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..........................................359 33. Holomorphic square roots and logarithms........................369 34. Bonus: Topological Abel-Ruffini Theorem........................375 X. Measure Theory 35. Measure spaces.................................................381 36. Constructing the Borel and Lebesgue measure....................389 37. Lebesgue integration...........................................399 38. Swapping order with Lebesgue integrals.........................405 39. Bonus: A hint of Pontryagin duality............................413 XI. Probability (TO DO) 40. Random variables (TO DO).......................................421 41. Large number laws (TO DO)......................................423 42. Stopped martingales (TO DO)....................................431 XII. Differential Geometry 43. Multivariable calculus done correctly..........................447 44. Differential forms.............................................455 45. Integrating differential forms.................................467 46. A bit of manifolds.............................................479 XIII. Riemann Surfaces 47. Basic definitions of Riemann surfaces..........................491 48. Morphisms between Riemann surfaces.............................497 49. Affine and projective plane curves.............................503 50. Differential forms.............................................513 51. The Riemann-Roch theorem.......................................517 52. Line bundles...................................................523 XIV. Algebraic NT I: Rings of Integers 53. Algebraic integers.............................................533 54. The ring of integers...........................................539 55. Unique factorization (finally!)................................547 56. Minkowski bound and class groups...............................559 57. More properties of the discriminant............................577 58. Bonus: Let's solve Pell's equation!............................579 XV. Algebraic NT II: Galois and Ramification Theory 59. Things Galois..................................................587 60. Finite fields..................................................599 61. Ramification theory............................................605 62. The Frobenius element..........................................615 63. Bonus: A Bit on Artin Reciprocity..............................627 XVI. Algebraic Topology I: Homotopy 64. Some topological constructions.................................643 65. Fundamental groups.............................................655 66. Covering projections...........................................667 XVII. Category Theory 67. Objects and morphisms..........................................679 68. Functors and natural transformations...........................691 69. Limits in categories (TO DO)...................................703 70. Abelian categories.............................................705 XVIII. Algebraic Topology II: Homology 71. Singular homology..............................................715 72. The long exact sequence........................................729 73. Excision and relative homology.................................741 74. Bonus: Cellular homology.......................................749 75. Singular cohomology............................................761 76. Application of cohomology......................................773 XIX. Algebraic Geometry I: Classical Varieties 77. Affine varieties...............................................793 78. Affine varieties as ringed spaces..............................803 79. Projective varieties...........................................813 80. Bonus: B\'ezout's theorem......................................821 81. Morphisms of varieties.........................................829 XX. Algebraic Geometry II: Affine Schemes 82. Sheaves and ringed spaces......................................841 83. Localization...................................................853 84. Affine schemes: the Zariski topology...........................863 85. Affine schemes: the sheaf......................................871 86. Interlude: eighteen examples of affine schemes.................881 87. Morphisms of locally ringed spaces.............................895 XXI. Set Theory I: ZFC, Ordinals, and Cardinals 88. Interlude: Cauchy's functional equation and Zorn's lemma.......909 89. Zermelo-Fraenkel with choice...................................917 90. Ordinals.......................................................925 91. Cardinals......................................................937 XXII. Set Theory II: Model Theory and Forcing 92. Inner model theory.............................................947 93. Forcing........................................................959 94. Breaking the continuum hypothesis..............................969 XXIII. Appendix