Applications of modern algebra

[quantumdude]

They do teach basic linear algebra in high school, but they lump it together with nonlinear algebra, and they just call it "algebra". It's all about solving equations though. No vector spaces or operators.

 

[matt grime]

I'm pretty sure this has been discussed numerous times on this board (and perhaps mathematics boards everywhere?), but what Americans call Linear Algebra and what people in the UK call it seem to differ. Based on the American definition, Linear Algebra is not taught in high schools.

It was not a serious comment. Even matrices are now disappearing from some high school courses in the UK. I just presumed (hoped) that at some point the OP had met vector spaces even if they didn't use that term for them.

 

[mathwonk]

back in the 1960's there was a book in the SMSG high school math series on matrices and linear algebra but i do not know of any schools that adopted it. i myself did teach linear algebra and vector calculus to high schoolers in atlanta one year, from the book by jerrold marsden, or maybe marsden and tromba, also used at berkeley. one of those former high school students is now full professor of math at brown university, with ba from yale, phd from berkeley.

 

[mathwonk]

i heard from my contacts that the high school administration would not allow me to teach it again, because it was over the heads of the mediocre stduents wanting only good grades and AP scores, rather than mastery of advanced material. they were apparently afraid a course too advanced would not be properly appreciated by the college admissions officials, and I did not give all A's. the graduates said my course (which included differential forms in high school, path integration, stokes theorem, complex numbers, and applications to fundamental theorem of algebra) made harvard's math courses survivable. but not everyone wants to go to harvard.

 

[stunner5000pt]

have been away from this board for a while and i have read some of these replies and they are quite useful.
So group adn rings - concepts from modern algebra - are used extensively in physics? Perhaps I'm not there yet...

what about analysis courses?? I have read a few papers (well morel ike glanced over them because the notation and concepts were wayyy beyond me) and they very similar to some proofs i have seen in math papers. I guess this is only the mathematical physics aspect of it? What about say Solid State Physics or ELementary particle physics though?

I wan to take this a course called vector Integral Calculus which basically presents the concepts of Calc 3 in rigorous theoretical detail as well as as intro to math analysis, but i am afriad that i am may not be able to handle it.

 

[matt grime]

1) Elementary Particle Physics.

Well, I'm going to ignore the term 'elementary' since one person's elementary is another person's advanced*, and you can't stay doing elementary all your life, and perhaps the elementary refers to the particle nature... (that's more likely, isn't it?)

Elementary particles 'are' elements of a group. The standard model. Which is something like

SO(3)xSU(2)xU(1)

You can't go very far in parts modern physics without meeting terms like 'symplectic form' or 'monoidal category' or 'n-categories'. See the nice articles of Jon Baez. A conformal string theory, or is it a topological string theory, is a functor from symmetric monoidal category to the category of Seagal's Riemann surfaces, and people like Borcherds are interested in generalizations of Lie Algebras (Vertex Algebras). A- and B-branes and the duality between them is (conjecturally) related to a result about quotients of a vector space by the action of a discrete subgroup of SL(n,C) (McKay Correspondence).

2) I know little about what constitutes solid state physics.

Other aspects of non-mathematical physics that involve mathematics:

Anything involving crystals might at some point benefit from group theory. There are relations between symmetry groups and parts of physics/chemistry (James and Liebeck Representations and Characters of Groups).

Signal processing is an application of group theory and analysis. (Sort of Fourier transforms/series but on discrete data sets, see eg Audrey Terras, Fourier Analysis on Finite Abelian Groups and Applications).

 

[dextercioby]

Just as a note to the post above: The gauge group for the SM of Particles & Interactions is $SU(3)_{C} \times SU(2)_{T} \times U(1)_{Y}$.

Daniel.

 

[stunner5000pt]

1) Elementary Particle Physics.

Well, I'm going to ignore the term 'elementary' since one person's elementary is another person's advanced*, and you can't stay doing elementary all your life, and perhaps the elementary refers to the particle nature... (that's more likely, isn't it?)

Elementary particles 'are' elements of a group. The standard model. Which is something like

SO(3)xSU(2)xU(1)

You can't go very far in parts modern physics without meeting terms like 'symplectic form' or 'monoidal category' or 'n-categories'. See the nice articles of Jon Baez. A conformal string theory, or is it a topological string theory, is a functor from symmetric monoidal category to the category of Seagal's Riemann surfaces, and people like Borcherds are interested in generalizations of Lie Algebras (Vertex Algebras). A- and B-branes and the duality between them is (conjecturally) related to a result about quotients of a vector space by the action of a discrete subgroup of SL(n,C) (McKay Correspondence).

2) I know little about what constitutes solid state physics.

Other aspects of non-mathematical physics that involve mathematics:

Anything involving crystals might at some point benefit from group theory. There are relations between symmetry groups and parts of physics/chemistry (James and Liebeck Representations and Characters of Groups).

Signal processing is an application of group theory and analysis. (Sort of Fourier transforms/series but on discrete data sets, see eg Audrey Terras, Fourier Analysis on Finite Abelian Groups and Applications).

so then it would be helpful to take an introductory course in modern (abstract) algebra... and maybe its follow up course as well?

what about analysis - introductory (i already did in my first year), multivariable calculus analysis(an introduction to manifolds and differential forms), real analysis and complex analysis.

I do have a strong interest in analysis however the way the courses are setup in my university, they often conflict with a physics course.