zlacker

>>qqqqqu+(OP)
* Lie algebras and Lie groups - I still don't understand this, what they're used for or how to use them in any practical sense.

* Galois Theory - I have a basic understanding of abstract algebra but for some reason Galois theory confounds me, especially as it relates to the inability of radical solutions to fifth and higher degree polynomials

* "State-of-the-art" Quantum Entanglement experiments and their purported success in closing all loopholes

* Babai's proof on graph isomorphism being (almost/effectively) in P - specifically how it might relate to other areas of group actions etc.

* Low density parity checks and other algorithms for reaching the Shannon entropy limit for communication over noisy channels

* Hash functions and their success as one-way(ish)/trapdoor(ish) functions - is SHA-2 believed to be secure because a lot of people threw stuff at the wall to see what stuck or is there a theoretical backpinning that allows people to design these hashes with some degree of certainty that they are irreversible?

>>abetus+zC
I think the best motivation for the theory of Lie groups and Lie algebras is representation theory. Like you don't need anything besides linear algebra to know what SO_n is, but if you want to know about how it can act on a vector space then you need to think about the Lie algebra.

The other great thing about Lie groups is you can discover new and valuable groups just from pretty basic topology. Like the Spin group, which you know has to be out there as soon as you know the fundamental group of SO_n, but otherwise would be very hard to think of.

The fancypants but I think most intuitive way to think about Galois theory is also with topology. It's an algebraic version of a much more geometric, visible story, the correspondence between {subgroups of the fundamental group} and {normal covering spaces}.