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.....................................109
  7. Topological spaces..............................................115
  8. Compactness.....................................................125

IV.    Linear Algebra
  9. Vector spaces...................................................139
  10. Eigen-things...................................................157
  11. Dual space and trace...........................................167
  12. Determinant....................................................175
  13. Inner product spaces...........................................183
  14. Bonus: Fourier analysis........................................191
  15. Duals, adjoint, and transposes.................................201

V.     More on Groups
  16. Group actions overkill AIME problems...........................211
  17. Find all groups................................................217
  18. The PID structure theorem......................................223

VI.    Representation Theory
  19. Representations of algebras....................................235
  20. Semisimple algebras............................................247
  21. Characters.....................................................255
  22. Some applications..............................................263

VII.   Quantum Algorithms
  23. Quantum states and measurements................................269
  24. Quantum circuits...............................................277
  25. Shor's algorithm...............................................285

VIII.  Calculus 101
  26. Limits and series..............................................293
  27. Bonus: A hint of p-adic numbers................................305
  28. Differentiation................................................317
  29. Power series and Taylor series.................................329
  30. Riemann integrals..............................................337

IX.    Complex Analysis
  31. Holomorphic functions..........................................347
  32. Meromorphic functions..........................................363
  33. Holomorphic square roots and logarithms........................373
  34. Bonus: Topological Abel-Ruffini Theorem........................379

X.     Measure Theory
  35. Measure spaces.................................................385
  36. Constructing the Borel and Lebesgue measure....................393
  37. Lebesgue integration...........................................403
  38. Swapping order with Lebesgue integrals.........................409
  39. Bonus: A hint of Pontryagin duality............................417

XI.    Probability (TO DO)
  40. Random variables (TO DO).......................................425
  41. Large number laws (TO DO)......................................427
  42. Stopped martingales (TO DO)....................................435

XII.   Differential Geometry
  43. Multivariable calculus done correctly..........................451
  44. Differential forms.............................................459
  45. Integrating differential forms.................................471
  46. A bit of manifolds.............................................483

XIII.  Riemann Surfaces
  47. Basic definitions of Riemann surfaces..........................495
  48. Morphisms between Riemann surfaces.............................503
  49. Affine and projective plane curves.............................509
  50. Differential forms.............................................519
  51. The Riemann-Roch theorem.......................................523
  52. Line bundles...................................................529

XIV.   Algebraic NT I: Rings of Integers
  53. Algebraic integers.............................................539
  54. The ring of integers...........................................545
  55. Unique factorization (finally!)................................553
  56. Minkowski bound and class groups...............................565
  57. More properties of the discriminant............................585
  58. Bonus: Let's solve Pell's equation!............................587

XV.    Algebraic NT II: Galois and Ramification Theory
  59. Things Galois..................................................595
  60. Finite fields..................................................607
  61. Ramification theory............................................613
  62. The Frobenius element..........................................623
  63. Bonus: A Bit on Artin Reciprocity..............................635

XVI.   Algebraic Topology I: Homotopy
  64. Some topological constructions.................................651
  65. Fundamental groups.............................................663
  66. Covering projections...........................................675

XVII.  Category Theory
  67. Objects and morphisms..........................................687
  68. Functors and natural transformations...........................699
  69. Limits in categories (TO DO)...................................711
  70. Abelian categories.............................................713

XVIII. Algebraic Topology II: Homology
  71. Singular homology..............................................723
  72. The long exact sequence........................................737
  73. Excision and relative homology.................................749
  74. Bonus: Cellular homology.......................................757
  75. Singular cohomology............................................769
  76. Application of cohomology......................................781

XIX.   Algebraic Geometry I: Classical Varieties
  77. Affine varieties...............................................801
  78. Affine varieties as ringed spaces..............................811
  79. Projective varieties...........................................821
  80. Bonus: B\'ezout's theorem......................................829
  81. Morphisms of varieties.........................................837

XX.    Algebraic Geometry II: Affine Schemes
  82. Sheaves and ringed spaces......................................849
  83. Localization...................................................861
  84. Affine schemes: the Zariski topology...........................871
  85. Affine schemes: the sheaf......................................879
  86. Interlude: eighteen examples of affine schemes.................889
  87. Morphisms of locally ringed spaces.............................905

XXI.   Set Theory I: ZFC, Ordinals, and Cardinals
  88. Interlude: Cauchy's functional equation and Zorn's lemma.......919
  89. Zermelo-Fraenkel with choice...................................927
  90. Ordinals.......................................................935
  91. Cardinals......................................................947

XXII.  Set Theory II: Model Theory and Forcing
  92. Inner model theory.............................................957
  93. Forcing........................................................969
  94. Breaking the continuum hypothesis..............................979

XXIII. Appendix