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
Probably the best place to get started with Lie groups is to wrap your head around SO(n), which has both a nice geometric interpretation as rotations of n-dimensional space as well as a concrete representation as the space of orthogonal (determinant 1) nxn matrices. With a little matrix calculus you can work out what the tangent directions are at the identity matrix: they’re precisely the skew-symmetric matrices. This is the Lie algebra so(n). Where the Lie group consists of rotations, the Lie algebra consists of directions in which you can rotate something, or really velocities of rotation. This is why classical angular momentum is actually an element of a Lie algebra.