2. new2y
3. new3z

Number theory applications

Cohomology theory

1. Cohomology group
2. Cohomology sequence
3. DeRham cohomology
4. new4

Homology theory

1. homology group
2. Homology sequence
3. Homology complex
4. new4

Duality in algebraic topology and category theory

1. Tanaka-Krein duality
2. Grothendieck duality
3. categorical duality
4. tangled duality
5. DA5
6. DA6
7. DA7

Category theory applications

1. abelian categories
2. topological category
3. fundamental groupoid functor
4. Categorical Galois theory
5. non-Abelian algebraic topology
6. Group category
7. groupoid category
8. category
9. topos and topoi axioms
10. generalized toposes
11. Categorical logic and algebraic topology
12. meta-theorems
13. Duality between spaces and algebras

Examples of Categories

1. Algebraic categories
2. Topological category
3. Category of sets, Set
4. Category of topological spaces
5. category of Riemannian manifolds
6. Category of CW-complexes
7. Category of Hausdorff spaces
8. category of Borel spaces
9. Category of CR-complexes
10. Category of graphs
11. Category of spin networks
12. Category of groups
13. Galois category
14. Category of fundamental groups
15. Category of Polish groups
16. Groupoid category
17. category of groupoids (or groupoid category)
18. category of Borel groupoids
19. Category of fundamental groupoids
20. Category of functors (or functor category)
21. double groupoid category
22. double category
23. category of Hilbert spaces
24. category of quantum automata
25. R-category
26. Category of algebroids
27. Category of double algebroids
28. Category of dynamical systems

Index of functors

1. Covariant functors
2. Contravariant functors
3. adjoint functors
4. preadditive functors
5. Additive functor
6. representable functors
7. Fundamental groupoid functor
8. Forgetful functors
9. Grothendieck group functor
10. Exact functor
11. Multi-functor
12. section functors
13. NT2
14. NT3

Index of natural transformations

1. Natural equivalence
2. Natural transformations in a 2-category
3. NT3
4. NT1

Grothendieck proposals

1. Esquisse d'un Programme
2. Pursuing Stacks
3. S2
4. S3

Descent theory

1. D1
2. D2
3. D3

Higher Dimensional Algebraic Geometry (HDAG)

1. Categorical groups and supergroup algebras
2. Double groupoid varieties
3. Double algebroids
4. Bi-algebroids
5. -algebroid
6. -category
7. -category
8. super-category
9. weak n-categories of algebraic varieties
10. Bi-dimensional Algebraic Geometry
11. Anabelian Geometry
12. Noncommutative geometry
13. Higher-homology/cohomology theories
14. H1
15. H2
16. H3
17. H4

Axioms of cohomology theory

1. A1
2. A2
3. A3

Axioms of homology theory

1. A1
2. A2
3. A3

Quantum algebraic topology (QAT)

(a). Quantum algebraic topology is described as the mathematical and physical study of general theories of quantum algebraic structures from the standpoint of algebraic topology, category theory and their non-Abelian extensions in higher dimensional algebra and supercategories

1. quantum operator algebras (such as: involution, *-algebras, or -algebras, von Neumann algebras, , JB- and JL- algebras, - or C*- algebras,
2. Quantum von Neumann algebra and subfactors; Jone's towers and subfactors
3. Kac-Moody and K-algebras
4. categorical groups
5. Hopf algebras, quantum Groups and quantum group algebras
6. quantum groupoids and weak Hopf -algebras
7. groupoid C*-convolution algebras and *-convolution algebroids
8. quantum spacetimes and quantum fundamental groupoids
9. Quantum double Algebras
10. quantum gravity, supersymmetries, supergravity, superalgebras and graded `Lie' algebras
11. Quantum categorical algebra and higher–dimensional, - Toposes
12. Quantum R-categories, R-supercategories and spontaneous symmetry breaking
13. Non-Abelian Quantum Algebraic Topology (NA-QAT): closely related to NAAT and HDA.

Quantum Geometry

1. Quantum Geometry overview
2. Quantum non-commutative geometry

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2. new2y

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Textbooks and bibliograpies

Bibliography on Category theory, AT and QAT

Textbooks and Expositions:

1. A Textbook1
2. A Textbook2
3. A Textbook3
4. A Textbook4
5. A Textbook5
6. A Textbook6
7. A Textbook7
8. A Textbook8
9. A Textbook9
10. A Textbook10
11. A Textbook11
12. A Textbook12
13. A Textbook13
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Bibliography

1

Alexander Grothendieck and J. Dieudonné.: 1960, Eléments de geometrie algébrique., Publ. Inst. des Hautes Etudes de Science, 4.

2

Alexander Grothendieck. Séminaires en Géometrie Algèbrique- 4, Tome 1, Exposé 1 (or the Appendix to Exposée 1, by `N. Bourbaki' for more detail and a large number of results. AG4 is freely available in French; also available here is an extensive Abstract in English.

3

Alexander Grothendieck. 1962. Séminaires en Géométrie Algébrique du Bois-Marie, Vol. 2 - Cohomologie Locale des Faisceaux Cohèrents et Théorèmes de Lefschetz Locaux et Globaux. , pp.287. (with an additional contributed exposé by Mme. Michele Raynaud)., Typewritten manuscript available in French; see also a brief summary in English . Available for free downloads at on the web.

4

Alexander Grothendieck, 1984. “Esquisse d'un Programme”, (1984 manuscript), finally published in “Geometric Galois Actions”, L. Schneps, P. Lochak, eds., London Math. Soc. Lecture Notes 242, Cambridge University Press, 1997, pp.5-48; English transl., ibid., pp. 243-283. MR 99c:14034 .

5

Qing Liu.2002. Algebraic Geometry and Arithmetic Curves, Oxford Graduate Texts in Mathematics 6, 2002. 300 pages on schemes followed by geometry and arithmetic surfaces. (Serre duality is approached via Grothendieck duality).

6

Igor Shafarevich, Basic Algebraic Geometry Vols. 1 and 2; Vol.2: Schemes and Complex Manifolds., Second Revised and Expanded Edition. Springer-Verlag; scheme theory, varieties as schemes, varieties and schemes over the complex numbers, and complex manifolds.

7

James Milne, Elliptic Curves, online course notes. Available at his website.

8

Joseph H. Silverman, The Arithmetic of Elliptic Curves. Springer-Verlag, New York, 1986.

9

Joseph H. Silverman, Advanced Topics in the Arithmetic of Elliptic Curves. Springer-Verlag, New York, 1994.

10

Goro Shimura, Introduction to the Arithmetic Theory of Automorphic Functions. Princeton University Press, Princeton, New Jersey, 1971.

11

David Mumford, Abelian Varieties, Oxford University Press, London, 1970. This book is a canonical reference on the subject. “It is written in the language of modern algebraic geometry, and provides a thorough grounding in the theory of abelian varieties.”