Exploring the Importance of Symmetry in Group Theory

[rudinreader]

I'm not an expert in Abstract Algebra, I am mainly an analyst. Is there anyone versed in Group Theory that can kind of discuss the theory and it's ramifications?

 

[rudinreader]

ROFLMAO3! I said relavent. I meant relevant...

 

[quasar987]

I'm certainly no expert either but the way I understand it, it's one of the main branch of mathematics, where by that I mean that it pops up everywhere. ODE, PDE, geometry, you name it. Maybe not group theory itself but a closely related ramification such as modules or whatnot.

For instance, I'm reading the PhD thesis of a student at my uni at the moment and he introduces a homological method to prove new existence and multiplicity theorems in critical point theory.

As always, maybe if you asked specific questions, you'd be more likely too get an answer. I think matt grime is an algebraist but I haven't seen him in a while.

 

[zhentil]

Functional analysis is basically a cross between analysis and algebra. In complex analysis, you study automorphisms of the complex plane. Anything that requires Hilbert space methods (e.g., in PDE, harmonic analysis, etc.) requires algebra. The modern approach to differential geometry is all algebra. How far are you in your career as an analyst that you haven't seen algebra pop up anywhere?

 

[John Creighto]

I know though that group theory is needed to prove certain theorems with regards to lattices in solid state physics. I know it is also quite important for symbolic root finding which is very relevant in computer algebra systems.

 

[morphism]

Group theory also has many applications to analysis, e.g. harmonic analysis, K-theory, etc.

 

[quasar987]

You consider K-theory to be analysis? Explain please.

 

[morphism]

I didn't mean K-theory was analysis, but it certainly can be applied to analysis (e.g. using K_0 and K_1 to study C*-algebras).

 

[quasar987]

Ok, I didn't know!

 

[mathwonk]

is symmetry important? answer: yes.