/sci/ - Science & Math

[Anonymous No.4141251]

okay so I'm a math major (algebraic geometry, to be specific) and I'm teaching some physicists this semester in basic mathematics. for cash obviously, because I'm a superficial hedonist. thing is, I know jack shit about physics.
I'd love to motivate some mathematical concepts though, and I wish to hear some good examples. of course there are the obvious ones like differential equations and manifolds, but what about some more algebraic stuff? like, group theory, homological algebra, varieties, commutative algebra.. even some applications of modules and rings would be enough

pic related. singular curve

[Anonymous No.4141256]

> basic mathematics
> group theory, homological algebra, varieties, commutative algebra

basic as in most abstract, or basic as in rudimentary tools like trigonometry and calculus?

[Anonymous No.4141262]

>and I'm teaching some physicists this semester in basic mathematics

for people in which semester?

And how in dept do you want to go? mention stuff, explain stuff, how many minutes/hours do you want to spent on topics?

[Anonymous No.4141265]

Well, here are two examples:

1) Whenever you talk about solid-state, you have to deal with crystals (metals, semiconductors, etc.). Their symmetry groups substantially restrict their internal structure, their response to x-ray diffraction and the like, their electronic properties, and a myriad of other attributes.

2) Quantum mechanics is completely built around Hilbert spaces, an extension of finite-dimensional complex vector spaces.

4th year engineering physics student ftw :)

[Anonymous No.4141268]

well basic as in first semester at university. they already know about dual spaces and implicit functions for instance, one semester ahead they'll be studying differential forms and smooth manifolds.
Okay even so, the topics I listed would be a bit far stretched. Let's stick to groups, rings and modules at the most fundamental level, is there any application for stuff like this?

[Anonymous No.4141270]

I forgot: general relativity. That's all about differential geometry and tensor algebra.

[Anonymous No.4141276]

>>4141251
First/second year physics students = integrate all day erry day. that and statistics imo are the best things to really get a handle on early on.

[Anonymous No.4141278]

>>4141268
Hm... that's more difficult. The basic problem is that math is used as a rigorous foundation for physics, but most physicists don't need to worry about the rigorous details so much. For example, you don't need to take real analysis to be able to take derivatives or have an intuitive understanding of derivatives, which is more or less fine unless you want to get into cutting edge theory...

[Anonymous No.4141283]

>>4141265

that sounds great, I'll read up on this. thanks

>>4141262
I'd only mention them briefly and state some facts. as in, "you'll see this reappear in a couple of months once you learn xyz".. The problem is, I keep getting asked why they have to learn about normal subroups, symmetric groups and thinks like that, and I can never give any good answer

[Anonymous No.4141290]

>>4141283
LOL, well, if you ever have ANYTHING to do with materials or chemistry, symmetry groups are gonna be useful :P

[Anonymous No.4141294]

>>4141276

that's what I suspected.. okay then I'll give some more background on those topics.
are mathematical pathologies relevant in any way? I mean in pure math lot of time is spent on stuff like space-filling curves, the weierstrass curve/related concepts, non measurable sets etc., is this negligible for physicists?

[Anonymous No.4141298]

>>4141262

by the way, who's that chick?

[Anonymous No.4141311]

>>4141294
Yea... not really. The problem is you really only get into the rigorous side of math in physics when you start trying to come up with rigorous theories that don't have all kinds of mathematical pathologies and so on :) I mean, if you have any aspiring theoretical physicists they definitely need to know the pure math side of things, but otherwise it's not really necessary.

In other words, mathematical pathologies aren't really too important in physics, simply because we deal with what's "real" - i.e. systems always tend to be well behaved, that sort of thing.

That being said, there's a lot of applications for various types of math other than basic algebra (Hilbert spaces for quantum mechanics, differential geometry for general relativity, symmetry groups for pretty much anything to do with materials + crystals, and statistics being just a few examples)

[Anonymous No.4141314]

>>4141311
Sorry, i meant "other than BASIC CALCULUS" (not basic algebra)

[Anonymous No.4141316]

>>4141283
okay, well as another person pointed out, the small finite groups find very much application in solid state physics.
You might start in this corner
http://en.wikipedia.org/wiki/Bravais_lattice

Groups, especially Lie Groups too, are essential in physicss.

As you said, there are billions of ways to motivate differential equations. You find a physical application for mery very many things in differential geometriy..

There are higher level motivations for the fundamental group and other topological stuff, but if you just mention things, then there are also some easier ones. One thing that has big concequences is the existence of universal covering groups. The example SU(2) containing SO(3) is especially important, since the additional phases get factored out in some hilbert spaces modeling QM, and therefore introduce spinors and, eventually, explain the stability of matter. I can go into more detail.

An example I would mention is deRham Cohomology, which finds a major application in Electrodynamics. The factor group (Cohomology group) physically factors out all possible gauges.
http://en.wikipedia.org/wiki/Gauge_theory
en.wikipedia.org/wiki/Maxwell_equations#Formulation_in_terms_of_differential_forms
(there are probably better links)
Cohomology plays also a role in supersymmetry theory.

>space filling curves
there are some cool concepts in lagrangian mechanics, maybe I write down the example, which motivate some math.

also, that's emma stone.

[Anonymous No.4141325]

>>4141316
Space filling curves in Lagrangian mechanics? Really? I never saw anything like that in my Lagrangian mechanics courses... How are they used?

[Anonymous No.4141346]

>>4141316

wow you seem to be quite competent, thanks so far.

I'll keep those things in mind, especially finite groups, de Rham cohomology, universal covering groups and lie groups. I'm pretty much an illiterate when it comes to real analysis - I only know stuff up to lebesgue integration, gauss/stokes - so diff geom and diff equations are not really my speciality.
complex analysis is another story, though

you don't have to completely write out the example, any sort of reference would be enough.

I'm just collecting informations. maybe I'll continue to teach them for a couple of semesters, it might be great to have some stuff in my back hand

[Anonymous No.4141357]

>>4141325

The 3 axioms of thermodynamics can be deduced from the fundamental postulate of statistical mechanics, which says that the possibility for all states you have no access to is equal (namely one over the phase space volume).

http://en.wikipedia.org/wiki/Statistical_mechanics#Fundamental_postulate

The Hamilton equations

http://en.wikipedia.org/wiki/Hamilton_equations

are first order in t, i.e. the diff.equ. has a unique solution, i.e. the curve never finds back to one place.

This is why the Liouville theorem is a justification for the fundamental postulate (I see they even mention it in the fund. postulate paragraph)

http://en.wikipedia.org/wiki/Liouville%27s_theorem_%28Hamiltonian%29

the whole thing is debated, however, because they involve unjustifyable/unphysical time scaler. (there are harmonic orscilator examples with periodicities which take 10^23 times the age of the universe etc., these are bad arguments). also, the Liouville theorem here is classical

[Anonymous No.4141384]

References? Well, I can only says books and scripts I guess. "Geometry, Topology and Physics" by Nakahara does 80 pages of physical motivation for 400 pages of math topics that follow. It's for physicists between semester 5 and 10 I'd say.

[Anonymous No.4141395]

Quaternions, bro.

Maxwell wrote his equations in them, then they were "revised" into vectors after his death.

[Anonymous No.4141425]

Okay, here is a (I think) really cool physical example for quite complex geometries:

Consider a child with a roud sled on a huge huge snow covered field, like this

http://www.youtube.com/watch?v=uYdtSBywHpg

What is the configuration space? The space of all states the kid can be in. Well the kid can be anywhere on the field, i.e. there are "R^2 places", and it can look in any of 360°, i.e. "2\pi directions". Therefore, the phase space is

R x R x S

where S is the 1-Sphere. Obviously all the points in this 3 dimensional space (where one dimension is compact/periodic) correspond to a position the kid can be in, and these are also already all possible configurations.
Okay, the space is 3-dimensional, but what about the tangent space?
The kid can only drive forward, backward maybe, and rotate. It cannot move sidewards. This restriction means that the tangent spece for the kid is 2-dimensional in every point.

Moving forward or backwards means moving forwards or backwards in R^2, and rotating means moving upwards or downwards along S. (Notice that rotation doesn't change the position in the real world, but it does in the configuration space)

Now physically it's obvious that the kid can get everywhere and look in any direction. Therefore you can foilate your R^2 x S with this restricted tangent spaces. Try to imagine what the kid has to do to get to a tangent space in configuration space right above/parallel to one it's in. It's pretty funny what you can do in this space.