Nov 29, 2024
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In my representation theory course this term we're moving onto representations of Lie Groups and I was wondering is there are results about how the homotopy groups/(co)homology groups of a Lie group affect what representations can exist of that Lie group?
[5/6/24] wow im back so soon? yes, it is to complain about matrix groups again. I may have girlbossed too far w my paper here but its too late to change. NEVER BACK DOWN NEVER WHATTTT (give up on my dreams and lie down)
I am GOING to finish this proof whether it kills me (I could never be a mathematician)
my friend latex’d a joke I made
Representation theory is the study of how symmetries occur in nature. Here, of course, "symmetries" means "groups", "occur in" means "act on" and "nature" means "finite dimensional vector spaces"
Rep Theory lecturer
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Eli Stein was the first to appreciate the interplay among representation theory, classical Fourier analysis, and partial differential equations, and to perceive the fundamental insights in each field arising from that interplay. How did Stein’s scope and originality contribute to numerical analysis?
Image: ‘E8 Petrie projection’ by Jgmoxness. CC BY-SA 3.0 via Wikimedia Commons.
Plato and Kepler were in the right ball-park, but not really right. Both the solar system and atoms are described pretty well by similar laws—the inverse-square force laws for gravity and electrostatics. And solving this problem (in either the classical or quantum case) does indeed require a deep understanding of rotations in 3-dimensional space. …[T]he Platonic solids have as their symmetries finite subgroups of the rotation group in 3 dimensions…. [T]he real reason the inverse square force law problem is exactly solvable is a surprising symmetry in four-dimensional rotations.
John Baez
citing
Victor Guillemin and Shlomo Sternberg, Variations on a Theme by Kepler, American Mathematical Society. Providence, Rhode Island, 1990.