40 written chapters · 0 published teaching videos

Graduate, one chapter at a time.

Meet selected advanced ideas in analysis, algebra, geometry, probability, and fluid mathematics. These short introductions emphasize definitions, hypotheses, and the structure of an argument; they are starting points for deeper study.

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Each chapter follows an existing academy lesson. Numbers match the written study order; grade placement is a guide and school sequences may vary.

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40 of 40 chapters · in study order

Chapter 1 · Algebra
Recover a missing eigenvector with a Jordan chain
Use a generalized eigenvector when an eigenvalue lacks a full eigenbasis.
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Chapter 2 · Algebra
Find the shortest polynomial that annihilates a matrix
Distinguish a minimal polynomial from a characteristic polynomial.
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Chapter 3 · Algebra
Identify linear measurements as dual vectors
Construct a dual basis that extracts nonstandard coordinates.
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Chapter 4 · Algebra
Build a bilinear interaction from elementary tensors
Expand an elementary tensor in a product basis.
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Chapter 5 · Algebra
Check whether cosets support a group operation
Use conjugation to test normality before defining a quotient group.
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Chapter 6 · Algebra
Constrain subgroups using Sylow counting
Combine divisibility and congruence conditions on Sylow subgroup counts.
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Chapter 7 · Algebra
Use an ideal to make quotient multiplication consistent
Connect polynomial evaluation with a quotient ring.
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Chapter 8 · Algebra
Find the symmetries of a quadratic field extension
Identify automorphisms that fix the rational base field.
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Chapter 9 · Algebra
See a linear operator as a polynomial-module action
Encode repeated operator application through multiplication by a formal variable.
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Chapter 10 · Algebra
Average a symmetry action to extract invariant vectors
Construct a projection by averaging a finite group representation.
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Chapter 11 · Geometry
Measure loop winding through a covering coordinate
Represent the fundamental group of a circle by integer winding.
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Chapter 12 · Geometry
Count sheets of a circle covering
Relate a covering's degree to its action on fundamental groups.
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Chapter 13 · Geometry
Compute torus homology from a cell model
Use boundary maps to distinguish cycles from boundaries.
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Chapter 14 · Geometry
Find a closed differential form with nonzero circulation
Use an integral obstruction to distinguish closed from exact forms.
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Chapter 15 · Geometry
Distinguish intrinsic curvature from visible bending
Compute a cylinder's metric and compare its two principal curvatures.
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Chapter 16 · Statistics
Specify which events a probability model can distinguish
Build a finite sigma-algebra from observable groups.
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Chapter 17 · Statistics
Project a random variable onto partial information
Compute conditional expectation as a measurable groupwise average.
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Chapter 18 · Statistics
Check fairness through conditional expectation
Verify the martingale property of a centered random walk.
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Chapter 19 · Statistics
Use optional stopping only with its conditions
Verify a stopped expectation at a bounded random time.
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Chapter 20 · Statistics
See the extra term in stochastic differentiation
Apply Itô's formula to the square of Brownian motion.
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Chapter 21 · Statistics
Accumulate squared increments instead of total displacement
Check Brownian quadratic variation along uniform partitions.
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Chapter 22 · Statistics
Distinguish rare large errors from small expected error
Construct convergence in probability without convergence in L¹.
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Chapter 23 · Statistics
Change probability measures with likelihood weights
Compute a Radon–Nikodym derivative on a finite sample space.
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Chapter 24 · Statistics
Keep the data summary a likelihood actually uses
Apply factorization to identify a sufficient statistic.
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Chapter 25 · Statistics
Update a probability distribution over an unknown probability
Combine a Beta prior with a Bernoulli likelihood.
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Chapter 26 · Calculus
Turn a contraction into a certified iterative solver
Check a contraction constant and derive a computable error bound.
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Chapter 27 · Calculus
Differentiate a corner through integration against tests
Identify a weak derivative without requiring a classical derivative everywhere.
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Chapter 28 · Calculus
Watch diffusion damp high spatial frequencies faster
Solve a heat equation through its Fourier eigenmodes.
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Chapter 29 · Calculus
Derive an equation from stationary energy
Use first variations to obtain an Euler–Lagrange equation.
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Chapter 30 · Calculus
Use an energy bound to support existence and uniqueness
Check continuity and coercivity for a model weak elliptic problem.
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Chapter 31 · Calculus
Move derivatives onto test functions
Derive a divergence-free weak formulation of incompressible flow.
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Chapter 32 · Calculus
Separate a smooth energy identity from a weak energy inequality
Derive the energy balance and understand what lower-regularity limits can retain.
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Chapter 33 · Calculus
Identify the three-dimensional vorticity stretching term
Distinguish vorticity transport, stretching, and diffusion.
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Chapter 34 · Calculus
Find which velocity norm survives Navier–Stokes scaling
Compute norm scaling in three spatial dimensions.
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Chapter 35 · Calculus
Remove the longitudinal component of a Fourier velocity
Compute the divergence-free projection of a single nonzero Fourier mode.
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Chapter 36 · Calculus
Recover pressure as a constraint-enforcing field
Derive the pressure Poisson equation by taking divergence.
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Chapter 37 · Calculus
Compare forced and unforced equations without changing the theorem
Keep the forcing class explicit when interpreting existence or breakdown claims.
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Chapter 38 · Calculus
Distinguish finite energy from bounded amplitude
Construct concentrating functions whose L² norms stay controlled while maxima grow.
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Chapter 39 · Calculus
See how one regular solution can control a comparison
Recognize the energy estimate behind a weak–strong uniqueness argument.
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Chapter 40 · Calculus
Read the September 2026 Navier–Stokes announcement precisely
Separate theorem hypotheses, formal verification, and institutional evaluation when reading a research announcement.
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Explore the complete written study sequence

Chapters and video availability

Recover a missing eigenvector with a Jordan chain

Find the shortest polynomial that annihilates a matrix

Identify linear measurements as dual vectors

Build a bilinear interaction from elementary tensors

Check whether cosets support a group operation

Constrain subgroups using Sylow counting

Use an ideal to make quotient multiplication consistent

Find the symmetries of a quadratic field extension

See a linear operator as a polynomial-module action

Average a symmetry action to extract invariant vectors

Measure loop winding through a covering coordinate

Count sheets of a circle covering

Compute torus homology from a cell model

Find a closed differential form with nonzero circulation

Distinguish intrinsic curvature from visible bending

Specify which events a probability model can distinguish

Project a random variable onto partial information

Check fairness through conditional expectation

Use optional stopping only with its conditions

See the extra term in stochastic differentiation

Accumulate squared increments instead of total displacement

Distinguish rare large errors from small expected error

Change probability measures with likelihood weights

Keep the data summary a likelihood actually uses

Update a probability distribution over an unknown probability

Turn a contraction into a certified iterative solver

Differentiate a corner through integration against tests

Watch diffusion damp high spatial frequencies faster

Derive an equation from stationary energy

Use an energy bound to support existence and uniqueness

Move derivatives onto test functions

Separate a smooth energy identity from a weak energy inequality

Identify the three-dimensional vorticity stretching term

Find which velocity norm survives Navier–Stokes scaling

Remove the longitudinal component of a Fourier velocity

Recover pressure as a constraint-enforcing field

Compare forced and unforced equations without changing the theorem

Distinguish finite energy from bounded amplitude

See how one regular solution can control a comparison

Read the September 2026 Navier–Stokes announcement precisely