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Re: [Phys-L] abstract algebra and physics?



On 01/14/2014 07:34 AM, David Ward wrote:

A math/physics double major just asked me about areas of physics
research involving abstract algebra. (He's torn between grad school
in physics vs mathematics, and he enjoys mathematics and mathematical
proof very much).

I am just coming up blank. Do any of you know of an area of research
and/or a physics graduate program that offers an opportunity to
employ and study abstract algebra? Do you have any recommendations?

I cannot imagine a graduate program in physics that does /not/
involve lots of algebra.

As a famous example from history, the invention of quarks and the
prediction of the Ω− particle was an exercise in group theory.
http://en.wikipedia.org/wiki/Eightfold_Way_%28physics%29

In the 20th century, a lot of mathematical research was done on
fiber bundles. Some of it looked like it was going to be highly
abstract, but then the physicists found applications for it. I
am not up to speed on the details, but my best guess is that there
are still open research questions in this area.

The gauge group (if any) of superstring theory is definitely an
open research question.
http://en.wikipedia.org/wiki/Superstring_theory

Crystallography is basically wall-to-wall group theory. The
fairly recent discovery of quasicrystals pushed this in new
directions.

Spectroscopy involves lots of group theory. I suspect this is mostly
a solved problem, but there may still be a research angle to it.

You can't go anywhere near modern statistical mechanics *or*
elementary particle theory without encountering renormalization group
ideas. It is of course a semigroup not a group, but it still counts
as abstract algebra.

The geometric algebra of Clifford and Grassmann has been relevant to
physics for a long time, even since before the invention of vectors,
but it has enjoyed a renaissance recently. I'm reasonably certain
we have not heard the last of this.

/Computer algebra/ is a hot topic. There is no shortage of physics-
related applications.

Liquid crystals. This includes some very abstract mathematics, and
some very applied engineering, and everything in between.

Fractals.

Wavelets.

============

You can find lots more on your own:
http://scholar.google.com/scholar?as_ylo=2013&q=physics+%22algebra%22