Date-first mathematical archive

INM: Informal Notes on Mathematics

A dated archive of mathematical notes reconstructed into an English TeX book.

158 dated notes15 thematic volumes in the book versionPDF + TeX for each note

Read full PDF Pre-TeX archive

Showing 158 notes, sorted by date. The full PDF is volume-organized; this website preserves the dated archive order.

2026

Jun

2026

N-20260621Complex Analysis12 sections

The Heat Equation, Diffusion, and Similarity Solutions

Contents preview

Diffusion versus wave motion
Heat kernel on the whole line
Similarity solution on a semi-infinite rod
Fourier series for a finite rod
Maximum principle
Energy decay on an interval
The maximum principle with proof idea
Smoothing and the heat semigroup

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Jun

2026

N-20260620Complex Analysis10 sections

The Second-Order Wave Equation and d'Alembert's Formula

Contents preview

The equation and its physical meaning
Factorization into transport operators
Initial-value problem on the whole line
Finite strings and Fourier series
Energy conservation
Inhomogeneous wave equation and Duhamel's principle
Boundary conditions on a finite interval
Deriving the Fourier coefficients

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Jun

2026

N-20260619Complex Analysis10 sections

First-Order Partial Differential Equations and Characteristics

Contents preview

What a first-order PDE asks for
The unforced transport equation
Example: translating a parabola
Forced transport
What can go wrong
General linear first-order equations
Quasilinear equations
Crossing characteristics and shocks

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Jun

2026

N-20260618Complex Analysis17 sections

First-Order and Higher-Order Differential Equations

Contents preview

Why the differential-equation part is one note
Basic concepts
Separable equations
Homogeneous first-order equations
First-order linear equations
Bernoulli equations
Exact equations
Reducible higher-order equations

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Jun

2026

N-20260617Complex Analysis20 sections

Fourier Series and Orthogonal Expansion of Functions

Contents preview

Why Fourier series is a separate theme
Orthogonality of trigonometric functions
Fourier coefficients
Dirichlet convergence theorem
Even and odd functions
Half-range expansions
Concrete coefficient models
Dirichlet kernel: what partial sums really do

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Jun

2026

N-20260616Complex Analysis14 sections

Numerical Series, Power Series, Taylor Series, and Improper Integrals

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Why these topics belong together
Numerical series
Positive-term series
Ratio and root tests
Alternating and arbitrary-sign series
Function series and uniform convergence
Power series
Taylor series

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Jun

2026

N-20260615Vector Calculus20 sections

Surface Integrals, Gauss Theorem, and Stokes Theorem

Contents preview

Why this note is split this way
Four similar-looking integrals
First-kind surface integrals
Template surfaces
Example pattern: spherical cap
Second-kind surface integrals as flux
Model flux computations: closed, capped, singular, and sheeted
Gauss theorem

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Jun

2026

N-20260604Linear Algebra16 sections

Solving Linear Systems: Direct, Iterative, and Optimization Methods

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The computational problem
Triangular solves
Gauss elimination
Pivoting
LU solution
Cholesky and LDL
Tridiagonal chasing
Stationary iteration

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Jun

2026

N-20260603Linear Algebra15 sections

Generalized Inverses, Projections, and Least-Squares Geometry

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Why ordinary inverse is not enough
Four regimes
Penrose equations
Fifteen generalized inverses
Projection transformations
Idempotence and orthogonality
Column and row-space projections
SVD computation

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Jun

2026

N-20260602Linear Algebra15 sections

Matrix Decompositions: LU, Cholesky, Full Rank, and SVD

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Why decompositions are algorithms
Doolittle LU
Existence and uniqueness
Elimination matrices
Doolittle formulas
Pivoting
Storage of LU
Cholesky decomposition

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Apr

2026

N-20260417Vector Calculus13 sections

Green's Theorem, Conservative Fields, and the Planar Stokes Principle

Contents preview

Green's theorem
Proof idea: rectangles and cancellation
Conservative fields and path independence
The punctured-plane warning
Area by Green's theorem
Differential forms notation
Closed forms, exact forms, and topology
Connection with de Rham cohomology

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Apr

2026

N-20260416Vector Calculus14 sections

Line Integrals and Surface Integrals of the First and Second Kind

Contents preview

Two meanings of integration on curves
Line integrals with respect to arc length
Line integrals of vector fields
Surface integrals of scalar fields
Flux integrals
First kind versus second kind
Parametrization invariance
Differential forms viewpoint

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Apr

2026

N-20260408Vector Calculus13 sections

Volumes, Spatial Regions, and Order of Integration

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Volume as an integral
Describing spatial regions
A reconstruction workflow
Changing order of integration
Cross-sections
Top minus bottom
Cylindrical and spherical descriptions
Absolute extrema over solids

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Apr

2026

N-20260406Vector Calculus14 sections

Triple Integrals: Solids, Cylindrical Coordinates, and Spherical Coordinates

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Triple integrals as weighted volume
Rectangular-coordinate example
Cylindrical coordinates
Spherical coordinates
Coordinate-choice table
Mass, center of mass, and moments
Symmetry
Three ways to understand volume factors

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Apr

2026

N-20260405Vector Calculus13 sections

Computing Double Integrals: Slicing, Polar Coordinates, and Change of Variables

Contents preview

The real task: describe the region
Changing the order of integration
Polar coordinates
Jacobian as local area scaling
Why the absolute value appears
A linear change example
Choosing coordinates
Slicing as pushforward of area

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Apr

2026

N-20260403Vector Calculus13 sections

Double Integrals: Riemann Sums, Area, Symmetry, and Average Value

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From area sums to double integrals
Geometric meanings
Basic properties
Existence of the integral
Symmetry before computation
Mean value theorem for double integrals
A worked average-value example
Relation with product measure

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Mar

2026

N-20260325Multivariable Calculus7 sections

Extrema of Two-Variable Functions and the Hessian Test

Contents preview

Local extrema
First-order necessary condition
Quadratic model and the Hessian
Second derivative test
A picture of the Hessian test
Constrained extrema
Endpoint and boundary warnings

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Mar

2026

N-20260318Linear Algebra14 sections

Matrix Computation III: Canonical Forms, Localization, and Iteration

Contents preview

Why canonical forms matter
Polynomial matrices
Smith normal form
Jordan data from elementary divisors
Jordan chains
Computing powers
Gershgorin discs
Gershgorin figure

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Mar

2026

N-20260314Multivariable Calculus8 sections

A Problem-Solving Guide for Multivariable Limits and Local Geometry

Contents preview

Purpose of this guide
Classifying points relative to a set
Limit templates
Degree comparison for rational expressions
Differentiability checklist
Tangent and normal templates
Local extrema template
Common failure modes

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Mar

2026

N-20260313Multivariable Calculus8 sections

Multivariable Chain Rule, Total Differentials, and Implicit Differentiation

Contents preview

The chain rule is composition of linear maps
One input variable, two intermediate variables
Two input variables
Total differentials
Implicit differentiation in one equation
Implicit differentiation for systems
A shape table
What to carry forward

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Mar

2026

N-20260312Linear Algebra13 sections

Matrix Computation II: Spectral Radius, Stability, and Conditioning

Contents preview

The long-term question
Spectral radius
Spectral radius is not a norm
Approximating spectral radius
Convergent matrices
Jordan boundary behavior
Neumann series
Matrix power series

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Mar

2026

N-20260311Linear Algebra12 sections

Matrix Computation I: Norms, Operators, and Numerical Size

Contents preview

Why size matters
Metrics and completeness
Normed vector spaces
Standard vector norms
Matrix norms
Unitary invariance
Induced norms
Normal matrices

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Mar

2026

N-20260306Multivariable Calculus8 sections

Multivariable Limits, Continuity, Differentiability, and the Total Derivative

Contents preview

What changes from one variable to many
How to prove and disprove limits
Continuity
Partial and directional derivatives
Differentiability as linear approximation
Tangent planes and normal vectors
Mixed partial derivatives
The main lesson

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Mar

2026

N-20260305Linear Algebra9 sections

Orthogonal Projection, Least Squares, and the Pseudoinverse

Contents preview

Projection onto a subspace
Projection matrix and least squares
Pseudoinverse
Projection geometry in the source diagram
Bridge to later ideas
Projection theorem
Normal equations
Pseudoinverse and singular values

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Mar

2026

N-20260304Multivariable Calculus8 sections

Point-Set Language in $\mathbb R^n$ : Neighborhoods, Boundaries, and Compactness

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Why point-set language appears
Balls and neighborhoods
Interior, exterior, and boundary
Open and closed sets
Accumulation points and closure
Boundedness and compactness
Connectedness and path-connectedness
What this vocabulary controls

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Mar

2026

N-20260303Linear Algebra8 sections

Two-Dimensional Linear Maps, Spectral Geometry, and Singular Value Decomposition

Contents preview

Concept map of the chapter
Five model transformations
A spectral example: a biological neural network
SVD as a sequence of geometric operations
SVD and latent structure in data
Rank-one blocks and image compression
A taxonomy of matrix properties
Closing perspective

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Mar

2026

N-20260302Machine Learning15 sections

Pure Mathematics as a Language for Machine Learning and Data Analysis

Contents preview

Backpropagation as cotangent pullback
Geometric data analysis
Manifold hypothesis and embeddings
Optimal transport and stochastic dynamics
Symmetry, equivariance, and category-like thinking
Further direction
Backpropagation and differential geometry
Manifold hypothesis and representation learning

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Jan

2026

N-20260106Calculus16 sections

Calculus Review III: Convexity, Integral Inequalities, Wallis Formula, and More

Contents preview

Convexity and the Hermite–Hadamard inequality
Kantorovich-type integral inequality
Integral mean value with odd symmetry
Symmetry of integrals about the midpoint
Midpoint and trapezoidal error terms
First special integral
Positivity of a Fresnel-type integral
A symmetry integral with arcsine

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Jan

2026

N-20260105Linear Algebra33 sections

Linear Algebra Review IV: Eigenvalue Tricks, Positive Definiteness, and More

Contents preview

Eigenvalues of matrix functions
No real eigenvector from a quadratic relation
Independence of eigenvectors
Real symmetric reconstruction from eigenvalues
Similarity invariants: trace and determinant
Trace shortcuts
Row-sum eigenvectors
Diagonalization problems

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Jan

2026

N-20260103Linear Algebra34 sections

Linear Algebra Review III: Determinant Tricks, Rank, Matrix Powers, and More

Contents preview

Determinant expansion and cofactors
Recursive determinants
Circulant-style determinant
Cancellation through orthogonal matrices
Determinant product and block simplification
Schur-complement style block transformation
Rank-two decomposition
Adjugate identities

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2025

Dec

2025

N-20251217Topology and Homotopy15 sections

A Nontrivial Loop at Zero: FinSet K-Theory and Stable Homotopy

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Why this second note exists
The lecture starts from ordinary equations
The Hopf fibration as the first nontrivial map
Homotopy groups and the first stable class
From natural numbers to finite sets
The space of finite sets
Why the first guess was close but reversed
Finite sets and stable spheres

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Dec

2025

N-20251210Calculus23 sections

Calculus Review II: Monotonicity, Integration Methods, Applications, and More

Contents preview

Strict monotonicity from the derivative
First derivative test for extrema
Second derivative test
Concavity and inflection points
Asymptotes
General procedure for sketching a curve
Basic antiderivative table
Substitution

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Dec

2025

N-20251202Linear Algebra28 sections

Linear Algebra Review II: Diagonalization, Inner Products, Quadratic Forms, and More

Contents preview

Diagonalizable matrices
Diagonalizability is independent from other properties
Algebraic and geometric multiplicity
Procedure for diagonalization
Inner product and vector length
Orthogonal and orthonormal systems
Schmidt orthogonalization
Example of Schmidt orthogonalization

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Nov

2025

N-20251128Combinatorics and Models9 sections

Self-Avoiding Walks III: Lace Expansion, Functional Integrals, and the Critical Dimension Four

Contents preview

The high-dimensional philosophy
The two-point function
The lace expansion as corrected random walk
Bubble diagrams and diagrammatic estimates
Differential inequalities
Functional integral representations
Dimension four and logarithmic corrections
How the three methods fit together

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Nov

2025

N-20251127Combinatorics and Models9 sections

Self-Avoiding Walks II: The Hexagonal Lattice, Holomorphic Observables, and the Exact Connective Constant

Contents preview

Why the hexagonal lattice is special
A domain version of the counting problem
The parafermionic observable
From local cancellation to a global identity
The upper bound on the connective constant
The lower bound on the connective constant
Relation with SLE
Loop models and the same mechanism

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Nov

2025

N-20251126Combinatorics and Models7 sections

Self-Avoiding Walks I: Connective Constants, Bridges, Polygons, and Critical Exponents

Contents preview

What the model is trying to measure
Submultiplicativity and the connective constant
Bridges and why they help
Polygons
Critical exponents
How dimension changes the story
The broader lesson

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Nov

2025

N-20251114Arithmetic Geometry12 sections

Teichmueller Theory Without One Universe: Classical Deformation, Hodge Theaters, and Identification

Contents preview

Why the word Teichmueller matters
Classical Teichmueller theory: fixed topology, variable complex structure
Distortion is measured in one universe
Why the arithmetic analogy cannot be literal
Hodge theaters as arithmetic stages
Etale-like and Frobenius-like directions
The theta-link is the point where one universe breaks
The log-link does not restore naive equality

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Nov

2025

N-20251113Arithmetic Geometry12 sections

When Addition Starts Floating: Theta-Links, Log-Links, and Arithmetic Deformation

Contents preview

What has to be explained
The nonarchimedean theta-function
The local theta-link as a non-ring-theoretic map
The bridge supplied by Kummer theory
Two symmetries and the bookkeeping of labels
Conjugate synchronization and the basepoint problem
The log-link: recovering additive structure from multiplicative structure
The log-theta-lattice

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Nov

2025

N-20251112Calculus18 sections

Calculus Review I: Inequalities, Limits, Derivatives, and More

Contents preview

Cauchy–Schwarz and Minkowski
Neighborhood notation
Trigonometric identities
Hyperbolic functions
Sequence limits: epsilon–N language
Infinite limits and one-sided limits
Heine criterion for function limits
Infinitesimals and infinitesimal orders

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Nov

2025

N-20251110Linear Algebra27 sections

Linear Algebra Review I: Determinants, Rank, Linear Systems, and More

Contents preview

Cofactor orthogonality
Vandermonde determinant
Block triangular determinants
General Laplace expansion
Determinant splitting
Adding rows or columns
Augmenting to a Vandermonde determinant
Row and column transformations

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Nov

2025

N-20251109Categories and Logic11 sections

Kernels, Cokernels, Images, and Coimages in Abelian Categories

Contents preview

The setting: maps with zero maps
Kernel: the universal object killed by a map
Cokernel: the universal quotient killing a map
Image: the part of the codomain actually reached
Coimage: the effective part of the domain
The standard factorization
A vector-space sanity check
A module example: multiplication by an integer

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Nov

2025

N-20251108Linear Algebra21 sections

Notes on MML: Vectors, the Minus-One Trick, Linear Maps, Basis Change, Kernel, and Image

Contents preview

Vectors beyond arrows
The minus-one trick
Example of the trick
Linear mappings
Complex numbers as a real vector space
Finite-dimensional isomorphism theorem
Basis change
Equivalence and similarity

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Oct

2025

N-20251011Machine Learning11 sections

The Reparameterization Trick and Auto-Encoding Variational Bayes

Contents preview

The gradient problem
Score-function estimator and pathwise estimator
Gaussian reparameterization
The stochastic ELBO estimator
Why implementations output log variance
Training as ELBO maximization
Decoder likelihoods and reconstruction losses
Generation after training

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Oct

2025

N-20251010Machine Learning11 sections

Variational Inference: KL Divergence, Jensen, and the ELBO

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The approximate posterior as an optimization problem
The ELBO identity
Jensen's inequality derivation
Reconstruction term and regularization term
The ELBO gap
Closed-form KL for diagonal Gaussians
Negative ELBO as a loss
Empirical lower bounds

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Oct

2025

N-20251009Machine Learning10 sections

Latent Variable Models: Priors, Decoders, and Intractable Posteriors

Contents preview

From autoencoders to latent variable models
The generative story
Graphical model notation
Marginal likelihood
The posterior problem
Why naive Monte Carlo is not enough
The recognition model
Amortized inference

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Oct

2025

N-20251008Machine Learning9 sections

Probability Language for Variational Autoencoders

Contents preview

Why probability enters generative modeling
Random variables, distributions, and densities
Joint, marginal, and conditional distributions
Bayes' rule as inversion of a generative story
Gaussian distributions
Expectation and Monte Carlo approximation
KL divergence
Jensen's inequality

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Sep

2025

N-20250926Machine Learning16 sections

Backpropagation in One MLP Layer: From a Three-Dimensional Tensor to an Outer Product

Contents preview

The setting
The theoretical tensor
Computing the sparse derivative
Contraction by the chain rule
Why the tensor disappears
Contravariant reading of the same calculation
Outer-product expression
Numerical example

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Jul

2025

N-20250707Curves and Structures8 sections

Nikonov's Paper at One Crossing: Region Probes, Partial Tribrackets, and What the Seminar Was About

Contents preview

Reading only one small part of a long paper
Where this sits inside Nikonov's paper
A region is not just a blank patch
At a crossing, four regions want to talk to each other
Why the operation is partial
Probe clearing and color rules
What about crossings themselves?
The paper in one playful sentence

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Jul

2025

N-20250705Curves and Structures7 sections

Nikonov's Seminar Prelude: Knots, Diagrams, Probes, and the Space Between the Lines

Contents preview

First orientation
The topological object and its planar shadow
Some basic knot-theory language
The pieces of a diagram
Nikonov's bridge: diagram elements as probes
Invariants, coinvariants, and labels
What this note is not trying to do

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Mar

2025

N-20250301Curves and Structures13 sections

Projective Schemes and Gluing: Building Projective Space from Affine Windows

Contents preview

Projective geometry returns with functions attached
The projective line as the model
Graded rings and homogeneous data
Projective space as Proj
The projective plane through three windows
A conic across charts
Projective closure with scheme memory
What Proj adds

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Feb

2025

N-20250211Classical Geometry17 sections

Transformations of Conics and the Erlangen View of Geometry

Contents preview

The organizing question
Affine transformations
Conics in matrix form
Projective geometry
Cross-ratio
Projective duality and conics
Inversion
Möbius transformations

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Jan

2025

N-20250118Curves and Structures12 sections

Schemes Gently: A Guided Tour of Spectrum, Nilpotents, and Local Rings

Contents preview

A new kind of point
The affine line spectrum
Closed sets and distinguished opens
Localization is zooming in
The structure sheaf
Nilpotents: invisible but geometric
The arithmetic curve
Maximal ideals are not enough

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2024

Oct

2024

N-20241026Curves and Structures14 sections

Zariski Geometry: When Functions Become the Space

Contents preview

Starting from functions, not pictures
Coordinate rings: a space through its functions
Nullstellensatz: the dictionary becomes reliable
The Zariski topology is coarse on purpose
Distinguished opens and localization
Sheaves: local data with a gluing rule
Stalks: looking through a microscope
A local example: two axes crossing

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Sep

2024

N-20240914Curves and Structures12 sections

Divisors on Curves: The Bookkeeping of Zeros, Poles, and Functions

Contents preview

A ledger for meromorphic functions
Principal divisors: the ledger of one function
Reading a divisor as permission
Line bundles: permission becomes geometry
The canonical divisor
Riemann–Roch without intimidation
Hurwitz as topology with branch points
A divisor game on the sphere

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Jul

2024

N-20240727Topology and Homotopy9 sections

Hopf Fibration, Homotopy, and the Infinite-Stability Reading

Contents preview

What this chapter is for
The first analytic reflex
Why the Hopf and K-theory hints changed the search
The Eilenberg swindle as the central guess
How this looked like a K-theory statement
The role I imagined for the Hopf fibration
The meme as a learning ladder
What the original interpretation claimed

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Jun

2024

N-20240608Curves and Structures14 sections

Plane Curves as Algebraic Creatures: Equations, Rings, and Singularities

Contents preview

The curve is not only the drawing
The function ring of a curve
When factorization becomes geometry
Smooth points and singular points
Tangent cones: the first nonzero shadow
The cusp as a ring
Intersections: one visible point may count twice
Projective closure repairs missing intersections

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May

2024

N-20240518Classical Geometry12 sections

Projective Geometry as the Art of Removing Exceptions

Contents preview

The annoyance of parallel lines
Homogeneous coordinates are ratios
A coordinate scene: where parallel lines meet
Duality: points and lines speak the same language
Transformations and what survives
A first invariant: the cross-ratio
Conics as one projective family
The projective plane as all lines through the origin

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Apr

2024

N-20240427Vector Calculus13 sections

Differential Forms and Stokes: The Geometry of Boundary Measurements

Contents preview

A theorem looking for its natural language
One-forms: measuring motion
The pullback experiment
Two-forms: measuring oriented area
Exterior derivative as one operator
A familiar theorem reappears
Orientation is not decoration
Why arc length is a different kind of geometry

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Apr

2024

N-20240413Multivariable Calculus13 sections

Local Linear Geometry: A Mapmaker's View of Derivatives and Tangent Spaces

Contents preview

The mapmaker's problem
Derivative first, partial derivatives second
The chain rule is not a formula trick
A small laboratory: polar coordinates
Tangent planes from equations
A manifold is a space with local Euclidean windows
Two pictures of tangent vectors
Regular values: one clean machine for making manifolds

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Mar

2024

N-20240330Combinatorics and Models20 sections

Combinatorial Geometry, Euler Formula, Platonic Solids, Degree, and Quotients

Contents preview

Good triangles
Why vertices must lie on the convex hull
Cyclic arc encoding
The geometric idea behind the proof
Euler formula for convex polyhedra
Regular convex polyhedra
Euler characteristic
Degree of a map

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Mar

2024

N-20240323Combinatorics and Models20 sections

Combinatorics: Identities, Generating Functions, Catalan Numbers, and More

Contents preview

Binomial identities
Good subsets
Generating functions
A probability example with repeated trials
Catalan numbers
Convex sets and convex hulls
Separating convex sets
Planar graphs and Euler-type counting

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Feb

2024

N-20240211Categories and Logic16 sections

Topos Theory: Categorical Preliminaries, Internal Logic, and the Geometry of Sets

Contents preview

Why topoi enter the story
Categories
Universal properties and limits
Exponentials and cartesian closure
Subobjects and the subobject classifier
Elementary topoi
A toy presheaf topos
Grothendieck topoi and sheaves

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2023

Aug

2023

N-20230801Curves and Structures18 sections

Reading Hitchin: Self-Duality Equations, Higgs Bundles, and Geometric Structures

Contents preview

Starting point
Connections and curvature
Higgs bundles
The moduli space
The Hitchin fibration
Spectral curves
Geometric Langlands perspective
Synthesis

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Jul

2023

N-20230710Sets and Foundations12 sections

Foundations of Mathematics: Set Theory, Category Theory, and Type Theory

Contents preview

The question of foundations
Set theory
Category theory
Type theory
Homotopy type theory and univalence
Comparative analysis
Modern applications
Toward a unified perspective

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Mar

2023

N-20230316Functions and Inequalities12 sections

Fixed Points, Iteration, and the Banach Contraction Principle

Contents preview

Fixed points
Iteration
Metric spaces and contractions
Banach fixed point theorem
Error estimate
Applications
Beyond Banach
What the example reveals

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Mar

2023

N-20230314Categories and Logic7 sections

Category Theory Continued: Free and Forgetful Functors, Concrete Categories, and Group Actions

Contents preview

Free and forgetful functors
Faithful and full functors
Concrete categories
Functors out of a one-object group category
Functors between one-object group categories
Why the note says “learn algebra first”
Looking ahead: natural transformations

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Mar

2023

N-20230313Categories and Logic10 sections

Category Theory Continued: Posets, Groups as One-Object Categories, Products, and Monomorphisms

Contents preview

What this note continues
Posets are categories
A small poset example
Isomorphisms in a poset category
Groups as one-object categories
Why this example matters
Products of categories
The opposite category

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Mar

2023

N-20230311Vector Calculus23 sections

Riemann Integration and Lebesgue Integration: From Partitions of the Domain to Partitions of Values

Contents preview

Why two theories of integration exist
Riemann sums
Darboux upper and lower sums
A simple integrable discontinuity
Dirichlet's function
The Lebesgue idea
Characteristic functions and sets
Convergence theorems

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Mar

2023

N-20230310Calculus16 sections

Derivatives, Tangents, Extremal Problems, Young–Holder–Minkowski, and Jensen

Contents preview

The derivative from secants to tangents
Derivative notation and local definition
Rules of differentiation
Elementary derivative formulas
Extrema, Fermat's lemma, and critical points
A derivative proof of a basic inequality
Young's inequality
Holder's inequality

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Feb

2023

N-20230225Sets and Foundations17 sections

Nonstandard Analysis and the Meaning of $0.999\ldots=1$

Contents preview

Why this topic appears
The real-number proof
Hyperreal numbers
Infinitesimals and finite hyperreals
Hyperfinite strings of nines
Transfer principle
Derivative through infinitesimals
Integral through hyperfinite sums

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Feb

2023

N-20230223Classical Geometry12 sections

Trigonometric Systems, Multiple-Angle Formulas, and a Determinant Recurrence

Contents preview

Correcting a trigonometric mistake
A trigonometric system
A rational-parametrization idea
A direct algebraic route
Multiple-angle formulas
The determinant recurrence
From technique to structure
Trigonometric systems as algebraic systems

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Feb

2023

N-20230220Classical Geometry10 sections

Special Proofs of Trigonometric Addition Formulas

Contents preview

Motivation
Sine difference formula by geometry
Rotation-matrix proof
Euler formula proof
Tangent formula
The method behind the trick
Rotation proof as the structural proof
Euler formula and characters

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Feb

2023

N-20230217Linear Algebra19 sections

A Brief Note on Tensors II: Tensor Products, Components, and More

Contents preview

Tensor products and bases
Bilinearity
Metric, dot product, and components
Reciprocal basis
Covariant and contravariant components
Vector product and Levi-Civita symbol
Scalar triple product
What changes next

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Feb

2023

N-20230209Linear Algebra11 sections

Linear Maps, Matrix Representations, and Tensor Invariance

Contents preview

From vector identities to linear representations
Coordinates as isomorphisms
The basis-change formula
A worked basis-change example
Which quantities are invariant?
Covectors and the inverse-transpose rule
Bilinear forms are not linear operators
Symmetric and skew parts

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Feb

2023

N-20230205Multivariable Calculus14 sections

Vector Triple Product, Exterior Algebra, and the Hodge Star

Contents preview

The vector triple product
Scalar triple product
From vector identities to exterior algebra
Levi-Civita notation
Lagrange identity
Why tensors enter naturally
Coordinate-free moral
The hidden structure

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Feb

2023

N-20230201Vector Calculus12 sections

Newton's Shell Theorem: Solid Angle, Potential, and Gauss Law

Contents preview

First orientation
Statement of Newton's shell theorem
Why symmetry alone is not enough
Solid-angle proof in full detail
Potential proof: doing the integral
The direct force integral
Gauss-law viewpoint
Harmonic potential

PDF TeX

Jan

2023

N-20230127Number Theory Basics15 sections

Factorization of $\sum_{i=0}^k x^{in}$ : Roots of Unity, False Generalization, and Cyclotomic Structure

Contents preview

The original problem and the first successful method
The direct factorization attempt
The attempted generalization
The computational experiment
Explicit root formula
Common real factors
The organizing idea
Cyclotomic summary of the experiment

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Jan

2023

N-20230123Vector Calculus15 sections

Definite Integrals, Line Integrals, Work, Green-Type Thinking, and Vector Operations

Contents preview

Indefinite and definite integrals
Area by slicing
Work as an integral
Line integrals
Green-type intuition
Polar coordinates
Spherical coordinates
Dot product and cross product

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Jan

2023

N-20230119Classical Geometry17 sections

Plane Geometry I: Ceva, Menelaus, Ptolemy, Simson Line, and Euler Line

Contents preview

Why this geometry chapter matters
Ceva's theorem
Menelaus' theorem
Ptolemy's theorem
Generalized Ptolemy inequality
Simson line
Euler line and Euler formula
Why the pattern matters

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Jan

2023

N-20230118Combinatorics and Models17 sections

Combinatorics: Pigeonhole Principle, Subsets, Binomial Sums, and Helly-Type Ideas

Contents preview

Pigeonhole principle
Subsets and binomial coefficients
Binomial identities by double counting
A subset-sum style example
Generating functions
Helly-type remark
The next layer
Pigeonhole as averaging

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Jan

2023

N-20230117Functions and Inequalities18 sections

More Function Exercises: Periodicity, Piecewise Definitions, Images, and Functional Equations

Contents preview

Continuing the function chapter
Piecewise functions
Periodicity from functional equations
Graph transformations
Image of an interval
Functional equations and substitution
Counting functions between finite sets
A compact bridge

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Jan

2023

N-20230116Functions and Inequalities17 sections

Functions, Mappings, Injections, Surjections, and Functional Equations

Contents preview

What a function is
Image and preimage
Injective and surjective
Composition and inverse functions
Functional equations on finite sets
Functional equations over real numbers
Looking ahead
Functions as morphisms

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Jan

2023

N-20230107Functions and Inequalities11 sections

Function Graphs, Symmetry Centers, and Transformations

Contents preview

A note on graph problems
Centers of symmetry
Symmetry about a line
Absolute value transformations
Zero counting by graphs
How to find a center of symmetry in practice
Absolute value as folding
Parameter motion and intersection counting

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2022

Dec

2022

N-20221229Topology and Homotopy19 sections

Algebraic Topology and Algebra: Products, Wedge Sums, CW Complexes, Cosets, and Surface Gluings

Contents preview

What this note is really doing
Disjoint union
Wedge sum
CW complexes: building spaces by cells
The torus as a quotient of a square
Cosets as algebraic quotients
The Klein bottle
Polygon schemata for surfaces

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Dec

2022

N-20221228Topology and Homotopy15 sections

Quotient Topology and Product Topology

Contents preview

Interval modulo endpoints
More quotient spaces
Definition of quotient topology
Product topology
Universal property of quotient topology
Endpoint identification in detail
Product topology via projections
Quotient and product as opposite directions

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Dec

2022

N-20221227Categories and Logic16 sections

Category Theory Continued and the Question Topology

Contents preview

More examples of categories
Subcategories and functors
Forgetful functors
A topology generated by a distance-like question
Posets as categories, with an example
Functorial thinking
Generated topology
Why category theory appears beside topology

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Dec

2022

N-20221223Topology and Homotopy17 sections

Homotopy, Simply Connected Spaces, Semigroups, Groups, and Homomorphisms

Contents preview

Homotopy intuition
Simply connected spaces
Semigroups, monoids, and groups
Congruence, quotients, and homomorphisms
Loops and the beginning of the fundamental group
Why the punctured plane has nontrivial loops
Homomorphisms as structure-preserving maps
Quotients in topology and algebra

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Dec

2022

N-20221222Topology and Homotopy17 sections

Topological Re-definitions, Subspaces, Connectedness, and Path-Connectedness

Contents preview

Re-defining continuity
Closed sets and closure
Subspaces
Connectedness
Path-connectedness
Continuity: why preimages appear
Closure through neighborhoods
Connectedness by continuous images

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Dec

2022

N-20221221Topology and Homotopy11 sections

Basic Topology II: Completion, Homeomorphism Warnings, and Quotient Intuition

Contents preview

Completion
Let the buyer beware
Homeomorphisms and embeddings
Quotient intuition
A more structural view
Completion and its universal property
Homeomorphism versus uniform equivalence
Continuous bijections that are not homeomorphisms

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Dec

2022

N-20221218Topology and Homotopy17 sections

Basic Topology I: Metric Spaces, Boundedness, Norms, Open Sets, and Completeness

Contents preview

Why the note starts with metrics
Boundedness: diameter and radius
Metric spaces
Sequences and continuity
Standard metrics
Norms and inner products
Open and closed balls
Open sets, closed sets, and limit points

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Dec

2022

N-20221217Functions and Inequalities22 sections

Function Exercises: Domains, Ranges, Dirichlet Function, Parameters, and Functional Equations

Contents preview

Structure of the exercise sheet
Domain problems
Floor functions
Range problems
Dirichlet function
Graph transformations and zero counting
Logarithmic parameter equation
Functional equations with period-like recurrence

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Dec

2022

N-20221214Curves and Structures18 sections

Tarski's Circle-Squaring Problem: From Polygon Dissection to Borel Equidecomposition

Contents preview

The route of the note
Equidecomposability and dissection congruence
Hilbert's third problem and the Dehn invariant
Measure-theoretic background
Tarski's circle-squaring problem
Scissors congruence and why nice pieces fail
Laczkovich's positive theorem
First idea: work in the torus

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Dec

2022

N-20221211Calculus16 sections

Derivatives: Definition, Rules, Elementary Functions, and Inverse Trigonometric Derivatives

Contents preview

Average velocity and instantaneous velocity
Derivative notation
Power functions
Exponential and logarithmic derivatives
Trigonometric derivatives
Inverse trigonometric derivatives
Examples
Context and outlook

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Dec

2022

N-20221210Combinatorics and Models15 sections

Binomial Theorem, Absolute-Value Estimates, and Set Exercises

Contents preview

Binomial theorem
Absolute value of binomial sums
A summation exercise
Set exercises
Counting subsets with properties
Set-builder notation
Closing perspective
Binomial identities as counting in two languages

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Dec

2022

N-20221209Combinatorics and Models18 sections

Set-Theoretic and Combinatorial Exercises after the First Lecture

Contents preview

What this page is doing
A floor-function example
A binomial counting identity
Solving a system by discriminants
A prime-number exercise
A nested-set exercise
A recurrence of sets
An intersection-counting problem

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Dec

2022

N-20221207Combinatorics and Models17 sections

Set Problems, Extremal Counting, and Venn-Diagram Reasoning

Contents preview

A set construction problem
What this problem teaches
A Venn-diagram necessary-and-sufficient problem
Difference identities
Coordinate and algebraic exercises
A note on proof writing
Extremal set problems: why constructions matter
Venn diagrams versus proofs

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Dec

2022

N-20221205Combinatorics and Models22 sections

Lecture Notes on Sets, Counting, and Binomial Coefficients

Contents preview

Summation and product notation
De Morgan's laws
Inclusion–exclusion
Pigeonhole principle
Addition and multiplication principles
Permutations
Combinations
Pascal identity

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Nov

2022

N-20221126Number Theory Basics18 sections

Congruence Classes, Complete Residue Systems, and Euler's Function

Contents preview

Congruence
Basic properties
Residue classes
Complete and reduced residue systems
Example: last digit
Example: reducing a long expression
A polynomial congruence by induction
Where the idea leads

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Nov

2022

N-20221120Arithmetic Geometry17 sections

Elliptic Curves: Nonsingularity, Group Law, and Cryptographic Motivation

Contents preview

Definition and nonsingularity
Why the discriminant appears
Cryptographic motivation
Geometric addition law
Algebraic formula
Special cases and the point at infinity
Why associativity is difficult
Elliptic curves over finite fields

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Nov

2022

N-20221119Number Theory Basics17 sections

Prime Numbers, Composite Numbers, Divisor Functions, Congruences, and Number-Theoretic Exercises

Contents preview

Prime and composite numbers
Euclid's theorem
Prime divisibility
Fundamental theorem of arithmetic
Perfect numbers and square numbers
Congruence exercises
Broader perspective
Prime exponents and valuations

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Nov

2022

N-20221107Calculus10 sections

Trigonometric Limits, Derivatives, and Inverse Trigonometric Examples

Contents preview

Trigonometric limits
Derivative of sine and cosine
Tangent, cotangent, secant, and cosecant
Inverse trigonometric derivatives
Examples from the handwritten page
A closing lens
The circle as a Lie group
Trigonometric derivatives from the exponential

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Nov

2022

N-20221105Functions and Inequalities10 sections

Function Exercises and Inequality Estimates

Contents preview

A short note about elementary methods
Periodicity from a transformed argument
A quadratic minimum
A logarithmic comparison
A parameterized polynomial
The conceptual turn
Periodicity from transformed arguments
Quadratic and logarithmic estimates

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Oct

2022

N-20221031Number Theory Basics20 sections

Divisibility, Euclidean Algorithm, Greatest Common Divisor, Bezout, and Congruences

Contents preview

Divisibility
Division algorithm
Greatest common divisor and least common multiple
Euclidean algorithm
Bezout theorem
Gauss lemma and divisibility consequences
Prime numbers
Congruences

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Oct

2022

N-20221030Number Theory Basics18 sections

Integer Arithmetic: Well-Ordering, Euclidean Algorithm, and the Euler Function

Contents preview

The well-ordering principle
Divisibility, gcd, and lcm
Division algorithm
Euclidean algorithm
Example
Fundamental theorem of arithmetic
The Euler phi function
The broader lesson

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Oct

2022

N-20221013Number Theory Basics14 sections

Shoelace Formula, Remainder Theorem, and Modular Arithmetic

Contents preview

Shoelace formula
Why the shoelace formula works
A circle-area maximum problem
Remainder theorem
A polynomial remainder example
Congruence arithmetic
Fermat and Euler
Shoelace formula as oriented area

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Oct

2022

N-20221012Functions and Inequalities15 sections

A Map of Classical Inequalities

Contents preview

Bernoulli's inequality
Power means
Cauchy–Schwarz
Minkowski inequality
A three-variable inequality
Chebyshev's inequality
Jensen's inequality
Young, Hölder, and Minkowski

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Oct

2022

N-20221005Categories and Logic11 sections

Linear Logic: Resources, Dialogue, Narrative, and Geometry

Contents preview

First orientation
From truth to resources
Narrative as a resource pipeline
Tensor, additives, and branching stories
Dialogue as a game
Schopenhauer's stratagem as a ludic interaction
Ludics: loci instead of signifieds
Orthogonality and negation

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Oct

2022

N-20221002Arithmetic Geometry21 sections

Finite Fields and Polynomial Factorization

Contents preview

Why finite fields matter
Fields
Field extensions
Constructing finite fields
Important properties of finite fields
Polynomial factorization over finite fields
Algorithms
Applications

PDF TeX

Sep

2022

N-20220918Number Theory Basics11 sections

Divisibility by Finite Differences and Pairing

Contents preview

A general odd-power divisibility principle
Why finite differences work
An alternating harmonic numerator
The next abstraction
Finite differences as discrete derivatives
The binomial basis and integer-valued polynomials
Pairing as group symmetry
P-adic valuation and lifting intuition

PDF TeX

Sep

2022

N-20220913Number Theory Basics15 sections

Divisibility, Congruences, and the Habit of Proving Necessity

Contents preview

From guessing to proving
Divisibility
Example: reducing the dividend
Example: polynomial divisibility in one variable
Congruences
A large power sum
A wider interpretation
Divisibility as ideal containment

PDF TeX

Sep

2022

N-20220911Classical Geometry23 sections

Analytic Geometry: Lines, Circles, and Coordinate Tricks

Contents preview

Line formulas
Parallel and perpendicular lines
Direction and normal vectors
Angles, distances, and parallel lines
Circle equations
A fractional-linear range problem
A graph-slope counting problem
A moving-line maximum

PDF TeX

Aug

2022

N-20220813Complex Analysis15 sections

Complex Contour Integrals and the Geometry of Integration

Contents preview

Why complex integration is different
Real integrals as a warm-up
Contours and parametrization
Riemann sums and midpoint sums
Complex Riemann sums
Integrals of powers on the unit circle
Cauchy–Goursat theorem
Contour integrals depend on paths

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Aug

2022

N-20220811Functions and Inequalities18 sections

Inequality Exercises: Cauchy, AM–GM, Vectors, and Optimization

Contents preview

How to read this note
A fraction problem with a constraint
A weighted square-root comparison
Maximizing a radical expression
Equality in Cauchy–Schwarz
Triangle-angle inequalities
Two spheres and a linear equation
A product inequality

PDF TeX

Aug

2022

N-20220810Functions and Inequalities11 sections

Basic Inequalities, Cauchy, Triangle Inequality, and Coordinate Geometry

Contents preview

Basic inequalities
AM–GM
Cauchy's inequality
Triangle inequality and its variants
Worked AM–GM examples
A Cauchy example
Coordinate geometry at the end of the note
What the examples reveal: three structures

PDF TeX

Aug

2022

N-20220808Functions and Inequalities16 sections

Set Counting, Modular Patterns, and Mathematical Induction

Contents preview

A set-counting problem
A modular arithmetic construction
Why not every construction is finished
Sequence exercises
Mathematical induction
Examples of induction
Induction structure
Modular patterns

PDF TeX

Aug

2022

N-20220806Functions and Inequalities23 sections

Set Theory Exercises, Closures, Functions, and Algebraic Side Notes

Contents preview

Images, preimages, and finite set examples
Closure under addition
A combinatorial maximal-set problem
Functional and algebraic exercises
Matrix and coordinate side computations
Images and preimages behave differently
Closure operations
The reusable pattern

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Aug

2022

N-20220803Number Theory Basics13 sections

Arithmetic and Geometric Sequence Exercises

Contents preview

A trigonometric opening example
Logarithmic sequences
Recovering terms from partial sums
Sums and recurrence relations
Geometric progression products
A harmonic-looking sequence
A final synthesis
Sequences as discrete dynamical systems

PDF TeX

Aug

2022

N-20220802Classical Geometry20 sections

Trigonometric Functions, Radians, Identities, and Triangle Formulas

Contents preview

Angles and radians
Sine, cosine, and tangent
Special angles
Addition formulas
Double-angle and half-angle formulas
Graphs of trigonometric functions
Triangle formulas
Radians are not a convention only

PDF TeX

Aug

2022

N-20220801Functions and Inequalities24 sections

Set Exercises, Functions, and Elementary Problem-Solving

Contents preview

Subsets and characteristic functions
Set laws
Venn diagram exercises
Inclusion–exclusion with literature examples
Set constructions with squares
Floor functions
Functions and monotonicity
Characteristic functions and subsets

PDF TeX

Jul

2022

N-20220728Multivariable Calculus19 sections

Spatial Vectors, Cross Products, and Categorical Products

Contents preview

Spatial vectors
Coordinate formulas
Cross product
From products to universal properties
Categories and groupoids
Cross product geometry
Categorical product versus vector product
What changes next

PDF TeX

Jul

2022

N-20220725Linear Algebra10 sections

Permutations, Inversions, and Determinants

Contents preview

Inversions and signs
Definition of determinant
Basic determinant properties
Sarrus and triangular matrices
Cofactor expansion
A structured determinant
The determinant line and exterior powers
Why permutation signs are forced

PDF TeX

Jul

2022

N-20220723Number Theory Basics7 sections

Roots of Unity, Decagons, and Cyclotomic Divisibility

Contents preview

The ant on a decagon
Roots of unity
A broader pattern
Cyclotomic viewpoint
Why this is more than a trick
Roots of unity viewpoint
Cyclotomic divisibility

PDF TeX

Jul

2022

N-20220721Functions and Inequalities12 sections

Probability, Conditional Probability, and Substitution Tricks in Inequalities

Contents preview

Random experiments and sample spaces
Classical model and Venn diagrams
Conditional probability
A transition to substitution tricks
A classical symmetric equation
Inequality examples
Jensen, trigonometry, and contest style
The hidden structure

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Jul

2022

N-20220720Combinatorics and Models15 sections

Counting Principles, Binomial Identities, and Mixed Problem Solving

Contents preview

Counting principles
A divisibility example
Permutations and combinations
Binomial theorem
Polynomial coefficients
Further contest examples
Binomial identities by stories
When to use multiplication and when to use addition

PDF TeX

Jul

2022

N-20220719Classical Geometry22 sections

Solid Geometry: Points, Lines, Planes, Parallelism, and Perpendicularity

Contents preview

Why solid geometry feels different
Basic axioms of incidence
Relations between two lines
A line and a plane
Parallel planes and parallel lines
Perpendicularity
Projection theorem
A coordinate viewpoint

PDF TeX

Jul

2022

N-20220718Functions and Inequalities12 sections

Nonlinear Recurrences, Fixed Points, and Metric-Space Exercises

Contents preview

Fixed-point iteration
Möbius recurrences
Example
Second-order linear recurrences
Dedekind cuts and metric exercises
Fixed points of recurrence maps
Monotone bounded sequences
After the examples

PDF TeX

Jul

2022

N-20220717Number Theory Basics16 sections

Recurrences, Arithmetic and Geometric Progressions, and the First Examples of Groups

Contents preview

A beginning point: recursive definitions
Arithmetic progressions
Determining an arithmetic progression
Geometric progressions
Converting recurrences
A factorial recurrence
Groups
Symmetric and dihedral groups

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Jul

2022

N-20220716Classical Geometry11 sections

Conic Exercises: Lines, Ellipses, Parabolas, and Tangents

Contents preview

Line and ellipse
Tangents from a point to an ellipse
Line and hyperbola
Parabola tangent through a point
A parabola and a circle
An ellipse with prescribed quadrilateral area
The next layer
Tangency via discriminants

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Jul

2022

N-20220715Classical Geometry14 sections

Conics and the Dedekind Construction of Addition

Contents preview

Conic sections
Latus rectum and tangent geometry
Rational and irrational density
Vector spaces
Dedekind cuts and order
Addition of Dedekind cuts
Additive inverses
Conics through focus and directrix

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Jul

2022

N-20220713Sets and Foundations14 sections

Completeness, Supremum, Dedekind Cuts, and Metric Spaces

Contents preview

Large and small positive real numbers
The geometric view of the real line
Supremum property
Nested intervals imply completeness
Dedekind cuts
Metric spaces
Subspaces and density
Supremum as least upper bound

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Jul

2022

N-20220710Sets and Foundations12 sections

Vectors, Ordered Fields, and the First Axioms of Real Numbers

Contents preview

Plane vectors
Field axioms
Ordered fields
Elementary consequences of the order axioms
Nested intervals
Ordered fields and the real line
Vectors as algebraicized geometry
The conceptual payoff

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Jul

2022

N-20220708Classical Geometry17 sections

Directed Angles, Reflections, and the Miquel Point

Contents preview

Reflecting the orthocenter
The incenter–excenter lemma
Miquel point of a triangle
Directed angles reduce case distinctions
Miquel point proof pattern
Further context
Directed angles as projective bookkeeping
Reflections and isogonal structure

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Jun

2022

N-20220628Calculus15 sections

Techniques of Integration: Partial Fractions, Parts, and Substitution

Contents preview

Linearity
Partial fractions
Integration by parts
Substitution
Partial fractions as algebra before calculus
Integration by parts as product rule reversed
A second lens
Partial fractions as algebra of poles

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Jun

2022

N-20220615Classical Geometry19 sections

A Russian-Style Early Olympiad Game: Coordinates, Packing, and Symmetry

Contents preview

What the note contains
Problem 1: two distances under a linear constraint
Problem 2: parabola and distance
Problem 3: packing unit circles
Problem 4: a symmetric Diophantine equation
The larger pattern
Coordinate choice and invariants
Packing and density

PDF TeX

May

2022

N-20220501Multivariable Calculus19 sections

Lagrange Multipliers, Hessians, Bordered Hessians, and Degenerate Extrema

Contents preview

Why this note was too compressed before
Lagrange multipliers: the basic idea
Example: minimizing distance to a line
Why the method is valid
Hessian matrix and the second derivative test
Sylvester criterion and eigenvalues
Bordered Hessian
Example revisited with a bordered Hessian

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Apr

2022

N-20220417Functions and Inequalities18 sections

Proof Strategies, Cyclic Sums, and Inequality Methods

Contents preview

Proof strategies
Cyclic and symmetric sums
AM–GM
Muirhead's inequality
Power means
Hölder and Cauchy
Jensen and Karamata
Cyclic sums

PDF TeX

Apr

2022

N-20220412Linear Algebra15 sections

Linear Algebra Continued: Indices, Rank, Transpose, Hermitian Conjugate, and Trace

Contents preview

Unit vectors and Kronecker delta
Einstein summation convention
Rank
Transpose and Hermitian conjugate
Trace
A more structural view
Transpose and dual maps
Hermitian adjoints in Hilbert spaces

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Apr

2022

N-20220411Linear Algebra18 sections

Invertible Matrices, Adjoints, Row Reduction, and Rotations

Contents preview

Invertibility
Diagonal matrices
Products of inverses
Adjugate formula
Row reduction for inverses
Rotation matrices
Beyond the first calculation
Invertibility as isomorphism

PDF TeX

Apr

2022

N-20220409Calculus13 sections

Faà di Bruno, Leibniz Rules, and Matrix Multiplication

Contents preview

Motivation
Faà di Bruno formula
Leibniz's differential notation
Parametric derivatives
Differentiating determinants
Matrix multiplication
Schur complement
What the pattern suggests

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Apr

2022

N-20220407Linear Algebra12 sections

Exercises on $\mathbb R^n$ , $\mathbb C^n$ , Vector Spaces, and Subspaces

Contents preview

Complex arithmetic
Vector space identities
A non-example with infinity
Subspaces
Bridge to later ideas
How to read basic vector-space exercises
Beyond real and complex scalar structures
Quotient spaces

PDF TeX

Apr

2022

N-20220405Multivariable Calculus10 sections

Fréchet Differentiability, Hessians, and Constrained Tests

Contents preview

Fréchet differentiability and complex differentiability
Fréchet derivative
Visualizing complex maps
Hessian matrices and Taylor expansion
Bordered Hessian
Example
Finite fields and Galois theory
Why complex analysis meets Hessians

PDF TeX

Mar

2022

N-20220323Complex Analysis7 sections

Complex Differentiability and the Rigidity of Holomorphic Functions

Contents preview

Why complex analysis feels different
Complex derivative
Basic examples
Closure properties
The main intuition
Complex versus real differentiability
Local power series intuition

PDF TeX

Mar

2022

N-20220321Complex Analysis16 sections

Some Pre-Complex Analysis: Motion on the Unit Circle and Roots of Unity

Contents preview

The complex exponential as motion
Roots and angles
A divisibility result
Closing perspective
Roots of unity as symmetry
Context and outlook
Roots of unity as a finite symmetry group
Cyclotomic factorization

PDF TeX

Mar

2022

N-20220320Calculus14 sections

A Review of Calculus: Mean Value Theorems and Continuity

Contents preview

From the mean value theorem to Cauchy's form
Continuity at a point
How this note fits the previous one
Rolle's theorem as the engine
Cauchy's theorem and L'Hopital's rule
Uniform continuity
Further direction
Mean value theorems as compactness plus local linearity

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Mar

2022

N-20220319Calculus12 sections

A Review of Calculus: Sequences, Limits, Derivatives, and Integrals

Contents preview

Sequences and convergence
Limits of functions
Derivatives
Small-o and big-O
Integration
Why the epsilon definition is shaped this way
Continuity by sequences
Derivative as best linear approximation

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Mar

2022

N-20220315Calculus11 sections

Quadratic Parameters, Tangency, and Constrained Extrema

Contents preview

A scattered algebraic note
A quadratic with parameters
A parabola and a hyperbola
A constrained extremum
Tangency as a double-root condition
Completing the square
A wider frame
Tangency as multiplicity

PDF TeX

Feb

2022

N-20220212Curves and Structures16 sections

Riemann Surfaces as One-Dimensional Complex Manifolds

Contents preview

Structure beyond topology
From smooth structure to complex structure
Definition of a Riemann surface
Examples
Holomorphic maps between Riemann surfaces
Meromorphic functions as maps to the sphere
Degree and multiplicity
Local power form is not global

PDF TeX

Feb

2022

N-20220205Sets and Foundations12 sections

A First Comparison of ZFC, NBG, KM, and New Foundations

Contents preview

Why there are several axiomatizations of set theory
The main systems
ZFC
NBG
Kelley–Morse set theory
New Foundations
How to compare the systems
What remains unclear

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Jan

2022

N-20220131Sets and Foundations13 sections

Rethinking Set Theory Through Functions Rather Than Membership

Contents preview

Why this translation/note matters
The discomfort with ZFC
Three misunderstandings about ETCS
Elements as special functions
The informal ETCS-style principles
Subsets via characteristic functions
Products, exponentials, and inverse images
Choice as right inverses

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Jan

2022

N-20220123Arithmetic Geometry17 sections

From Algebraic Number Theory to the First Shadow of Langlands

Contents preview

Why this note starts with algebraic number theory
Number fields and algebraic integers
Ideals and factorization
Quadratic fields and class number
First understanding of Langlands
Local and global correspondences
Cross-field intuitions
Numerical experiments

PDF TeX

2021

Oct

2021

N-20211007Arithmetic Geometry11 sections

From Diophantine Inequalities to Arithmetic Deformation

Contents preview

The question is an inequality, not a slogan
A first chain of rigidity results
From Mordell finiteness to effective inequalities
The Szpiro inequality as a concrete target
Vojta's inequality and the hyperbolic curve viewpoint
Arithmetic fundamental groups
The local elliptic curve model: Tate curves and q-parameters
Normalised degree as log-volume

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Aug

2021

N-20210806Topology and Homotopy9 sections

Group Actions, Homotopy, and a Rough Road to the Hopf Fibration

Contents preview

Purpose of the note
Group actions
Homotopy
Cofibrations, fibrations, and deformation retracts
Spheres and homotopy groups
The Hopf fibration
References to return to
Group actions as symmetry bookkeeping

PDF TeX

Aug

2021

N-20210804Categories and Logic11 sections

Boolean Algebras, Boolean Rings, and the First Hint of Stone Duality

Contents preview

Boolean algebra beyond computing
A first definition via Heyting algebras
A small amount of algebra
Boolean rings
Examples of Boolean rings
Boolean algebras directly
Moving between the two structures
The logical shadow

PDF TeX

Jul

2021

N-20210721Categories and Logic11 sections

A Gateway to Dualities: Space, Algebra, and the Dualizing Object

Contents preview

Starting point
The table is not a list
The dualizing object
Stone duality as the first complete example
What Stone duality says in one sentence
Gelfand duality as a continuous analogue
Affine schemes and the affine line
Locales and the Sierpi'nski space

PDF TeX

Jun

2021

N-20210621Linear Algebra14 sections

Prop Presentations for Linear Algebra: Diagrams, Matrices, and Relations

Contents preview

First orientation
Props as typed wiring diagrams
The alphabet: copy, discard, add, zero, and scalars
Two colours: a comonoid and a monoid
The bialgebra law
Scalars as ring operations
How a matrix becomes a diagram
Completeness: a proof system for matrices

PDF TeX

Feb

2021

N-20210207Sets and Foundations8 sections

Choice, Well-Ordering, and the Arithmetic of Cardinal Size

Contents preview

Choice enters the story
Partially ordered sets
Well-ordering
Equipollence and cardinal numbers
Cantor's theorem
Comparing cardinals
Infinite cardinals and finite disturbance
Concrete encodings behind the arithmetic

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Feb

2021

N-20210206Sets and Foundations6 sections

Relations, Products, and the First Appearance of Universal Properties

Contents preview

Relations and equivalence classes
Partitions
Indexed products and projections
The first universal property
Integers and induction
Division, gcd, and congruence

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Feb

2021

N-20210204Sets and Foundations7 sections

Sets, Classes, and the First Warnings About Size

Contents preview

Why classes enter before functions
Russell's class as the first warning
Power sets and indexed families
Set identities
Images, inverse images, and functions
One-sided inverses
Set-level bookkeeping behind functions

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2026

The Heat Equation, Diffusion, and Similarity Solutions

The Second-Order Wave Equation and d'Alembert's Formula

First-Order Partial Differential Equations and Characteristics

First-Order and Higher-Order Differential Equations

Fourier Series and Orthogonal Expansion of Functions

Numerical Series, Power Series, Taylor Series, and Improper Integrals

Surface Integrals, Gauss Theorem, and Stokes Theorem

Solving Linear Systems: Direct, Iterative, and Optimization Methods

Generalized Inverses, Projections, and Least-Squares Geometry

Matrix Decompositions: LU, Cholesky, Full Rank, and SVD

Green's Theorem, Conservative Fields, and the Planar Stokes Principle

Line Integrals and Surface Integrals of the First and Second Kind

Volumes, Spatial Regions, and Order of Integration

Triple Integrals: Solids, Cylindrical Coordinates, and Spherical Coordinates

Computing Double Integrals: Slicing, Polar Coordinates, and Change of Variables

Double Integrals: Riemann Sums, Area, Symmetry, and Average Value

Extrema of Two-Variable Functions and the Hessian Test

Matrix Computation III: Canonical Forms, Localization, and Iteration

A Problem-Solving Guide for Multivariable Limits and Local Geometry

Multivariable Chain Rule, Total Differentials, and Implicit Differentiation

Matrix Computation II: Spectral Radius, Stability, and Conditioning

Matrix Computation I: Norms, Operators, and Numerical Size

Multivariable Limits, Continuity, Differentiability, and the Total Derivative

Orthogonal Projection, Least Squares, and the Pseudoinverse

Point-Set Language in Rn\mathbb R^nRn: Neighborhoods, Boundaries, and Compactness

Two-Dimensional Linear Maps, Spectral Geometry, and Singular Value Decomposition

Pure Mathematics as a Language for Machine Learning and Data Analysis

Calculus Review III: Convexity, Integral Inequalities, Wallis Formula, and More

Linear Algebra Review IV: Eigenvalue Tricks, Positive Definiteness, and More

Linear Algebra Review III: Determinant Tricks, Rank, Matrix Powers, and More

2025

A Nontrivial Loop at Zero: FinSet K-Theory and Stable Homotopy

Calculus Review II: Monotonicity, Integration Methods, Applications, and More

Linear Algebra Review II: Diagonalization, Inner Products, Quadratic Forms, and More

Self-Avoiding Walks III: Lace Expansion, Functional Integrals, and the Critical Dimension Four

Self-Avoiding Walks II: The Hexagonal Lattice, Holomorphic Observables, and the Exact Connective Constant

Self-Avoiding Walks I: Connective Constants, Bridges, Polygons, and Critical Exponents

Teichmueller Theory Without One Universe: Classical Deformation, Hodge Theaters, and Identification

When Addition Starts Floating: Theta-Links, Log-Links, and Arithmetic Deformation

Calculus Review I: Inequalities, Limits, Derivatives, and More

Linear Algebra Review I: Determinants, Rank, Linear Systems, and More

Kernels, Cokernels, Images, and Coimages in Abelian Categories

Notes on MML: Vectors, the Minus-One Trick, Linear Maps, Basis Change, Kernel, and Image

The Reparameterization Trick and Auto-Encoding Variational Bayes

Variational Inference: KL Divergence, Jensen, and the ELBO

Latent Variable Models: Priors, Decoders, and Intractable Posteriors

Probability Language for Variational Autoencoders

Backpropagation in One MLP Layer: From a Three-Dimensional Tensor to an Outer Product

Nikonov's Paper at One Crossing: Region Probes, Partial Tribrackets, and What the Seminar Was About

Nikonov's Seminar Prelude: Knots, Diagrams, Probes, and the Space Between the Lines

Projective Schemes and Gluing: Building Projective Space from Affine Windows

Transformations of Conics and the Erlangen View of Geometry

Schemes Gently: A Guided Tour of Spectrum, Nilpotents, and Local Rings

2024

Zariski Geometry: When Functions Become the Space

Divisors on Curves: The Bookkeeping of Zeros, Poles, and Functions

Hopf Fibration, Homotopy, and the Infinite-Stability Reading

Plane Curves as Algebraic Creatures: Equations, Rings, and Singularities

Projective Geometry as the Art of Removing Exceptions

Differential Forms and Stokes: The Geometry of Boundary Measurements

Local Linear Geometry: A Mapmaker's View of Derivatives and Tangent Spaces

Combinatorial Geometry, Euler Formula, Platonic Solids, Degree, and Quotients

Combinatorics: Identities, Generating Functions, Catalan Numbers, and More

Topos Theory: Categorical Preliminaries, Internal Logic, and the Geometry of Sets

2023

Reading Hitchin: Self-Duality Equations, Higgs Bundles, and Geometric Structures

Foundations of Mathematics: Set Theory, Category Theory, and Type Theory

Fixed Points, Iteration, and the Banach Contraction Principle

Category Theory Continued: Free and Forgetful Functors, Concrete Categories, and Group Actions

Category Theory Continued: Posets, Groups as One-Object Categories, Products, and Monomorphisms

Riemann Integration and Lebesgue Integration: From Partitions of the Domain to Partitions of Values

Derivatives, Tangents, Extremal Problems, Young–Holder–Minkowski, and Jensen

Nonstandard Analysis and the Meaning of 0.999…=10.999\ldots=10.999…=1

Trigonometric Systems, Multiple-Angle Formulas, and a Determinant Recurrence

Special Proofs of Trigonometric Addition Formulas

A Brief Note on Tensors II: Tensor Products, Components, and More

Linear Maps, Matrix Representations, and Tensor Invariance

Vector Triple Product, Exterior Algebra, and the Hodge Star

Newton's Shell Theorem: Solid Angle, Potential, and Gauss Law

Point-Set Language in $\mathbb R^n$ : Neighborhoods, Boundaries, and Compactness

Nonstandard Analysis and the Meaning of $0.999\ldots=1$

Factorization of $\sum_{i=0}^k x^{in}$ : Roots of Unity, False Generalization, and Cyclotomic Structure

Exercises on $\mathbb R^n$ , $\mathbb C^n$ , Vector Spaces, and Subspaces