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Welcome to Covariance

Before functoriality, people lived in caves.

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Past exams	12:47 PM - August 28, 2026
Problem bank	12:47 PM - August 25, 2026
The Algebra Qual	12:34 PM - August 25, 2026

Knowledge

Mathematics

Algebra theory

Not to be confused with the general concept of algebra (abstract or otherwise), an algebra is an algebraic structure similar to a module, only with two internal binary operations.

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Algebras	2:03 PM - October 30, 2025
Exterior algebras	2:27 PM - November 06, 2025
Symmetric algebras	7:39 PM - November 02, 2025
Symmetric and alternating tensors	1:45 PM - November 04, 2025
Tensor algebras	7:42 PM - November 02, 2025

Algebraic geometry

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Examples of classical conic duality	7:24 AM - March 31, 2025

Category theory

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Ab-categories	6:51 AM - December 01, 2025
Abelian categories	6:55 AM - December 01, 2025
Additive categories	6:53 AM - December 01, 2025
Adjoints	8:45 PM - October 07, 2025
Categories	7:55 PM - October 07, 2025
Chain complexes	12:34 PM - March 31, 2025
Diagram chases without elements	7:26 PM - November 23, 2024
Diagram lemmas	12:34 PM - March 31, 2025
Double complexes and mural maps	6:40 AM - December 04, 2025
Exact sequences and chain homology	12:35 PM - March 31, 2025
Examples of adjoints	9:00 PM - October 07, 2025
Functor categories	8:42 PM - October 07, 2025
Functors	8:12 PM - October 07, 2025
Kan extensions	11:12 AM - November 24, 2025
Natural transformations	8:38 PM - October 07, 2025
Properties of adjoints	12:39 PM - March 31, 2025
Special morphisms	2:21 PM - November 18, 2025
Spectral sequences	12:37 PM - March 31, 2025
The Salamander Lemma	3:38 PM - December 09, 2025
Universal arrows and elements	12:48 PM - March 31, 2025
Universal Properties I - Inspiring Examples	7:56 PM - October 07, 2025
Universal Properties II - Commutative diagrams, cones and limits	3:08 PM - August 06, 2025
Universal Properties III - Yoneda's Lemma	10:14 AM - October 17, 2025

Differential equations

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Fourier series solutions I - A problem with power series	3:38 PM - August 02, 2025
Fourier series solutions II - Periodic functions	9:03 AM - August 06, 2025
Fourier series solutions III - Series representations of periodic functions	11:40 AM - August 05, 2025
Fourier series solutions IV - Computing Fourier coefficients	3:17 PM - August 06, 2025
Fourier series solutions V - Computing some Fourier series	8:24 AM - August 07, 2025
Fourier series solutions VI - Inner product spaces	12:49 PM - August 08, 2025
Fourier series solutions VII - Solving differential equations with Fourier series	9:25 AM - August 13, 2025
Fourier transform I - Pushing Fourier series to the limit	2:45 PM - August 18, 2025
Fourier transform II - The Fourier transform and inverse transform	3:21 PM - August 18, 2025
Fourier transform III - Properties of the Fourier transform	9:52 AM - August 19, 2025
Fourier transform IV - Derivatives	2:06 PM - August 19, 2025
Fourier transform V - Convolution	2:37 PM - August 19, 2025
Fourier transform VI - Solving differential equations	3:20 PM - August 19, 2025
Frobenius Series Solutions I - Slightly generalizing power series	3:30 PM - July 30, 2025
Frobenius Series Solutions II - Some illustrative examples	3:30 PM - July 30, 2025
Frobenius Series Solutions III - Frobenius theory	3:23 PM - August 02, 2025
Laplace transform I - Desperate times	1:41 PM - August 21, 2025
Laplace transform II - The inverse Laplace transform	2:10 PM - August 21, 2025
Laplace transform III - The Shifting Theorems	1:17 PM - August 22, 2025
Laplace transform IV - Convolution redux	8:42 AM - August 25, 2025
Laplace transform V - Solving differential equations	2:54 PM - August 23, 2025
Power Series Solutions I - From polynomials to power series	7:16 PM - July 28, 2025
Power Series Solutions II - First steps with power series	3:48 PM - July 28, 2025
Power Series Solutions III - A more representative example	8:21 AM - July 29, 2025
Power Series Solutions IV - Analytic functions and ordinary points	9:04 AM - July 31, 2025

Field theory

Note	Last Updated
A question about finite fields	10:26 AM - May 30, 2025

Group theory

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Normal subgroups	7:25 AM - March 31, 2025

Module theory

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Bimodule morphisms	1:02 PM - October 13, 2025
Bimodules	8:52 AM - October 14, 2025
Direct products of modules	1:51 PM - October 08, 2025
Direct products vs. direct sums vs. sums	2:29 PM - October 08, 2025
Direct sums of modules	2:11 PM - October 08, 2025
Exact Sequences I - Illustrative Examples	6:46 AM - October 24, 2025
Exact Sequences II - Exact Sequences	2:38 PM - October 23, 2025
Exact Sequences III - Morphisms of Exact Sequences	7:26 AM - October 24, 2025
Exact Sequences IV - Exact Sequences and Functors	6:39 AM - November 10, 2025
Examples of free modules	2:25 PM - October 09, 2025
Free modules	2:17 PM - October 09, 2025
Generators for modules and submodules	2:31 PM - October 09, 2025
Jordan Canonical Form I - Definition	10:59 AM - November 14, 2025
Jordan Canonical Form II - Computation	10:12 AM - November 18, 2025
Linear independence, rank and the structure of free modules	4:50 PM - November 05, 2025
Module morphisms	10:28 AM - October 08, 2025
Module morphisms and submodules	10:43 AM - October 08, 2025
Modules	10:15 AM - October 08, 2025
Modules over a PID - The Fundamental Theorem	1:01 PM - March 31, 2025
Noetherian modules	1:01 PM - March 31, 2025
Quotient modules	10:47 AM - October 08, 2025
Rational Canonical Form I - Definition	1:02 PM - March 31, 2025
Rational Canonical Form II - Additional Properties	1:02 PM - March 31, 2025
Rational Canonical Form III - Computation	6:50 AM - November 18, 2025
Submodules	10:36 AM - October 08, 2025
Sums of submodules	2:14 PM - October 08, 2025
Tensor Products I - Extending scalars	6:48 AM - October 13, 2025
Tensor Products II - Tensor products of bimodules	2:19 PM - October 14, 2025
Tensor Products III - Balanced Maps and a Universal Property of the Tensor Product	6:51 AM - October 17, 2025
Tensor Products IV - The Adjoint Property	8:48 AM - October 24, 2025
The 2-category of bimodules	6:46 AM - October 10, 2024
The Isomorphism Theorems for Modules	10:13 AM - October 10, 2025

Representation theory

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Representations of groups	9:26 AM - July 08, 2025

Ring theory

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Chinese Remainder Theorem	9:26 AM - July 15, 2025
Division rings	9:54 AM - July 15, 2025
Graded rings	7:25 AM - March 31, 2025
Least common multiples	7:34 PM - July 21, 2024
Principal ideal domains (PIDs)	6:29 AM - May 07, 2024

Tropical algebraic geometry

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Investigation into tropical tangency loci	7:27 AM - March 31, 2025
Linear tropical varieties	7:27 AM - March 31, 2025
Quadratic tropical varieties	7:27 AM - March 31, 2025
Tropical varieties	7:27 AM - March 31, 2025

Software

Blender

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Houdini

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L A T E X

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Systems of equations	2:56 PM - August 06, 2024