the first time I saw the tensor product was in linear algebra but the prof just copied the construction for modules and then talked about kronecker products for a day so I didn't really see the point
@TedShifrin it'd be interesting to compare where those concepts emerge in the physicist line of math (supposing a lack of an explicit diff-geo course, to be sure)
How could I simplify $H_d(d,k)/k!$ if $$H_d(d,k) = (k-1)!\ \sum_{j=0}^{k-1} \frac{(-1)^j}{(k-1-j)!} \sum_d H_d(n/d, k-1-j)$$ I stupidly tried to replace $(k-1)!$ with $1/k$ but it's clearly not the way to do it...
you do see tensors in places when you need to represent anisotropic responses---say, an electric field giving rise to an electric current which isn't parallel with the field
well, obviously, for $m = n = 1$, the map $k \otimes_k k \to k$ is $x \otimes y \mapsto xy$. the generalization is $k^m \otimes k^n \to k^{mn}$ by $(x_1, \cdots, x_n) \otimes (y_1, \cdots, y_m) \mapsto (x_1y_1, \cdots, x_ny_m)$
seems like it should be an isomorphism. shouldn't be hard to verify. let me see
(Recall, @Balarka, I also said you should try to prove it as many ways as possible. :) But your 30 minutes are up and it's past your bedtime. You need to be awake to see mr. prof.
@TedShifrin Hey man do you have any knowledge on how I can start this problem? Let n and k be integers such that n > k ≥ 0. Show that nCk + nCk + 1 = n + 1Ck + 1? I'm completely lost and I need somewhere to start.
but sources can also define it using a formula involving factorials, or as coefficients appearing in expansions of binomials raised to powers (a la binomial theorem)
@Semiclassical: the only place i know it showing up in physics is seiberg-witten theory, where you need a clifford multiplication to define the equations
but that probably speaks more to my taste and ignoramosity than it does to the applications of clifford algebras
@TedShifrin: 1/2 of the SW eqns is $D^A\varphi = 0$, where $D^A$ is the Dirac operator assoc'd to a connection $A$ and $\varphi$ is a section of the bundle in question. the correct statement there is that you need a Clifford multiplication to define $D^A$, and of course L/M are big fans of Dirac operators
they show up in the study of topological insulators in physics, with kitaev providing a classification in terms of K-theory (real and complex). (i don't know that story so well, since there are other ways to formulate that same classification)
just another real vs complex thing... I think the real reason it's introduced is because you can't actually put a spin structure on every 4-manifold, but they all have spin^c structures
@robjohn Hi. Are you familiar with epsilon-delta proofs? Let's say I want to prove $\displaystyle\lim_{x\to 0}e^x=1$. We have that if $|x|<\delta$ then $|e^x-1|<\varepsilon$. Manipulating this last inequality I got $\ln(-\varepsilon+1)<x<\ln(\varepsilon+1)$ but I'm stuck there.
@KhallilBenyattou find some keywords and branch out from there. hyperbolic trig functions, riemann zeta function, L-functions, modular forms, weierstrass p functions, elliptic functions, hypergeometric functions, etc.
(many of those concepts are probably beyond your current level, but the basics in others might be approachable)
