I also know that a function is Riemann integrable iff the set of discontinuities is Lebesgue measure 0. And I can kinda see how you'd prove it, just partition around it and the function is continuous everywhere else
And lol @Mike I'm pretty accustomed to invoking OP machinery. Really early last quarter Soug got sidetracked and proved that if a function's derivative was in $L^p$, the function itself was Hölder-continuous with exponent $\frac{1}{p}$. I used that to prove that a $C^1$ function was Lipschitz in a compact set
We started with some stuff on hyperplanes and functionals, then spent some time on the dual of $\ell^p$, the big 3 theorems, Hilbert spaces, spectra, and spectral theorem for compact operators
The first half of the quarter was just random stuff
Polar decomposition and spectral theorem in finite dimensions, FTA, differential forms in the plane, ODEs, and a bit on submanifolds of $\mathbb{R}^n$
Hey folks, does anyone have a special method of writing fraktur letters by hand? Every way I tried to write down e.g. the lie algebra \mathfrak{g} to a lie Group G was ugly and/or really slow.
I think german mathematicians never write fraktur in that script on the board. They use a different script whose name I always forget. I usually stylize Lie algebras as a lowercase g, where the end of the g curls back through the vertical part.
@Krijn That would be sweet! @MikeMiller I see your Point, but that's effectively just the difference between typeset and handwritten font, the same way nobody would write down a g like the little Google-g with serifs and all those complicated arcs
@MikeMiller Hm, I see. What a pity, I kind of hoped to find that the art of reproducing fraktur by hand in high-speed would have been over-engineered to perfection by some people here ;)
I do know they let someone who did algebraic geometry with Farb first quarter hop directly into second quarter of algebra, but I imagine that's because it doesn't rely on rep theory as much. Still, if Neves thinks you're in good shape then you are
my very first interaction with the man was him reprimanding me for not having taken the class, and then he went off on a really wide tangent about algebraic geometry that I didn't understand
Demonark: I think it would be insanely boring just to teach courses in one's area. Algebraists at UGA were generally like that (other than calculus). I couldn't stand it.
It was about these three important theorems in geotop. One I think was about Euler Characteristic, I forget the second, and the third I think had something to do with Gauss-Bonnet somehow
I actually never had time to discuss Morse theory (other than defining Morse functions, differently from G&P) when I taught undergrad diff top. There just isn't time for everything.
I meant seeing different handlebody decompositions from the different Morse indices ... but not a big difference from Poincaré-Hopf, admittedly.
@Ted I like three proofs: one via handle swapping (it's so trivial!), one via uniformization and Fuchsian groups, and one via prime decomposition ideas.
I don't know anywhere the last one is written down, though surely it's well known.
@TedShifrin Well, ok, gradient transverse to zero section means the Hessian (derivative of gradient) condition near the critical points (where the gradient is zero). So, yeah.
GP and Milnor does everything near a chart on the critical points I think
