Some more References

I found a number of free references on various subjects in mathematical physics, available free, legally, online.

Mathematics

Partial Differential Equations

1. Alexander Komech, and Andrew Komech,
   "Book of Practical PDEs."
   Eprint Lecture note 33/2007, 125 pages.
2. Alexander Komech,
   "Lectures on elliptic partial differential equations (Pseudodifferential operator approach)".
   Eprint Lecture note 32/2007, 44 pages.
3. Norbert Ortner, and Peter Wagner,
   "Distribution-Valued Analytic Functions - Theory and Applications".
   Eprint Lecture note 37/2008, 133 pages.
4. A.D.R. Choudary, Saima Parveen, Constantin Varsan,
   "Partial Differential Equations An Introduction".
   Eprint arXiv:1004.2134v1 [math.AP], 204 pages.
5. Robert Geroch,
   "Partial Differential Equations of Physics".
   Eprint arXiv:gr-qc/9602055v1, 57 pages.

Differential Geometry

1. Jürgen Jost,
   "The principles and concepts of geometric analysis".
   Eprint Lecture note 12/2001, 15 pages.
   Note: "geometric analysis" studies things like Morse functions, etc.

Homological Algebra

1. Volker Runde,
   "Abstract harmonic analysis, homological algebra, and operator spaces".
   Eprint arXiv:math/0206041v6 [math.FA], 12 pages.
2. Mohamed Barakat, Markus Lange-Hegermann,
   "An Axiomatic Setup for Algorithmic Homological Algebra and an Alternative Approach to Localization".
   Eprint arXiv:1003.1943v3 [math.AC], 30 pages.
3. Joseph Krasil'shchik, Alexander Verbovetsky,
   "Homological Methods in Equations of Mathematical Physics".
   Eprint arXiv:math/9808130v2 [math.DG], 150 pages.
4. R. P. Thomas,
   "Derived categories for the working mathematician".
   Eprint arXiv:math/0001045v2 [math.AG], 13 pages.
5. Semen Podkorytov,
   "On homology of map spaces".
   Eprint arXiv:1102.1645v1 [math.AT], 9 pages.

K-Theory

1. Max Karoubi,
   "K-theory. An elementary introduction".
   Eprint arXiv:math/0602082v1 [math.KT], 22 pages
2. Ioannis P. Zois,
   "18 Lectures on K-Theory".
   Eprint arXiv:1008.1346v1 [math.KT], 137 pages.

Mathematical Physics

This is kind of physics stuff, kind of mathematics stuff, and doesn't really fit cleanly into either category.

1. Andreas Knauf,
   "Number theory, dynamical systems and statistical mechanics".
   Eprint: Lecture note 3/1998, 41 pages.
2. Bruce Hunt,
   "Geometry of super Yang-Mills and supergravity".
   Eprint: Lecture note 4/1999, 59 pages.
3. Bruce Hunt,
   "Conformal and gauge symmetry in D = 2 QFT".
   Eprint: Lecture note 5/1999, 115 pages.
4. Friedemann Brandt,
   "Lectures on Supergravity".
   Eprint: Lecture note 13/2002, 50 pages.
5. Alexander Komech,
   "Lectures on Quantum Mechanics (nonlinear PDEs point of view)".
   Eprint: Lecture note 25/2005, 231 pages.

Physics

1. Christopher J. Fewster,
   "Lectures on quantum field theory in curved spacetime".
   Eprint Lecture note 39/2008, 62 pages.

Quantifiers

I have been thinking about quantifiers, and the notation around them...since most books use differing notation that is not easy to read.

Since I am trying to be consistent throughout all of mathematics, it seems natural to suggest that the colon ":" should be read as "such that".

In this case, one should write "∃x : P(x)" since one usually writes in natural language "There is some x such that P(x)".

Likewise we often find expressions "If x∈X, then P(x)"...so it would be natural to use the notation "∀x∈X, P(x)" or "∀ x, x∈X and P(x)". After all, "x∈ X" is really a predicate "isIn(x,X)"...

I'm currently revising my notes on logic in Fascicles 0 of my Elements of Mathematics, which is why I bring this notational problem up! Hopefully, I will be done with revising and improving my logic chapter soon.