mathphysics

I will try to collect my notes and solutions on math and physics, and links to them here.

Open-source; PayPal only

From the beginning of 2016, I decided to cease all explicit crowdfunding for any of my materials on physics, math. I failed to raise any funds from previous crowdfunding efforts. I decided that if I was going to live in abundance, I must lose a scarcity attitude. I am committed to keeping all of my material open-sourced. I give all my stuff for free.

In the beginning of 2017, I received a very generous donation from a reader from Norway who found these notes useful, through PayPal. If you find these notes useful, feel free to donate directly and easily through PayPal, which won’t go through a 3rd. party such as indiegogo, kickstarter, patreon.

Otherwise, under the open-source MIT license, feel free to copy, edit, paste, make your own versions, share, use as you wish.

Algebraic Geometry

(symbolic computational) Algebraic Geometry with Sage Math on a jupyter notebook

cf.

https://github.com/ernestyalumni/mathphysics/blob/master/AG_sage.ipynb

http://nbviewer.jupyter.org/github/ernestyalumni/mathphysics/blob/master/AG_sage.ipynb

I did a Google search for “Sage Math groebner” and I came across Martin Albrecht’s slides on “Groebner Bases” (22 October 2013).  I implemented fully on Sage Math all the topics on the slides up to the F4 algorithm.  In particular, I implemented in Sage Math/Python the generalized division algorithm, and Buchberger’s Algorithm with and without the first criterion (I did plenty of Google searches and couldn’t find someone who had a working implementation on Sage Math/Python).  Another bonus is the interactivity of having it on a jupyter notebook.  If this jupyter notebook helps yourself (reader), students/colleagues, that’d be good, as I quickly picked up the basic and foundations of using computational algebraic geometry quickly (over the weekend) from looking at the slides and working it out running Sage Math on a jupyter notebook.

I’ll update the github file as much as I can as I’m going through Cox, Little, O’Shea (2015), Ideals, Varieties, and Algorithms, and implementing what I need from there.

Algebraic Geometry and Algebraic Topology dump (AGDT_dump.tex and DGDT_dump.pdf)

20171002 – I’ve consolidated by notes on Algebraic Geometry and Algebraic Topology.  Because central extensions of groups, Lie group, Lie algebras play an important role in Conformal Field Theory, I include notes on Conformal Field Theory (CFT) in these notes.

Of note, I compare 2 definitions of semi-direct product and show how they’re related and the same.

Differential Geometry and Differential Topology dump (DGDT_dump.tex and DGDT_dump.pdf)

I continue to take notes on differential geometry and differential topology and its relation to physics, with an emphasis on topological quantum field theory.  I dump all my note and thoughts immediately in the LaTeX and compiled pdf file here and here.  I don’t try to polish or organize these notes in any way, as I am learning at my own pace.  I’ve put this out there, with a permanent home on github, to invite any one to copy, edit, reorganize, and use these notes in anyway they’d like (the power of crowdsourcing).

Manifolds

20171203 update

I added notes on immersions, submersions, and some immersed submanifolds.  Absil, Mahony, and Sepulchre (2008)’s book Optimization algorithms on Matrix Manifolds had clearer, more concise, and more to the point definitions in its manifolds review than many of these classic pure math textbooks (Lees).

I finally understood clearly immersions, submersions, and their differences after writing (and drawing) this down:

Holonomy

20170423 update.

I have been reviewing holonomy by reading Conlon (2008), Clarke and Santoro (2012, 1206.3170 [math.DG]), and Schreiber and Waldorf (2007, 0705.0452 [math.DG]) concurrently.  I’ve already put these notes on my github repository mathphysics , in DGDT_dump.tex and DGDT_dump.pdf.