An Infinitely Large Napkin RZP28ERRZV

URI	`m2://res/RZP28ERRZV`
Kind	textbook
Creators	Evan Chen
Scope	deferred (no test decides the record, and the classifier has no model)

Expressions

e1: edition

Contents

chapter 1: Group actions overkill AIME problems
section 1.1: Definition of a group action
section 1.2: Stabilizers and orbits
section 1.3: Burnside's lemma
section 1.4: Conjugation of elements
chapter 2: Groups
section 2.1: Definition and examples of groups
section 2.2: Properties of groups
section 2.3: Isomorphisms
section 2.4: Orders of groups, and Lagrange's theorem
section 2.5: Subgroups
section 2.6: Groups of small orders
section 2.7: Unimportant long digression
section 2.8: \problemhead
chapter 3: Homomorphisms and quotient groups
section 3.1: Generators and group presentations
section 3.2: Homomorphisms
section 3.3: Cosets and modding out
section 3.4: (Optional) Proof of Lagrange's theorem
section 3.5: Eliminating the homomorphism
section 3.6: (Digression) The first isomorphism theorem
chapter 4: Flavors of rings
section 4.1: Fields
section 4.2: Integral domains
section 4.3: Prime ideals
section 4.4: Maximal ideals
section 4.5: Field of fractions
section 4.6: Unique factorization domains (UFD's)
section 4.7: Extra: Euclidean domains
section 4.8: \problemhead
chapter 5: Rings and ideals
section 5.1: Some motivational metaphors about rings vs groups
section 5.2: (Optional) Pedagogical notes on motivation
section 5.3: Definition and examples of rings
section 5.4: Fields
section 5.5: Homomorphisms
section 5.6: Ideals
section 5.7: Generating ideals
section 5.8: Principal ideal domains
section 5.9: Noetherian rings
section 5.10: \problemhead
chapter 6: The PID structure theorem
section 6.1: Finitely generated abelian groups
section 6.2: Some ring theory prerequisites
section 6.3: The structure theorem
section 6.4: Reduction to maps of free $R$-modules
section 6.5: Uniqueness of primary form
section 6.6: Smith normal form
chapter 7: Find all groups
section 7.1: Sylow theorems
section 7.2: (Optional) Proving Sylow's theorem
subsection 7.2.1: Definitions
subsection 7.2.2: Step 1: Prove that a Sylow $p$-subgroup exists
subsection 7.2.3: Step 2: Any two Sylow $p$-subgroups are conjugate
subsection 7.2.4: Step 3: Showing $n_p \equiv 1 \pmod p$
subsection 7.2.5: Step 4: $n_p$ divides $m$
section 7.3: (Optional) Simple groups and Jordan-H\"older
chapter 8: Bonus: A Bit on Artin Reciprocity
section 8.1: Overview
section 8.2: Infinite primes
section 8.3: Modular arithmetic with infinite primes
section 8.4: Infinite primes in extensions
section 8.5: Frobenius element and Artin symbol
section 8.6: Artin reciprocity
section 8.7: Application: Generalization of sum of two squares
section 8.8: \problemhead
chapter 9: Minkowski bound and class groups
section 9.1: The class group
section 9.2: The discriminant of a number field
section 9.3: The signature of a number field
section 9.4: Minkowski's theorem
section 9.5: The trap box
section 9.6: The Minkowski bound
section 9.7: The class group is finite
section 9.8: Computation of class numbers
section 9.9: Optional: Proof that $\OO_K$ is a free $\ZZ$-module
subsection 9.9.1: First proof
subsection 9.9.2: Second proof
subsection 9.9.3: Third proof
chapter 10: Unique factorization (finally!)
section 10.1: Motivation
section 10.2: Ideal arithmetic
section 10.3: Dedekind domains
section 10.4: Unique factorization works
section 10.5: The factoring algorithm
section 10.6: Fractional ideals
section 10.7: The ideal norm
chapter 11: More properties of the discriminant
chapter 12: Finite fields
section 12.1: Example of a finite field
section 12.2: Finite fields have prime power order
section 12.3: All finite fields are isomorphic
section 12.4: The Galois theory of finite fields
section 12.5: Extra: The multiplicative group of a finite field
chapter 13: The Frobenius element
section 13.1: Frobenius elements
section 13.2: Conjugacy classes
section 13.3: Chebotarev density theorem
section 13.4: Example: Frobenius elements of cyclotomic fields
section 13.5: Frobenius elements behave well with restriction
section 13.6: Application: Quadratic reciprocity
subsection 13.6.1: Step 1: Setup
subsection 13.6.2: Step 2: Reformulation
subsection 13.6.3: Step 3: Introducing the Frobenius
subsection 13.6.4: Finishing up
section 13.7: Frobenius elements control factorization
section 13.8: Example application: IMO 2003 problem 6
chapter 14: Things Galois
section 14.1: Motivation
section 14.2: Field extensions, algebraic extension, and splitting fields
section 14.3: Embeddings into algebraic closures for number fields
section 14.4: Everyone hates characteristic 2: separable vs irreducible
section 14.5: Automorphism groups and Galois extensions
section 14.6: Fundamental theorem of Galois theory
section 14.7: (Optional) Proof that Galois extensions are splitting
chapter 15: The ring of integers
section 15.1: Norms and traces
section 15.2: The ring of integers
section 15.3: On monogenic extensions
section 15.4: \problemhead
chapter 16: Algebraic integers
section 16.1: Motivation from high school algebra
section 16.2: Algebraic numbers and algebraic integers
section 16.3: Number fields
section 16.4: Primitive element theorem, and monogenic extensions
section 16.5: \problemhead
chapter 17: Bonus: Let's solve Pell's equation!
section 17.1: Units
section 17.2: Dirichlet's unit theorem
section 17.3: Finding fundamental units
section 17.4: Pell's equation
chapter 18: Ramification theory
section 18.1: Ramified / inert / split primes
section 18.2: Primes ramify if and only if they divide $\Delta_K$
section 18.3: Inertial degrees
section 18.4: The magic of Galois extensions
section 18.5: (Optional) Decomposition and inertia groups
section 18.6: Tangential remark: more general Galois extensions
chapter 19: Affine varieties
section 19.1: Affine varieties
section 19.2: Naming affine varieties via ideals
section 19.3: Radical ideals and Hilbert's Nullstellensatz
section 19.4: Pictures of varieties in $\Aff^1$
section 19.5: Prime ideals correspond to irreducible affine varieties
section 19.6: Pictures in $\Aff^2$ and $\Aff^3$
section 19.7: Maximal ideals
section 19.8: Motivating schemes with non-radical ideals
chapter 20: Bonus: B\'ezout's theorem
section 20.1: Non-radical ideals
section 20.2: Hilbert functions of finitely many points
section 20.3: Hilbert polynomials
section 20.4: B\'ezout's theorem
section 20.5: Applications
chapter 21: Localization
section 21.1: Spoilers
section 21.2: The definition
section 21.3: Localization away from an element
section 21.4: Localization at a prime ideal
section 21.5: Prime ideals of localizations
section 21.6: Prime ideals of quotients
section 21.7: Localization commutes with quotients
section 21.8: \problemhead
chapter 22: Morphisms of locally ringed spaces
section 22.1: Morphisms of ringed spaces via sections
section 22.2: Morphisms of ringed spaces via stalks
section 22.3: Morphisms of locally ringed spaces
section 22.4: A few examples of morphisms between affine schemes
subsection 22.4.1: One-point schemes
subsection 22.4.2: Examples of constant maps
subsection 22.4.3: The map $t \mapsto t^2$
subsection 22.4.4: An arithmetic example
section 22.5: The big theorem
section 22.6: More examples of scheme morphisms
section 22.7: A little bit on non-affine schemes
section 22.8: Where to go from here
section 22.9: \problemhead
section 22.10: Projective scheme
section 22.11: Morphisms of sheaves
chapter 23: Projective varieties
section 23.1: Graded rings
section 23.2: The ambient space
section 23.3: Homogeneous ideals
section 23.4: As ringed spaces
section 23.5: Examples of regular functions
chapter 24: Morphisms of varieties
section 24.1: Defining morphisms of baby ringed spaces
section 24.2: Classifying the simplest examples
subsection 24.2.1: Affine classification
subsection 24.2.2: Projective classification
section 24.3: Some more applications and examples
section 24.4: The hyperbola effect
chapter 25: Sheaves and ringed spaces
section 25.1: Motivation and warnings
section 25.2: Pre-sheaves
subsection 25.2.1: Usual definition
subsection 25.2.2: Categorical definition
section 25.3: Stalks and germs
section 25.4: Sheaves
section 25.5: For sheaves, sections ``are'' sequences of germs
section 25.6: Sheafification (optional)
chapter 26: Interlude: eighteen examples of affine schemes
section 26.1: Example: $\Spec k$, a single point
section 26.2: $\Spec \CC[x]$, a one-dimensional line
section 26.3: $\Spec \RR[x]$, a one-dimensional line with complex conjugates glued (no fear nullstellensatz)
section 26.4: $\Spec k[x]$, over any ground field
section 26.5: $\Spec \ZZ$, a one-dimensional scheme
section 26.6: $\Spec k[x] / (x^2-7x+12)$, two points
section 26.7: $\Spec k[x]/(x^2)$, the double point
section 26.8: $\Spec k[x]/(x^3-5x^2)$, a double point and a single point
section 26.9: $\Spec \Zc{60}$, a scheme with three points
section 26.10: $\Spec k[x,y]$, the two-dimensional plane
section 26.11: $\Spec \ZZ[x]$, a two-dimensional scheme, and Mumford's picture
section 26.12: $\Spec k[x,y]/(y-x^2)$, the parabola
section 26.13: $\Spec \ZZ[i]$, the Gaussian integers (one-dimensional)
section 26.14: Long example: $\Spec k[x,y]/(xy)$, two axes
subsection 26.14.1: Picture
subsection 26.14.2: Throwing out the $y$-axis
subsection 26.14.3: Stalks above some points
subsection 26.14.4: Stalk above the origin (tricky)
section 26.15: $\Spec k[x,x\inv]$, the punctured line (or hyperbola)
section 26.16: $\Spec k[x]_{(x)}$, zooming in to the origin of the line
section 26.17: $\Spec k[x,y]_{(x,y)}$, zooming in to the origin of the plane
section 26.18: $\Spec k[x,y]_{(0)} = \Spec k(x,y)$, the stalk above the generic point
section 26.19: \problemhead
chapter 27: Affine schemes: the sheaf
section 27.1: A useless definition of the structure sheaf
section 27.2: The value of distinguished open sets (or: how to actually compute sections)
subsection 27.2.1: A basis of the Zariski topology
subsection 27.2.2: Sections are computable
subsection 27.2.3: They are affine
subsection 27.2.4: Classic example: the punctured plane
section 27.3: The stalks of the structure sheaf
subsection 27.3.1: They are localizations
subsection 27.3.2: Motivating local rings: germs should package values
section 27.4: Local rings and residue fields: linking germs to values
subsection 27.4.1: Localizations give local rings
subsection 27.4.2: Computing values: a convenient square
section 27.5: Recap
section 27.6: Functions are determined by germs, not values
section 27.7: \problemhead
chapter 28: Affine schemes: the Zariski topology
section 28.1: Some more advertising
section 28.2: The set of points
section 28.3: The Zariski topology on the spectrum
section 28.4: Krull dimension
section 28.5: On radicals
section 28.6: \problemhead
chapter 29: Affine varieties as ringed spaces
section 29.1: Synopsis
section 29.2: The Zariski topology on $\Aff^n$
section 29.3: The Zariski topology on affine varieties
section 29.4: Coordinate rings
section 29.5: The sheaf of regular functions
section 29.6: Regular functions on distinguished open sets
section 29.7: Baby ringed spaces
chapter 30: Differentiation
section 30.1: Definition
section 30.2: How to compute them
section 30.3: Local (and global) maximums
section 30.4: Rolle and friends
section 30.5: Smooth functions
section 30.6: \problemhead
chapter 31: Riemann integrals
section 31.1: Uniform continuity
section 31.2: Dense sets and extension
section 31.3: Defining the Riemann integral
section 31.4: Meshes
section 31.5: \problemhead
chapter 32: Limits and series
section 32.1: Completeness and inf/sup
section 32.2: Proofs of the two key completeness properties of $\RR$
section 32.3: Monotonic sequences
section 32.4: Infinite series
section 32.5: Series addition is not commutative: a horror story
section 32.6: Limits of functions at points
section 32.7: Limits of functions at infinity
section 32.8: \problemhead
chapter 33: Bonus: A hint of $p$-adic numbers
section 33.1: Motivation
section 33.2: Algebraic perspective
subsection 33.2.1: Definition of $\ZZ_p$
subsection 33.2.2: Base $p$ expansion
subsection 33.2.3: Constructing $\QQ_p$
section 33.3: Analytic perspective
subsection 33.3.1: Definition
subsection 33.3.2: Ultrametric space
subsection 33.3.3: More fun with geometric series
subsection 33.3.4: Completeness
subsection 33.3.5: Philosophical notes
section 33.4: Mahler coefficients
section 33.5: \problemhead
chapter 34: Power series and Taylor series
section 34.1: Motivation
section 34.2: Power series
section 34.3: Differentiating them
section 34.4: Analytic functions
section 34.5: A definition of Euler's constant and exponentiation
section 34.6: This all works over complex numbers as well, except also complex analysis is heaven
section 34.7: \problemhead
chapter 35: Abelian categories
section 35.1: Zero objects, kernels, cokernels, and images
section 35.2: Additive and abelian categories
section 35.3: Exact sequences
section 35.4: The Freyd-Mitchell embedding theorem
section 35.5: Breaking long exact sequences
chapter 36: Objects and morphisms
section 36.1: Motivation: isomorphisms
section 36.2: Categories, and examples thereof
section 36.3: Special objects in categories
section 36.4: Binary products
section 36.5: Monic and epic maps
chapter 37: Functors and natural transformations
section 37.1: Many examples of functors
section 37.2: Covariant functors
section 37.3: Covariant functors as indexed family of objects
section 37.4: Contravariant functors
section 37.5: Equivalence of categories
section 37.6: (Optional) Natural transformations
section 37.7: (Optional) The Yoneda lemma
chapter 38: Limits in categories (TO DO)
section 38.1: Equalizers
section 38.2: Pullback squares (TO DO)
section 38.3: Limits
section 38.4: \problemhead
chapter 39: Holomorphic functions
section 39.1: The nicest functions on earth
section 39.2: Complex differentiation
section 39.3: Contour integrals
section 39.4: Cauchy-Goursat theorem
section 39.5: Cauchy's integral theorem
section 39.6: Holomorphic functions are analytic
section 39.7: Optional: Proof that holomorphic functions are analytic
subsection 39.7.1: Proof of Cauchy-Goursat theorem
subsection 39.7.2: The rest
chapter 40: Holomorphic square roots and logarithms
section 40.1: Motivation: square root of a complex number
section 40.2: Square roots of holomorphic functions
section 40.3: Covering projections
section 40.4: Complex logarithms
section 40.5: Some special cases
chapter 41: Meromorphic functions
section 41.1: The second nicest functions on earth
section 41.2: Meromorphic functions
section 41.3: Winding numbers and the residue theorem
section 41.4: Argument principle
section 41.5: Digression: the Argument Principle viewed geometrically
section 41.6: Philosophy: why are holomorphic functions so nice?
chapter 42: Bonus: Topological Abel-Ruffini Theorem
section 42.1: The Game Plan
section 42.2: Step 1: The Simplest Case
section 42.3: Step 2: Nested Roots
section 42.4: Step 3: Normal Groups
section 42.5: Summary
chapter 43: Differential forms
section 43.1: Pictures of differential forms
section 43.2: Pictures of exterior derivatives
section 43.3: Differential forms
section 43.4: Exterior derivatives
section 43.5: Digression: $\bigwedge^k(V^\vee)$ versus $(\bigwedge^k(V))^\vee$
section 43.6: Tangential remark: Arc length $ds$ is not a $1$-form
section 43.7: Closed and exact forms
chapter 44: A bit of manifolds
section 44.1: Topological manifolds
section 44.2: Smooth manifolds
section 44.3: Regular value theorem
section 44.4: Differential forms on manifolds
section 44.5: Orientations
section 44.6: Stokes' theorem for manifolds
section 44.7: (Optional) The tangent and cotangent space
subsection 44.7.1: Tangent space
subsection 44.7.2: The cotangent space
subsection 44.7.3: Sanity check
chapter 45: Multivariable calculus done correctly
section 45.1: The total derivative
section 45.2: The projection principle
section 45.3: Total and partial derivatives
section 45.4: (Optional) A word on higher derivatives
section 45.5: Towards differential forms
chapter 46: Integrating differential forms
section 46.1: Motivation: line integrals
section 46.2: Pullbacks
section 46.3: Cells
section 46.4: Boundaries
section 46.5: Stokes' theorem
section 46.6: Back to Earth: A comparison to what you learned in vector calculus
chapter 47: Advice for the reader
section 47.1: Prerequisites
section 47.2: Deciding what to read
section 47.3: Questions, exercises, and problems
section 47.4: Paper
section 47.5: On the importance of examples
section 47.6: Conventions and notations
subsection 47.6.1: Natural numbers are positive
subsection 47.6.2: Sets and equivalence relations
subsection 47.6.3: Functions
subsection 47.6.4: Cycle notation for permutations
subsection 47.6.5: Rings
subsection 47.6.6: Choice
section 47.7: Further reading
chapter 48: Preface
section: About this book
section: Source code
section: Philosophy behind the Napkin approach
section: More pedagogical comments and references
section: Historical and personal notes
section: Acknowledgements
chapter 49: Sales pitches
section 49.1: The basics
section 49.2: Abstract algebra
section 49.3: Real and complex analysis
section 49.4: Algebraic number theory
section 49.5: Algebraic topology
section 49.6: Algebraic geometry
section 49.7: Set theory
chapter 50: Bonus: Cellular homology
section 50.1: Degrees
section 50.2: Cellular chain complex
section 50.3: Digression: why are the homology groups equal?
section 50.4: Application: Euler characteristic via Betti numbers
section 50.5: The cellular boundary formula
chapter 51: Singular cohomology
section 51.1: Cochain complexes
section 51.2: Cohomology of spaces
section 51.3: Cohomology of spaces is functorial
section 51.4: Universal coefficient theorem
section 51.5: Explanation for universal coefficient theorem
section 51.6: Example computation of cohomology groups
section 51.7: Visualization of cohomology groups
section 51.8: Relative cohomology groups
chapter 52: Application of cohomology
section 52.1: Poincar\'e duality
section 52.2: de Rham cohomology
section 52.3: Graded rings
section 52.4: Cup products
section 52.5: Relative cohomology pseudo-rings
section 52.6: Wedge sums
section 52.7: Cross product
subsection 52.7.1: Motivation
subsection 52.7.2: Cross product on singular homology
subsection 52.7.3: Cross product is not a $\ZZ$-module homomorphism
subsection 52.7.4: Cross product on cellular homology
subsection 52.7.5: Cross product on cellular cohomology
subsection 52.7.6: Motivation: cross product of differential forms
subsection 52.7.7: Piecing the cohomology groups together
section 52.8: K\"unneth formula
chapter 53: Excision and relative homology
section 53.1: Motivation
section 53.2: The long exact sequences
section 53.3: The category of pairs
section 53.4: Excision
section 53.5: Some applications
section 53.6: Invariance of dimension
chapter 54: The long exact sequence
section 54.1: Short exact sequences and four examples
section 54.2: The long exact sequence of homology groups
section 54.3: The Mayer-Vietoris sequence
chapter 55: Singular homology
section 55.1: Simplices and boundaries
section 55.2: The singular homology groups
section 55.3: The homology functor and chain complexes
subsection 55.3.1: Algebra of chain complexes
subsection 55.3.2: Geometry of chain complexes
section 55.4: More examples of chain complexes
chapter 56: Determinant
section 56.1: Wedge product
section 56.2: The determinant
section 56.3: Characteristic polynomials, and Cayley-Hamilton
chapter 57: Dual space and trace
section 57.1: Tensor product
section 57.2: Dual space
section 57.3: $V^\vee \otimes W$ gives matrices from $V$ to $W$
section 57.4: The trace
section 57.5: \problemhead
chapter 58: Eigen-things
section 58.1: Why you should care
section 58.2: Warning on assumptions
section 58.3: Eigenvectors and eigenvalues
section 58.4: The Jordan form
section 58.5: Nilpotent maps
section 58.6: Reducing to the nilpotent case
section 58.7: (Optional) Proof of nilpotent Jordan
section 58.8: Algebraic and geometric multiplicity
chapter 59: Bonus: Fourier analysis
section 59.1: Synopsis
section 59.2: A reminder on Hilbert spaces
section 59.3: Common examples
subsection 59.3.1: Binary Fourier analysis on $\{\pm1\}^n$
subsection 59.3.2: Fourier analysis on finite groups $Z$
subsection 59.3.3: Fourier series for functions $L^2([-\pi, \pi])$
section 59.4: Summary, and another teaser
section 59.5: Parseval and friends
section 59.6: Application: Basel problem
section 59.7: Application: Arrow's Impossibility Theorem
section 59.8: \problemhead
chapter 60: Inner product spaces
section 60.1: The inner product
subsection 60.1.1: For real numbers: bilinear forms
subsection 60.1.2: For complex numbers: sesquilinear forms
subsection 60.1.3: Inner product space
section 60.2: Norms
section 60.3: Orthogonality
section 60.4: Hilbert spaces
section 60.5: \problemhead
chapter 61: Duals, adjoint, and transposes
section 61.1: Dual of a map
section 61.2: (Optional) A cautionary tale on what goes wrong if you don't have a natural isomorphism $V \cong V^\vee$
section 61.3: Identifying with the dual space
section 61.4: The adjoint (conjugate transpose)
section 61.5: Eigenvalues of normal maps
section 61.6: \problemhead
chapter 62: Vector spaces
section 62.1: The definitions of a ring and field
section 62.2: Modules and vector spaces
section 62.3: Direct sums
section 62.4: Linear independence, spans, and basis
section 62.5: Linear maps
section 62.6: What is a matrix?
subsection 62.6.1: Extended example with $\RR^2$, suitable for the general public
subsection 62.6.2: General discussion, back to Napkin levels of abstraction
section 62.7: Subspaces and picking convenient bases
section 62.8: A cute application: Lagrange interpolation
section 62.9: Pedagogical digression: Arrays of numbers are evil
section 62.10: A word on general modules
section 62.11: \problemhead
chapter 63: Constructing the Borel and Lebesgue measure
section 63.1: Pre-measures
section 63.2: Outer measures
section 63.3: Carath\'{e}odory extension for outer measures
section 63.4: Defining the Lebesgue measure
section 63.5: A fourth row: Carath\'{e}odory for pre-measures
section 63.6: From now on, we assume the Borel measure
section 63.7: \problemhead
chapter 64: Large number laws (TO DO)
section 64.1: Notions of convergence
subsection 64.1.1: Almost sure convergence
subsection 64.1.2: Convergence in probability
subsection 64.1.3: Convergence in law
section 64.2: Weak law of large numbers
subsection 64.2.1: Application: Weierstrass approximation
section 64.3: Strong law of large numbers
subsection 64.3.1: Motivation: Biased random walk
subsection 64.3.2: Statement
subsection 64.3.3: Proof for finite-variance case
subsection 64.3.4: The general proof
section 64.4: \problemhead
chapter 65: Lebesgue integration
section 65.1: The definition
section 65.2: An equivalent definition
section 65.3: Relation to Riemann integrals (or: actually computing Lebesgue integrals)
section 65.4: \problemhead
chapter 66: Stopped martingales (TO DO)
section 66.1: How to make money almost surely
section 66.2: Sub-$\sigma$-algebras and filtrations
section 66.3: Conditional expectation
section 66.4: Supermartingales
section 66.5: Optional stopping
section 66.6: Fun applications of optional stopping (TO DO)
subsection 66.6.1: The ballot problem
subsection 66.6.2: ABRACADABRA
subsection 66.6.3: USA TST 2018
section 66.7: \problemhead
chapter 67: Measure spaces
section 67.1: Letter connotations
section 67.2: Motivating measure spaces via random variables
section 67.3: Motivating measure spaces geometrically
section 67.4: $\sigma$-algebras and measurable spaces
section 67.5: Measure spaces
section 67.6: A hint of Banach-Tarski
section 67.7: Measurable functions
section 67.8: On the word ``almost''
section 67.9: \problemhead
chapter 68: Bonus: A hint of Pontryagin duality
section 68.1: LCA groups
section 68.2: The Pontryagin dual
section 68.3: The orthonormal basis in the compact case
section 68.4: The Fourier transform of the non-compact case
section 68.5: Summary
section 68.6: \problemhead
section 68.7: Peter-Weyl
chapter 69: Random variables (TO DO)
section 69.1: Random variables
section 69.2: Distribution functions
section 69.3: Examples of random variables
section 69.4: Characteristic functions
section 69.5: Independent random variables
section 69.6: \problemhead
chapter 70: Swapping order with Lebesgue integrals
section 70.1: Motivating limit interchange
section 70.2: Overview
section 70.3: Fatou's lemma
section 70.4: Everything else
section 70.5: Fubini and Tonelli
section 70.6: \problemhead
chapter 71: Working with general schemes
chapter 72: Projective schemes
chapter 73: Quasicoherent sheaves
chapter 74: Sheaves of $\OO_X$-modules
chapter 75: Quantum circuits
section 75.1: Classical logic gates
section 75.2: Reversible classical logic
section 75.3: Quantum logic gates
section 75.4: Deutsch-Jozsa algorithm
chapter 76: Shor's algorithm
section 76.1: The classical (inverse) Fourier transform
section 76.2: The quantum Fourier transform
section 76.3: Shor's algorithm
chapter 77: Quantum states and measurements
section 77.1: Bra-ket notation
section 77.2: The state space
section 77.3: Observations
section 77.4: Entanglement
chapter 78: Some applications
section 78.1: Frobenius divisibility
section 78.2: Burnside's theorem
section 78.3: Frobenius determinant
chapter 79: Characters
section 79.1: Definitions
section 79.2: The dual space modulo the commutator
section 79.3: Orthogonality of characters
section 79.4: Examples of character tables
chapter 80: Representations of algebras
section 80.1: Algebras
section 80.2: Representations
section 80.3: Direct sums
section 80.4: Irreducible and indecomposable representations
section 80.5: Morphisms of representations
section 80.6: The representations of $\Mat_d(k)$
chapter 81: Semisimple algebras
section 81.1: Schur's lemma continued
section 81.2: Density theorem
section 81.3: Semisimple algebras
section 81.4: Maschke's theorem
section 81.5: Example: the representations of $\CC[S_3]$
chapter 82: Affine and projective plane curves
section 82.1: Affine plane curves
section 82.2: The projective line $\CP^1$
section 82.3: Projective plane curves
section 82.4: Filling in the holes
section 82.5: Nodes of a plane curve
chapter 83: Basic definitions of Riemann surfaces
section 83.1: Complex structures
section 83.2: Riemann surface
section 83.3: Complex manifold
section 83.4: Examples of Riemann surfaces
chapter 84: Differential forms
section 84.1: Differential form on $\CC$
section 84.2: Visualization of differential forms
subsection 84.2.1: Holomorphic forms
subsection 84.2.2: Putting the pieces together: $1$-forms on a Riemann surface
chapter 85: Line bundles
section 85.1: Overview
section 85.2: Definition
section 85.3: Visualizing a line bundle
section 85.4: Morphisms between line bundles
section 85.5: Relation to invertible sheaves
chapter 86: Morphisms between Riemann surfaces
section 86.1: Definition
section 86.2: Functions to the Riemann sphere
section 86.3: Some other nice properties
section 86.4: Multiplicity of a map
section 86.5: The sum of the orders of a meromorphic function
section 86.6: The Hurwitz formula
section 86.7: The identity theorem
chapter 87: The Riemann-Roch theorem
section 87.1: Motivation
section 87.2: Divisors
section 87.3: Degree of a divisor
section 87.4: The principal divisor of a meromorphic function
section 87.5: The Riemann-Roch theorem
chapter 88: Breaking the continuum hypothesis
section 88.1: Adding in reals
section 88.2: The countable chain condition
section 88.3: Preserving cardinals
section 88.4: Infinite combinatorics
chapter 89: Cardinals
section 89.1: Equinumerous sets and cardinals
section 89.2: Cardinalities
section 89.3: Aleph numbers
section 89.4: Cardinal arithmetic
section 89.5: Cardinal exponentiation
section 89.6: Cofinality
section 89.7: Inaccessible cardinals
chapter 90: Forcing
section 90.1: Setting up posets
section 90.2: More properties of posets
section 90.3: Names, and the generic extension
section 90.4: Fundamental theorem of forcing
section 90.5: (Optional) Defining the relation
section 90.6: The remaining axioms
chapter 91: Inner model theory
section 91.1: Models
section 91.2: Sentences and satisfaction
section 91.3: The Levy hierarchy
section 91.4: Substructures, and Tarski-Vaught
section 91.5: Obtaining the axioms of $\ZFC$
section 91.6: Mostowski collapse
section 91.7: Adding an inaccessible
section 91.8: FAQ's on countable models
section 91.9: Picturing inner models
chapter 92: Ordinals
section 92.1: Counting for preschoolers
section 92.2: Counting for set theorists
section 92.3: Definition of an ordinal
section 92.4: Ordinals are ``tall''
section 92.5: Transfinite induction and recursion
section 92.6: Ordinal arithmetic
section 92.7: The hierarchy of sets
chapter 93: Zermelo-Fraenkel with choice
section 93.1: The ultimate functional equation
section 93.2: Cantor's paradox
section 93.3: The language of set theory
section 93.4: The axioms of $\ZFC$
section 93.5: Encoding
section 93.6: Choice and well-ordering
section 93.7: Sets vs classes
chapter 94: Interlude: Cauchy's functional equation and Zorn's lemma
section 94.1: Let's construct a monster
section 94.2: Review of finite induction
section 94.3: Transfinite induction
section 94.4: Wrapping up functional equations
section 94.5: Zorn's lemma
section 94.6: \problemhead
chapter 95: Compactness
section 95.1: Definition of sequential compactness
section 95.2: Criteria for compactness
section 95.3: Compactness using open covers
section 95.4: Applications of compactness
section 95.5: (Optional) Equivalence of formulations of compactness
section 95.6: \problemhead
chapter 96: Some topological constructions
section 96.1: Spheres
section 96.2: Quotient topology
section 96.3: Product topology
section 96.4: Disjoint union and wedge sum
section 96.5: CW complexes
section 96.6: The torus, Klein bottle, $\RP^n$, $\CP^n$
subsection 96.6.1: The torus
subsection 96.6.2: The Klein bottle
subsection 96.6.3: Real projective space
subsection 96.6.4: Complex projective space
chapter 97: Covering projections
section 97.1: Even coverings and covering projections
section 97.2: Lifting theorem
section 97.3: Lifting correspondence
section 97.4: Regular coverings
section 97.5: The algebra of fundamental groups
section 97.6: \problemhead
chapter 98: Fundamental groups
section 98.1: Fusing paths together
section 98.2: Fundamental groups
section 98.3: Fundamental groups are invariant under homeomorphism
section 98.4: Higher homotopy groups
section 98.5: Homotopy equivalent spaces
section 98.6: The pointed homotopy category
chapter 99: Properties of metric spaces
section 99.1: Boundedness
section 99.2: Completeness
section 99.3: Let the buyer beware
section 99.4: Subspaces, and (inb4) a confusing linguistic point
section 99.5: \problemhead
chapter 100: Metric spaces
section 100.1: Definition and examples of metric spaces
section 100.2: Convergence in metric spaces
section 100.3: Continuous maps
section 100.4: Homeomorphisms
section 100.5: Extended example/definition: product metric
section 100.6: Open sets
section 100.7: Closed sets
section 100.8: \problemhead
chapter 101: Topological spaces
section 101.1: Forgetting the metric
section 101.2: Re-definitions
subsection 101.2.1: Continuity
subsection 101.2.2: Closed sets
subsection 101.2.3: Properties that don't carry over
section 101.3: Hausdorff spaces
section 101.4: Subspaces
section 101.5: Connected spaces
section 101.6: Path-connected spaces
section 101.7: Homotopy and simply connected spaces
section 101.8: Bases of spaces
section 101.9: \problemhead

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