“The thing to keep in mind is that all these “basics” were once cutting-edge research. And the people who got to the bottom of these matters were not trivial idiots. They were great masters.

The more you look at this material in this way, the more you’ll find you can tolerate it in good humor, and without fearing its deadening effects on your strange, wild mind.”

— Laurens Gunnarsen


Math Girls

Introduction to Real Analysis(the Lebesgue integral — warning: hilarious)
An Illustrated Handbook of Theoretical Physics: Introduction to Mathematical Structures
All of the Mathematics You Missed (And Need To Know For Graduate School) (not free)
Algebra: Chapter 0 (I highly recommend doing the exercises, it goes from set theory to basic homological algebra)
The Algebra II Lectures of the Poetic Kazuya Kato

Pictorial basic topology via physics and engineering examples
Topology for Physicists
Kindergarten Quantum Mechanics

Rational Points on Elliptic Curves (not free)
An Invitation to Algebraic Geometry (not free)

A great resource for learning Haskell for experienced programmers: Haskell Fast & Hard.
Category Theory & Programming (includes kitten pictures and bad puns)
Functional Programming with Bananas, Lenses, Envelopes and Barbed Wire
When is one thing equal to another thing?

Hitchhiker’s Guide to Math (In Progress)

the nLab (note: many terms used on this site are not well known to the general mathematical community)

Medium spice

A User’s Guide to Spectral Sequences (a nice reference, not great for first time) (not free)
Mosher and Tangora
Informal Introduction to Topos Theory

(Differential) Geometry Primer
Mathematical Resources of Jinhyun Park 
Geometric Langlands Seminar

Sundry Topology

Forms of K-theory
Stable homotopy groups of spheres

The Brain:

FAQ: Atoms of Neural Computation
Towards a Thermodynamic Theory of Nerve Pulse Propogation (Soliton model in neuroscience)
Physical Principles for Scalable Neural Recording

Comparison and Enumeration of Chemical Graphs
An Updated Account of the Integrated Information Theory of Consciousness


Line Segment Detection (Works without parameter tuning!)
Mechanical Engineer’s Handbook
Calculations for Structures Under Mechanical Load