My topic will be on Lie groups.
Lie groups are groups that are also smooth manifolds such that the smooth structure is compatible with the group operation, that is where the group operation and inversion are smooth maps. If one has a Lie group, then they can construct a Lie algebra from the group.
I chose this topic because I thought it would be interesting and because I have decent experience with algebra. We are considering a group that has some additional structure.
The topic is interesting because of the way that it combines topology with abstract algebra. I know that the topic is very useful in physics although I do not know how (I do know that if you are going to do particle physics, that you take Lie algebra, but I don’t know about this either).
hduan2
February 20, 2018 — 08:51
I am also interested in such topic. Lie group is the most important group in physics, such as SU(2)and SO(3) for Spin, SU(3) for strong interaction, and people also try to use higher symmetry like SU(4) looking for weird phenomena. To study the connection between topology and lie group would help us to have a better understanding of symmetry in physics.
I wanted to do some work about spin structure. Due to my algebra foundation is not very solid, I have to change the topic. Looking forward your final project!
gbarkle
February 20, 2018 — 02:57
Interesting topic! From a physics standpoint, I know that angular momentum acts as a generator of rotations in quantum mechanics, which comes from the fact that the angular momentum operators form a Lie algebra which is derived from the Lie group of rotations in 3-space. This can be generalized, so that every symmetry that can be parameterized (which means it forms a manifold!) has a corresponding operator. I would be very interested in seeing more how topology plays in with this subject.