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
Channels
MIT, Borcherds,
Numberphile, Mathemaniac,
Faculty,
Schuller
Funny/fun things
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"
Abstract algebra
"Isomorphism"
Topology
"Invariants"
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
-
Number theory
- Borcherds
Mathematical Logic
"Proofs"
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.
Contest math
Contests
Waterloo contests, COMC contest, AMC
contest, MIT integration bee, Miklos
Schweitzer contest, Putnam contest, APMO
contest, IMO
contest
Resources
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