I.     Starting Out
  0. Sales pitches....................................................37
  1. Groups...........................................................43
  2. Metric spaces....................................................55

II.    Basic Abstract Algebra
  3. Homomorphisms and quotient groups................................69
  4. Rings and ideals.................................................81
  5. Flavors of rings.................................................93

III.   Basic Topology
  6. Properties of metric spaces.....................................107
  7. Topological spaces..............................................113
  8. Compactness.....................................................123

IV.    Linear Algebra
  9. Vector spaces...................................................137
  10. Eigen-things...................................................155
  11. Dual space and trace...........................................165
  12. Determinant....................................................173
  13. Inner product spaces...........................................181
  14. Bonus: Fourier analysis........................................189
  15. Duals, adjoint, and transposes.................................199

V.     More on Groups
  16. Group actions overkill AIME problems...........................209
  17. Find all groups................................................215
  18. The PID structure theorem......................................221

VI.    Representation Theory
  19. Representations of algebras....................................233
  20. Semisimple algebras............................................245
  21. Characters.....................................................253
  22. Some applications..............................................261

VII.   Quantum Algorithms
  23. Quantum states and measurements................................267
  24. Quantum circuits...............................................275
  25. Shor's algorithm...............................................283

VIII.  Calculus 101
  26. Limits and series..............................................291
  27. Bonus: A hint of p-adic numbers................................303
  28. Differentiation................................................315
  29. Power series and Taylor series.................................327
  30. Riemann integrals..............................................335

IX.    Complex Analysis
  31. Holomorphic functions..........................................345
  32. Meromorphic functions..........................................361
  33. Holomorphic square roots and logarithms........................371
  34. Bonus: Topological Abel-Ruffini Theorem........................377

X.     Measure Theory
  35. Measure spaces.................................................383
  36. Constructing the Borel and Lebesgue measure....................391
  37. Lebesgue integration...........................................401
  38. Swapping order with Lebesgue integrals.........................407
  39. Bonus: A hint of Pontryagin duality............................415

XI.    Probability (TO DO)
  40. Random variables (TO DO).......................................423
  41. Large number laws (TO DO)......................................425
  42. Stopped martingales (TO DO)....................................433

XII.   Differential Geometry
  43. Multivariable calculus done correctly..........................449
  44. Differential forms.............................................457
  45. Integrating differential forms.................................469
  46. A bit of manifolds.............................................481

XIII.  Riemann Surfaces
  47. Basic definitions of Riemann surfaces..........................493
  48. Morphisms between Riemann surfaces.............................501
  49. Affine and projective plane curves.............................507
  50. Differential forms.............................................517
  51. The Riemann-Roch theorem.......................................521
  52. Line bundles...................................................527

XIV.   Algebraic NT I: Rings of Integers
  53. Algebraic integers.............................................537
  54. The ring of integers...........................................543
  55. Unique factorization (finally!)................................551
  56. Minkowski bound and class groups...............................563
  57. More properties of the discriminant............................583
  58. Bonus: Let's solve Pell's equation!............................585

XV.    Algebraic NT II: Galois and Ramification Theory
  59. Things Galois..................................................593
  60. Finite fields..................................................605
  61. Ramification theory............................................611
  62. The Frobenius element..........................................621
  63. Bonus: A Bit on Artin Reciprocity..............................633

XVI.   Algebraic Topology I: Homotopy
  64. Some topological constructions.................................649
  65. Fundamental groups.............................................661
  66. Covering projections...........................................673

XVII.  Category Theory
  67. Objects and morphisms..........................................685
  68. Functors and natural transformations...........................697
  69. Limits in categories (TO DO)...................................709
  70. Abelian categories.............................................711

XVIII. Algebraic Topology II: Homology
  71. Singular homology..............................................721
  72. The long exact sequence........................................735
  73. Excision and relative homology.................................747
  74. Bonus: Cellular homology.......................................755
  75. Singular cohomology............................................767
  76. Application of cohomology......................................779

XIX.   Algebraic Geometry I: Classical Varieties
  77. Affine varieties...............................................799
  78. Affine varieties as ringed spaces..............................809
  79. Projective varieties...........................................819
  80. Bonus: B\'ezout's theorem......................................827
  81. Morphisms of varieties.........................................835

XX.    Algebraic Geometry II: Affine Schemes
  82. Sheaves and ringed spaces......................................847
  83. Localization...................................................859
  84. Affine schemes: the Zariski topology...........................869
  85. Affine schemes: the sheaf......................................877
  86. Interlude: eighteen examples of affine schemes.................887
  87. Morphisms of locally ringed spaces.............................903

XXI.   Set Theory I: ZFC, Ordinals, and Cardinals
  88. Interlude: Cauchy's functional equation and Zorn's lemma.......917
  89. Zermelo-Fraenkel with choice...................................925
  90. Ordinals.......................................................933
  91. Cardinals......................................................945

XXII.  Set Theory II: Model Theory and Forcing
  92. Inner model theory.............................................955
  93. Forcing........................................................967
  94. Breaking the continuum hypothesis..............................977

XXIII. Appendix