Check the differentiability and continuity of the function at $(0,0)$
$$ f(x,y)= \begin{cases} x^2\sin\frac{1}{x}+y^2\sin\frac{1}{y},x\neq
0, y\neq 0\\\\ x^2\sin\frac{1}{x},x\neq 0, y= 0\\
y^2\sin\frac{1}{y},y\neq 0, x= 0\\ 0, x=0,y=0 \end{cases} $$
In all cases, $|f(x,y)|\le|x^2+y^2|\leq...
hmm, I think Weil restriction should be irrelevant
Given K/F finite Galois, there should be an antiequivalence of categories between 1. Torus over F that splits over K 2. finitely generated free abelian groups with a right $\Gamma$-action
the latter should just be homomorphisms $\Gamma \to \operatorname{GL}_n(\Bbb Z)$
@TobiasKildetoft we kind of did modular representation theory in ANT today, although we didn't call it that. We used the standard argument that if $G$ is a $p$-group, then every representation on a finite-dimensional $\Bbb F_p$ vector space has nontrivial fixed points to see that $\Bbb F_p[G]$ is a local ring. This allowed us to do some Nakayama arguments and in the end we obtained some cool results on cohomologically trivial $G$-modules
@TobiasKildetoft For a $p$ group $G$, the following are equivalent and a $\Bbb Z[G]$-module $A$ such that $A$ is $p$-torsion, the following are equivalent: - $A$ is a free $\Bbb F_p[G]$-module - There exists a $i \in \Bbb N$ such that $\operatorname{Tor}^i_{\Bbb Z[G]}(\Bbb Z,A) = 0$ - There exists a $i \in \Bbb N$ such that $\operatorname{Ext}^i_{\Bbb Z[G]}(\Bbb Z,A) = 0$ - For every subgroup $H \subset G$ and all $i \in \Bbb N$, we have $\operatorname{Ext}^i_{\Bbb Z[H]}(\Bbb Z,A) = 0$ and $\operatorname{Tor}^i_{\Bbb Z[H]}(\Bbb Z,A) = 0$
@LeakyNun there's a more elementary way to see this: any element in $K^\times$ can be uniquely written as $\pi^n u$ for $n \in \Bbb Z$ and $u \in \mathcal O^\times$ (just by unique factorization)
when you put the discrete topology on $\Bbb Z$ and the one induced by $v$ on $K^\times$ and $\mathcal{O}^\times$ this is even an isomorphism of topological groups
If you throw in Hausdorff too this situation is common enough there's a special name for compact, totally disconnected Hausdorff spaces: they're known as Stone spaces
more precisely, you can write down a functor $\mathbf{Ab}^{op} \to \mathbf{Ab}$ that sends colimits to limits and restricts to an equivalence between finite groups and their own opposite category
(possibly useful hint: why is the category of finite-dimensional vector spaces over a field equivalent to its opposite category?)
@MatheinBoulomenos If $T$ is a split torus over $K$, is $T$ and $\operatorname{Spec}(K[\operatorname{Hom}(T,\operatorname{GL}_{1,K})])$ canonically isomorphic?
just started helping some fresh blood with p-adic numbers, planning a potential lie theory thing next semester that hopefully I can trick some physics students into joining
the putnam group is mostly disbanded for the summer
@Cows there's no shortcut. If you're serious, learn some serious abstract algebra, commutative algebra, maybe even start with linear algebra first. (And do some other maths related like algebraic topology, algebraic number theory and manifolds) It's like if you're asking a physicist to explain string theory without knowing mechanics or calculus
Both Abraham-Marsden and Da Silva seem to imply that given a symplectomorphism $g:T^\ast X\to T^\ast X$ which preserves the tautological $1$-form $\alpha$, it must be that $g$ is fibre preserving.
In fact, this is addressed in the question
Cotangent bundle lift theorem.
The answer given to thi...
I think he does bring up the point on math being disconnected with others (even within STEM) which I think is a valid point and something should be attempted to be addressed, although I don't really know how this could be improved
On this note.. @Cows if you want to get an 'idea' on why schemes are developed you should learn a little bit about varieties first - then you'll see how there are certain things which are rather awkward to talk about (for example - multiplicities) and become more natural if you know schemes
At least from the geometric perspective, people in algebraic geometry really care about things like curves, surfaces and higher dimensional varieties - so I think it's a good idea to get a feel of them first.
@MatheinBoulomenos thanks for displaying arrogance. 1) I have taken most of the classes you listed in you mockery of my mathematical capabilities. 2) the reason I am not in school right now is a bit personal and I suggest you reconsider and your words and show decorum next time.
@MatheinBoulomenos I attempted showing humility by asking in a meek manner
I am going to immediately step out of this conversation, but while I respect a desire for respect Mathei did not say anything arrogant. At most, he made suggestions that did not know your background.
I just want to learn some about schemes and you just start running up about my deficiencies and other things, I mean dude, you don't even know my background
I think the problem is that some math people grow up with everyone telling them they are the brightest people on the planet and lavishing praise. I'm not hatting but don't . . . talk to people like they are dogs
@MatheinBoulomenos learn some related math like manifolds? wtf
Check the differentiability and continuity of the function at $(0,0)$
$$ f(x,y)= \begin{cases} x^2\sin\frac{1}{x}+y^2\sin\frac{1}{y},x\neq
0, y\neq 0\\\\ x^2\sin\frac{1}{x},x\neq 0, y= 0\\
y^2\sin\frac{1}{y},y\neq 0, x= 0\\ 0, x=0,y=0 \end{cases} $$
In all cases, $|f(x,y)|\le|x^2+y^2|\leq...
Both Abraham-Marsden and Da Silva seem to imply that given a symplectomorphism $g:T^\ast X\to T^\ast X$ which preserves the tautological $1$-form $\alpha$, it must be that $g$ is fibre preserving.
In fact, this is addressed in the question
Cotangent bundle lift theorem.
The answer given to thi...
