Mhm. At some point tho the industry has to change what it's paying people. The skill that I see being exhibited vs the salaries people are getting is... concerning sometimes. I felt like people cared more about doing excellent work simply because excellence is good (as opposed to "do what you can to make it appear like you did good work") when I was at school, but idk if that's all of academia or just the professors I was around.
Do you know of any good books/resources that might help someone figure out whether they'd enjoy being a professor? The one concern I have right now is that I do kinda want to stay close-ish to the east coast but a lot of good CS schools are over on the other side of the continent... but there are a few places I'd consider for grad school over here, too.
I did a research project in undergrad that resulted in a conference paper, then I worked with a professor on a conference workshop after I graduated, too --- both of those experiences I really liked, which I think/been told is somewhat representative of what researchers do in their day-to-day
More on the west coast, yeah. No, there are two basic issues to contemplate. One is research and pressure to publish. That's significant. The other is teaching and working with students. And, there's also general university responsibilities, committee work, etc.
I'm pretty confident I'd enjoy the teaching aspect, at least from what I saw when I was a TA and worked closely with a couple professors supporting their classes
But that changed with the changes in society, etc. The first 25+ years of my career, granted plenty of students avoided me, but I motivated most of my students to work their butts off and do better in my classes than they did with most teachers. That changed toward the end. I still had amazing students, but others I couldn't get to do anything.
^^ I can also see the impact of this change in society where I work --- there are a lot of my peers who are super lazy, while it's less common to see in the people who've been around a while (of course, this snapshot has some massive sampling bias)
It's kinda scary when I think about how some of the people who I TA'ed managed to squeak by with a degree, and are now probably working on software running my bank accounts, flying planes, etc.
Speaking as someone who hasn't always been the best student, I think the change has a lot to do with a sort of general malaise or exhaustion with the socioeconomic state of things. College is "what you're supposed to do" for many, and it's seen as a job factory. Society in general is also much more atomized and individualized than it used to be, as a result of the same factors.
In my specific case, it was very different---the fact that I was seen as a precocious young brat meant that I was, until a certain point, literally allowed to be lazy. Broader changes in educational standards have had something like that effect for many people---people being pushed forward through school even though they may not have adequately learned what they needed to. Many people who get pushed forth in this way mentally check out even before high school.
And I don't blame them.
NCLB and the things it precipitated are kind of what I'm thinking of.
And then there's something to be said for the erosion of the "triangle of educational responsibility" between teacher, student, and guardian.
hey what on earth is the map $F^\times \to F^\times \cap K^{\times n}$ if not still $x \mapsto x^n$? I guess we can't just change that term to $F^{\times n}$ because we would lose exactness
@Fargle @apnorton: What's worse is that so many high school teachers and college teachers just give up and have no standards, because it's just easier to avoid all the whining and parents complaining. So they teach easy courses and give high grades. Guess who didn't fit into that one?
YEs, @apnorton. When I was a kid, the teacher had authority and parents always backed up the teacher. Now parents attack teachers — even up to and including college teachers.
Most universities have very careful academic dishonesty policies. You can't do that sort of thing without going through a hearing by the academic dishonesty board. That protects against that sort of thing.
well we know that $1 \to \mu_n \to F^\times \to F^{\times n} \to 1$ is exact so we can "divide" that exact sequence by this exact sequence to get $1 \to F^{\times n} \to F^\times \cap K^{\times n} \to \operatorname{Hom}(G,\mu_n) \to 1$
i.e. $\operatorname{Hom}(G,\mu_n) = (F^\times \cap K^{\times n}) / F^{\times n}$
according to wiki if $G=\operatorname{Gal}(K/F)$ and $H = GL_n(K)$ or $H = SL_n(K)$ then $H^1(G,K) = 1$ also [where this special case is $H = GL_1(K) = K^\times$]
Conjugacy classes of complements of split extensions $1 \to N \to A \to H \to 1$, where $N$ is a $H$-module, are in 1-1 correspondence with $H^1(H; N)$, no?
@BalarkaSen let $\varphi : G \to L^\times$ be a cocycle. By Dedekind's independence, $\{\sigma \mid \sigma \in G\}$ are linearly independent, so $\sum_{\sigma \in G} \varphi(\sigma) \sigma$ is not the zero function, so there is $a$ such that $b := \sum_{\sigma \in G} \varphi(\sigma) \sigma(a) \ne 0$. Then for any $\tau \in G$ we have $\tau(b) = \varphi(\tau)^{-1} b$, i.e. $\varphi(\tau) = \tau(c)/c$ where $c := b^{-1}$, so $\varphi$ is a coboundary
I don't know what that means. If you precompose an $n$-cochain in $H$ with values in $M$ with the map $\phi\colon G\to H$, you don't get an $n$-cochain in $G$? ... I don't think about group cohomology.
any idea how I would go about proving f(b) = f(a) + f'(a)(b-a) + f''(c)/2(b-a)^2 as an extended or modified version of MVT? looks a lot like a taylor expansion but it's not quite
I'm looking at Rudin's thm. 5-15 for that Taylor Remainder thm question, but the problem is, is that all I have to work with is f and f' are continuous on [a,b], and f'' exists on (a,b). So conclude from those cond. that there exists a c in (a,b) s.t. f(b) = f(a) + f'(a)*(b-a) + f''(c)/2*(b-a)^2
@CaptainAmerica16 its hat, winter bash. I don't know what are qualifications to get one. There was a message to me from stackexchange, and I wore that hat.
