News
My basil finally developed roots! I'm currently reading about quantum mechanics, ferrofluids, and language models.
Blog
[08/09/2024:] Motivating Ladder Operators II
[08/08/2024:] A Cool Way to Garden
[08/08/2024:] Life and Basil Limeade
[08/05/2024:] Motivating Ladder Operators
[08/04/2024:] Migrated to AstroJS
[07/31/2024:] The Classical in the Quantum
Notes
Working on notes on the quantum mechanics, derivatives, and uploading my previous course notes onto this blog!
Projects
Finally started a projects page! I've recently made some nice upgrades to my post component, so it looks pretty clean! ;)
π
I'm considering whether or not to continue this project using WebGL or Three.js.
I'm also researching methods for generating the 3D scenes I want for this project automatically.
In the meantime, I've decided to proceed with some preliminary prototypes of the other interactive parts of this project.
Orange Juice
I like orange juice. :)
Mlog
Homotopy Groups
September 24, 2024
By Aathreya Kadambi
Expanding on Lectures by Professor Alexander Givental and Fomenko and Fuchs
. Note that is the number of path-connected components of .
He then explained the pictures from Fig 37 and Fig 38 in the book and basically section 8.1.
Depednence on Base Points
If we have another base point and a path from to , we get a map .
We introduced many algebraic invariants but we canβt compute many of them unless the space is trivial, so letβs look at a covering . In the theory of coverings we showed that mappings are injective. This is because if a loop is contractible in , we can use the homotopy lifting lemma to lift this homotopy up to , and then because the fiber is discrete, we get the same point up there so that the other loop upstairs is also contractible. From this, we get that is injective for . It turns out for , it is also surjective.
Relative Homotopy Groups
He discussed some stuff from this section.
The Reason This Was introduced
This isnβt necessarily helpful for computations or anything, but it letβs us make the long exact homotopy sequence.
He wrote something about and .