Notes on Topological K-Theory

Hello All! These are some notes on Topological K-Theory that I wrote by combining stuff from various sources on the internet and some books. Sorry I won't be explaining anything here - but I hope this is useful! :) 

Stuff I used: Some stuff from John Baez and Complex topological K-theory by Efton Park. I'm not an expert on this stuff, but its useful for classifying topological insulators and superconductors. There's a paper by Kitaev on this - and my motivation for studying K-theory was to understand this paper. I hope these notes are useful for you if you wanna get a basic idea of K-theory (whether from the Physics or the Math perspective) I apologise to Theoretical Computer Scientists, I don't know how to explain this nicely - this stuff requires a bit of thinking! 

Note: The Black Board Images are the notes simplified. 

Getting SageMath to work on Jupyter (again)

I recently reinstalled the MacOS on my Macintosh when my system was taking up way too much space. It was taking up about 50 Gigs and that too when I literally had nothing. So I decided to refresh the whole thing. On the downside, I had to go through the painful process of reinstalling everything from Skype to Anaconda etc...

The worst of all was SageMath. I'd forgotten how I got Sage to work the first time but now, I found a new way. First of all, you need to use python 2.7 for sage to work. This isn't an issue if you have Anaconda - just create a virtual environment in the environments tab in Anaconda Navigator. Now, make sure you've got your python 2.7 environment selected before you proceed further - once you've downloaded Sage from SageMath and transferred it to the applications section of your Mac, you need to do three things:

1) Run the following command:
sudo jupyter kernelspec install /Applications/SageMath/local/share/jupyter/kernels/sagemath
where /Applications/SageMath is the directory in which SageMath is saved (at least on Macs)

2) You need to run Sage from the terminal file in the SageMath folder

3) Once you've opened the sage command line prompt, just type the following command:
!sage -n jupyter

After that, Sage should start as a Kernel in your Jupyter notebook. Python2.7 support runs out in 2020 (Jan 1'st actually) so hopefully Sage transfers to Python 3 use by then...

Also, will try to post more frequently on this blog from now on - as a parting note, some of my old diagrams have been lost due to the fact that Blogger was connected with my Google+ account which has been deleted since Google+ ended. So sorry about that - I'll try to post Latex notes for stuff when I have time (and in general from now on). Cheers folks!