Mathematics

Remarks

If a link does not work, search the textbook on LibGen... they probably have it.

Multiple books on the same topic is better than a single "best" book on the topic.

Self-learning comes with self-research outside of textbooks or lectures.

Calculus and linear algebra are pretty much required to understand the rest of math. Everything after that is a matter of personal choice.

Useful references

• Programs - MIT
• Textbooks - Reddit
• Documentation - LaTeX
• Applied

Channels

MIT, Borcherds, Numberphile, Mathemaniac, Faculty, Schuller

Funny/fun things

• Euclidea
• Best proof methods
• 99 Variations of proof - Ording
• How to catch lions
• Diagonal arguments
• Halting problem

Marianopolis

• Library: Math section on 1st floor.
• Articles: Any online database like EBSCO and JSTOR.
• Computer lab: Usually vacant.
• Teachers
• Peers

Introduction to Mathematics

"Calculus, Linear Algebra"

• Precalculus - Khan What is to be known before calculus.
• Essence of calculus - 3B1B Beautiful and informal exposition to calculus.
• Essence of linear algebra - 3B1B Beautiful and informal exposition to linear algebra.
• Calculus, linear algebra - Khan
• Start math - Bright side Naive set theory.
• Beginning logic - Lemmon Proof theory: Natural deduction

Analysis

"Continuity"

• (Understanding analysis + solutions) - Abbott Structure of the real numbers, cardinality; rigorous treatment of sequences, series, limits, differentiation, integration; convergence and limits of functions.
• Measure theory - Bright side Measure spaces, Lebesgue integrals.
• Complex analysis.com Beautiful and informal exposition to complex numbers, complex differentiation and integration; complex series.
• Variational calculus - Faculty
• Nonlinear dynamics and chaos - Faculty
• ODE - Arnold

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Abstract algebra

"Isomorphism"

• Topics in algebra - Hernstein Groups, rings, fields, vector spaces.
• Linear algebra - Axler Vector spaces, linear functions, determinants, eigen-everything, inner product spaces.
• Group theory - Borcherds
• Tensors - Porat

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Topology

"Invariants"

• Topology and geometry - Tokieda Exposition to algebraic and differential topology/geometry.
• (Topology - Bruno Zimmerman) + (Topology - Munkres) Point-set topology, topological invariants, covers; metric and function spaces; introductory algebraic topology.
• (Algebraic topology - Albin) + (Algebraic topology - Hatcher) Fundamental group, homology, homotopy. Requires understanding of group theory and point-set topology.
• Lie algebra - Mathemaniac
• Lie algebra - Kirillov

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Number theory

"Primes"

• Number theory - Michael Penn Euclidean algorithm, modular arithmetics, Chinese remainder theorem, Fermat's little theorem, quadratic forms, generating functions.
• Number theory - Hardy

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• Number theory - Borcherds

Mathematical Logic

"Proofs"

• Beginning logic - Lemmon Proof theory: Natural deduction
• Set theory - Enderton Set theory: ZFC.
• Model theory - Stanford Philosophy
• Study guide - Smith

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Misc

• (Elements - Euclid) + (Euclid and beyond - Hartshorne) Analysis of Euclid's Elements and learning Euclidean geometry.
• Combinatorics - Cameron
Permutations; some number theory, some group theory, some set theory; finite geometry; graph theory.

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Contest math

Contests

Waterloo contests, COMC contest, AMC contest, MIT integration bee, Miklos Schweitzer contest, Putnam contest, APMO contest, IMO contest

Resources

• Evan Chen OTIS Articles Excerpt
• CAN handouts
• IMO more problems

Articles and scholarly

Not exhaustive: SciHub (our holy savior), Arxiv.org, MathSciNet, Zentralblatt, Worldcat, Google scholar, MIT newd, Phys.org, Nature, Science, PNAS, Quanta magazine, MIT libguides, Oxford libguides, Stanford libguides, Princeton libguides, Cambridge libguides, Harvard libguides