@MikeMiller My comment was more of a rant. I see lots of really good teachers who care but their broad rewards are miniscule compared with the value added to society.
unless I'm misreading his question, why do you even need to go with that, Ted?
@AttractorNotStrangeAtAll it's not continuous on the irrationals. but you can extend Riemann integrability to functions whose set of discontinuities has measure $0$.
I've shown that $f(x)=\sin{\frac{1}{x}}$ is not uniformly continuous on $(0,+\infty)$, but I can't show that $f$ is uniformly continuous (maybe even Lipschitz?) on $[a,+\infty)$ for $a>0$. Any hints?
Say I have 2 equal sets of colored points, and I want to pair them up while minimizing the number of unique edges, where 2 edges are considered equal if they connect the same colors. Is this a hard problem?
You know how some general values of $\pi(n)$ are easy to calculate, such as $\pi(p_{\pi(n)})$ just equals $n$. Well, I present (another in a long list of formulas), a setwise recursive formula for the first few prime numbers with only an elementwise product needed on sets. This took a few hours...
When we define 'Affine n-space over $k$' where $k$ is a field, do we assume $k$ be algebraically closed? Book says we allow any field but professor said assume $k$ be algebrically closed
Algebraic geometry with varieties works best for algebraically closed fields. Keeping track of things when the field is not algebraically closed without the language of schemes is usually not worth it
If I could ask a question quick about showing a basis change on a Lie algebra, if we have the bracket given by: $$[T_a,T_b]=f^c_{ab}T_c,$$ and we make a change of basis with some matrix on the left we get (supressing summations on repeated indices): $$A_{ka}A_{hb}[T_a,T_b]=A_{ka}A_{hb}f^c_{ab}T_c$$ however the notes that I'm following then insert the inverse of $A$ to get: $$A_{ka}A_{hb}[T_a,T_b]=A_{ka}A_{hb}f^e_{ab}A^{-1}_{ec}T_c$$ why do we use the inverse here?
@Thorgott the elementary functions with riemann integral 0. Unless I misread the question, we don't need anything fancy for this, any point-symmetric function about (a+b)/2 suffices, e.g. x - (a+b)/2
So with the full power of elementary functions, we obtain that any linear combination of odd-order monomials, sines, cosines with an appropriately scaled cutoff etc. satisfies this
(since the integral of the cosine from -pi to pi is zero)
which is not point symmetric, so that we can also have nonzero values at (a+b)/2
@AttractorNotStrangeAtAll This is why Thorgott mentioned composites of Lipschitz functions. That those are Lipschitz does follow immediately from the definition.
And then you just check that sin x is Lipschitz on R and 1/x is Lipschitz in [a, inf)
@MikeMiller TOP = DIFF in dimension 2 philosophically because of Schoenflies. What's the fundamental reason that TOP = DIFF in dimension 3?
I'm flubbing, TOP = Locally flat is Schoenflies, but you can believe Locally flat = PL = DIFF or whatever
TOP = PL in dim 2 should be Schoenflies because you can always triangulate a chart, transition functions to still get well-defined Jordan arc bounding Jordan domains on a part of another chart, extend this triangulation, etc
I would say the key point in turning that into a triangulation is that embedded discs can't intersect too badly; it's easy in the Topological category to reduce to countably many intersection points and deal with that
In higher dimensions the intersections will look probably like countable sets of manifolds which accumulate somewhere or another
I can believe this is manageable in 3D with more work (lots of circles) bit that past that topology of manifolds is too complicated
Oh no it was a final exam. It wasn't too hard, you wouldn't be excited by it. The problem I used LCF on said curvature of projection to osculating plane is the same as curvature of the original curve, at the point.
Ah. That's one of my exercises. Any way you do it, the only subtlety is that when you project you lose arclength parametrization, but at the point in question you're OK.
I had plenty of colleagues who would get waylaid going off on tangents or bogging down for unprepared students. I tried to keep in mind where I wanted to get and cut down on things in the middle if I needed to.
Well, some of our algebraists complained bitterly when I taught algebra (and then — gadzooks — wrote the textbook for the course), but algebraists tend to have little feel for things they don't use all the time.
I remember a well-regarded algebraist taught the Honors multivariable calculus when I was a grad student at Berkeley. He used Spivak's little book. It was a horrid course. The kids in the course came to me every day to find out what was actually going on.
@Astyx, even over $\Bbb C$ sometimes you play with things that are not smooth manifolds. When there are singularities you need more structure (and recall that the "variety" given by $x^2=0$ is different from the "variety" given by $x=0$; the difference is precisely scheme structure).
Oh, good; my hat is showing up.
@Balarka: Well, lucky for you, you already know Gauss-Bonnet and you can always work more good exercises in my book :D
@Balarka: I still remember years and years ago one of my students coming into my office, drawing a helicoid at the top of the board, and explaining to other students what the tangent plane was doing moving along a ruling. Of course, he got a Ph.D. in geometric PDE later. :P
I also try to train students (and I've done so here, too) that if they have $f\colon X\to Y$, they should use $x$ for an element of $X$, not for an element of $Y$, etc.
I didn't pay attention to his differential topology rant in the course so I wasn't sure if he actually showed the Mobius strip does not admit an oriented atlas.
I only realized recently that the correct basic understanding of orientation of topological manifolds is still in terms of local orientation, so that you can make sense of w_1-homomorphism.
I thought it was funny that our friend considers representation theory to be rigid enough that group actions are not representation theory, but not so rigid that group actions on linear 2-categories are still representation theory
Basic representation theory is too mind-numbingly concrete. I quit auditing the course this semester after the guy started listing down characters of $\text{SL}_2(\Bbb F_p)$
Like Balarka's diff geo teacher, I presented a representation-theoretic proof of the Kähler identities when I taught complex manifolds, just to "learn" it. It was actually sorta cool.
it's a general fact that characters of S_n have integral values and it's nice to observe explicitly how this fails explicitly in A_n precisely on those S_n-conjugacy classes that split into two A_n-conjugacy classes
it made sense in the context of the talk, which was about Burnside's theorem
one can show that groups whose order has at most 3 prime factors are solvable by induction and showing they contain a normal Sylow subgroup, the remark says this method doesn't generalize
the crazy part is that if $\chi$ is the character of an irrep $\rho$ and $g$ a group element such that the conjugacy class of $g$ has size coprime to the degree of the rep, then either $\chi$ vanishes on $g$ or $\rho(g)$ is a multiple of the identity
I've internalized the proof of this fact to the point where it's absolutely clear why it's true, yet it's still magic
@Thorgott I don't remember any details, but intuitively $\rho$ is one of the matrix algebras in the AW decomposition of $\Bbb CG$, and the conjugacy class gives rise to an irrep hence another one of the matrix algebras in the AW decomposition (by the bijection b/w conjugacy classes and irreps).
If they are different guys, $\chi$ should vanish on $g$. If they are the same guys, $\rho(g)$ should be a multiple of the identity by Schur.
Why doesn't this "work", in the sense of, why can't you make this into a proof?
@BalarkaSen somehow, I have way more problems with these kinds of situations in algebra than in analysis
And a particularly weird thing was the yoneda lemma, which in summer I could just reprove if I wanted to (which is ofc not that great of a feat, since it's only one trick between the detail murkiness), but still had no intuition for what so ever
Where in contrast I can often confidently use results from analysis that I definitely can't reprove
You know how some general values of $\pi(n)$ are easy to calculate, such as $\pi(p_{\pi(n)})$ just equals $n$. Well, I present (another in a long list of formulas), a setwise recursive formula for the first few prime numbers with only an elementwise product needed on sets. This took a few hours...
