@TedShifrin Hello Ted. Finished my first measure theory exam yesterday. I am glad that he put up some facts from topology in the hints, otherwise I wouldn't have gotten the questions right. I think I got 100%. But wow, has this class been tough.
I've seen several posts on Reddit this semester of students talking about how hard it is to take measure theory remotely. I can relate.
I've been seeing why since the beginning of the semester.
I think one person once told me that when you're in math grad school, topology is more or less assumed. I guess I didn't take that to heart since I thought this is more a probability course than a measure theory course, but wow, was I wrong.
Let $\{a_k\}_{k=0}^{\infty}$ be a sequence of complex numbers. Is it true that $\sum_{k=0}^{\infty} \frac{2k^2}{4k^2 + 8k +3} |a_k|^2 < \infty$ if and only if $\sum_{k=0}^{\infty} |a_k|^2 < \infty$.
Note that $\frac{2k^2}{4k^2+8k+3}<\frac{1}{2}$, so if the latter converges, so does the former. On the other hand, $3\frac{2k^2}{4k^2+8k+3}>1$ for large enough $k$, so if the former converges, so does the latter.
Let $f$ be an entire function such that $f(z_0) = 0$. If $$\int_{\Bbb{C}} |f(z)|^p e^{- \frac{p \alpha}{2} |z|^2} dA(z) < \infty$$, how do I argue that $$\int_{\Bbb{C}} |\frac{f(z)}{z-z_0}|^p e^{- \frac{p \alpha}{2} |z|^2} dA(z) < \infty$$?
Sanity check: If I have a partially ordered set $\mathcal{I}$ and $I$ has an initial object $e$ in it. Suppose $\mathcal{C}$ is any category that is indexed by $\mathcal{I}$, then $\underset{i \in \mathcal{I}}{\varprojlim}A_i$ always exists right because it is precisely the indexed object in $C$ corresponding to $e$?
Yeah @Edward. I had to take classes at the same time slot this semester because I had committed to both of them beforehand without paying attention to the timings. Another reason because one of the classes was Symplectic Geometry and Classical Mechanics and I badly wanted to take that.
What does one do in that situation? Just visit one on alternating weeks or whut? The problem is that one of the classes is a Seminar and the other is a problem class for my most difficult course, so I'm not sure I wanna skip either of those on any week lol
I was mostly paying attention to the symplectic geometry course and reading the material for the other course before hand sometimes so that I can answer questions if something came up lol
it depends what the program is. It might be that the punchline is a correspondence between p-adic Galois representations and a weird category of modules
in which case it should be okay
if the punchline is some correspondence between étale cohomology and algebraic de Rham cohomology then I'ma ditch it hahaha
The tutor says "in principle you should be fine with ANT1 and Alg2"
yeah you're right, but I looked at the master's theses of the lecturer's PhD students that focus on aspects of the first punchline and then CTRL-F'd some algebraic geometry buzzwords and got 0 results so
Man I hate courses. Because of them my reading list just keeps on increasing and the lockdown isn't helping in decreasing it at all. Currently I have to read Arnold, read some Vakil, get to Lurie and field theories someday, more symplectic geometry, and decide a thesis topic
My notes say that if $X$ is irreducible then $\mathcal{O}_X(U)$ is the set of all rational functions $k(X)$, which are defined at every point in $U$. I do not see why do you need $X$ to be irreducible here
Let $f$ be an entire function such that $f(z_0) = 0$. If $$\int_{\Bbb{C}} |f(z)|^p e^{- \frac{p \alpha}{2} |z|^2} dA(z) < \infty$$, how do I argue that $$\int_{\Bbb{C}} |\frac{f(z)}{z-z_0}|^p e^{- \frac{p \alpha}{2} |z|^2} dA(z) < \infty$$?
math.stackexchange.com/questions/2534369 For this question, I want to show that the set of points of discontinuity is exactly equal to $[0,1]\times [0,1]\cap \mathbb{Q}\times \mathbb{Q}$. But when I use the sequential continuity definition, I get a larger set. For eg: $p_n$ be a sequence of rational points converging to $\frac{1}{\sqrt{2}}$ but the sequence $f(\frac{1}{2},p_n)$ doesn't converge to zero. What is wrong with this method?
@LeakyNun sorry, I messed up. It is the other way round i.e. $f(p_n,\frac{1}{2})=\frac{1}{2}$ for all $n\in\mathbb{N}$ but $f(\frac{1}{\sqrt{2}},\frac{1}{2})=0$
@LeakyNun Maybe here's why you maybe need $X$ to be irreducible. To define $k(X)$ you need to have your coordinate ring $\Gamma(X)$ be an integral domain. That only happens when $I(X)$ is a prime ideal which only happens when $X$ is irreducible.
math.stackexchange.com/questions/2534369 For this question, I want to show that the set of points of discontinuity is exactly equal to $[0,1]\times [0,1]\cap \mathbb{Q}\times \mathbb{Q}$. But when I use the sequential continuity definition, I get a larger set. For eg: $p_n$ be a sequence of rational points converging to $\frac{1}{\sqrt{2}}$ but the sequence $f(p_n,\frac{1}{2})$ doesn't converge to zero i.e. $f(\frac{1}{\sqrt{2}},\frac{1}{2})$ What is wrong with this method?
@AlessandroCodenotti Your recipe is too hard, just do it inductively. It suffices to prove that the compactly supported homeomorphism group of a connected topological manifold acts transitively. And sure, you can try some geometric argument like that, but I don't know how you'll get a nice embedded path; it sounds very fiddly.
Just use the standard argument: there is an equivalence relation, where two points are equivalent if there is a compactly supported homeomorphism taking one to the other. Equivalence classes are open since you know how to do this in open balls via explicit formula. Connectedness implies there is only one equivalence class.
@LeakyNun Why use alternating column sums in the definition of Specht module (or rather, polytabloids, which span the Specht module), instead of (non-alternating) row sums?
@Lukas I'm still yet to find a satisfactory topic for a master's thesis, I know I wanna do something Iwasawa theory-y but classical Iwasawa theory won't cut it for a master's thesis lol
@AlessandroCodenotti My Master's thesis and my PhD dissertation ended up with about the same number of pages of actual content. My master's thesis just had a bunch of extra pages with some code and stuff
Some 18 year old just moved into the flat opposite me and I don't think he knows that you can see through net curtains in the evening when his light is on
Errr I'm basically just preparing for the semester that starts on monday; smooth representations, recapping some stuff on local fields and homological algebra
The proper definition of the Riemann zeta function is $\zeta(s)=\int_{\widehat{\Bbb Z}^\times} |x|^s d^\times x$ where $d^\times x$ is a Haar measure on $\widehat{\Bbb Z}^\times$ and $|x|$ is the restriction of the adele norm
One thing that's kinda cool I've been looking at (by which I mean I wrote about for a fellowship research statement and will table it for now) is arithmetic quantum chaos
Pretty much the setup is, let's say you're on a Riemannian manifold. Classical dynamics on it is given by geodesic flow, quantum mechanics (up to some simplifications) by the Laplacian
@Sayan idk there was some thing released years ago about it and I read it for like 10 seconds and was like "whats a Hamiltonian" and then never looked at it again
So you sorta expect from physics that in the high energy limit, quantum mechanics starts to resemble classical. In particular, if you're on a hyperbolic manifold, then classical dynamics is chaotic
Let's say $X$ is a compact Riemannian surface, and let's say $\Delta \phi + \lambda \phi = 0$. We know in general that $\|\phi\|_{\infty} \le C\lambda^{1/4}$
($\phi$ here is $L^2$-normalized)
This bound is sharp for the sphere but we expect better in the hyperbolic case. Some mix of numerical evidence and that this might be what one would consider "chaotic", that function kinda evens out or smth
But for general hyperbolic surfaces all you can really get (at least at the time I-S was written) is $\|\phi\|_{\infty} \le C\frac{\lambda^{1/4}}{\log(\lambda)}$
But if you're dealing with arithmetic surfaces you're able to get a power saving bound by playing with Hecke operators
In particular $\|\phi\|_{\infty} = O(\lambda^{5/24 + \varepsilon})$
@loch That's definitely not unproductive! I need to figure out that stuff as well, in particular (prob not this summer but in a future one) I may apply for finance-type internships just to be safe lol
What are the maps $f$ such that $f\circ g$ continuous implies $g$ continuous?
I start by naming open injections
Anonymous
Hi. If the arithmetic mean $\sigma_n = \frac{1}{n} (s_1 + s_2 + \cdots + s_N)$ converges to some finite value as $n \to \infty$, can we also say that $(s_n)$ and $(\sum_n s_n)$ converge?
I want to show that if S is a regular surface and p\in S and if P\subset R^3 is a plane such that P contains p and S lies on one side of P then T_pS = P.
My definition of T_p S is {f'(0): f is a regular curve in S with f(0)=p}
@love_sodam both are two-dimensional subspaces, so it's enough to show containment in one direction, I believe $T_pS\subseteq P$ is the easier one
Anonymous
@Thorgott I can't think of any strong reason why those two implications should hold, but it seems like they've used that fact here in the 2nd page, to prove that Cesaro convergence => Abel convergence
Anonymous
Notice the statement: "If σn converges to a limit and r < 1, then both (N + 1)σN+1r N and sN r N+1 will vanish"
@Thorgott Agree. Maybe up to rigid motion, we may assume p is the origin and P is the xy-plane(@TedShifrin). If f'(0)\in T_pS then f'(0) should contained in xy-plane why..?
