Show that for any irrational $x \in \mathbb{R}$ and positive integer $n$ $\text { there exists a rational number } \frac { p } { q } \text { with } 1 \leq q \leq n \text { such that } \left| x - \frac { p } { q } \right| < \frac { 1 } { n q }$
I have done this so far
We can manipulate inequality to get $|xq - p| < \frac{1}{n}$
Now we want to show that there is some $q$ between 1 and $n$ such that $xq$ is within $\frac{1}{n}$ of some integer
@TedShifrin But then in calculating $\frac{\partial f}{\partial r}(a)$, $f(a + te_j)$ wouldn't be increasing or decreasing $f$ in the $r$ "direction" if that makes sense
Let $n\in \mathbb N$, and let $\mathscr P$ denote the collection of all polynomials in $n$ variables. For $p\in \mathscr P$, let $Z(p)=\{(x_1,x_2,...,x_n)|p(x_1,x_2,...,x_n)=0\}$.
Show that $\{Z(p):p\in \mathscr P\}$ is a basis for the closed sets of some topology (Called the Zariski topology) on...
@SharathZotis Every element of $\Bbb R/\Bbb Z$ is of the form $a+\Bbb Z$, yes
Imagine a circle. Pick a starting point. The point $a$ degrees away from that starting point is the same as the point $a+360^\circ$ away from that starting point, is the same as the point $a+720^\circ$ away from that starting point…
just over half an hour until my students' exam starts. I really hope my phone doesn't ring, because that might be the proctors calling to ask about something which might turn out to be an error in the exam.
Well, past half-way in the exam and only one call so far (which was 5 minutes before the exam and was about a student asking whether they were allowed to answer in English).
I must prove that an hypersurface $M$ on $\mathbb{R}^{n+1}$ that is Einstein and compact can be only the $n-$dimensional sphere when $n>2$
The Einstein condition we permits to say that scalar curvature of $M$ is costant because $n>2$.
The fact that it is an hypersfurface of $\mathbb{R}^{n+1}$ c...
Is this statement correct: (n+1)-th singular homology of (suspension(X), suspension (A)) is isomorphic to n-th singular homology of (X,A) - I am aware of the usual suspension iso and trying to prove that H_n(D^m, S^(m-1)) = 0 if n is not equal to m
Hello there, I would like to know if 200(x+8)(x-1) in Z[x] (integers) is already factored in irreducible factors. I think not since 200 can be factored further in 2^3*5^2 but I am not sure.
That's not a triviality with 70 students. I remember you told me about the external person. This is with one exam only? Like at the end of the course? I never understand the European systems.
Yeah. He is not only to ensure fairness but also to ensure quality in some sense (i.e. that the university does not start lowering its standards to get students through)
I too dislike having the entire grade based on a single exam, but there is not anything I can do to change it.
Of course, the external person might have slacker standards. But I understand this completely. I used to complain that my colleagues gave A's to some students I would probably give a C or D to.
Since Mathei is nowhere to be found I'm going to bug you @Ted :P If you have time for this can you look at the course summary here? How much diffgeo does it look like I should already know?
You need to know the Levi-Civita connection, parallel transport, curvature. The point is that for a (locally) symmetric space one can tie all this stuff into the Lie algebra structure of a symmetric pair.
You can understand the algebraic stuff, but you won't have any idea of what it's measuring or telling you.
I mean, it is Diff Geo 2. That means Diff Geo 1 should be a prerequisite.
I wonder if there are almonds and raisins in Müsli.
@AlessandroCodenotti That might be because they want you to have a semester in between to make sure you actually understood the first one before tackling the second one
@TedShifrin Hmmm I see, if I pass all of my exams on the first attempt I'll have around 6 free weeks in which I could do some Riemannian geometry I guess... I'm not sure if I want to attempt this though
Not that professors always provide intuition or understanding, but diff geo can be thick and there's lots of room for intuition/understanding to be provided.
I like to believe I provided more insight than a textbook, anyhow.
@TedShifrin Yeah that's probably the best choice. But I still think I have more chances at surviving diffgeo than harmonic analysis in the next semester
I recently learned what a vector bundle is. Just a locally free $\mathcal O_X$-module of finite constant rank!
@TedShifrin Advanced geometry I is "Geometry of Classical Field Theory"... Diffgeo is an undergrad course that I cannot take for credits because I'm already doing algebraic topology which is also an undergrad credit in the same area
@TobiasKildetoft that's because you're really tensoring over $\mathcal O_X$ so it makes sense that $\mathcal O_X$ acts as the identity (it actually is the identity in the Picard group)
I answered some MO question recently that wanted to know the SW classes of $E \otimes \det(E)$. May as well get the formula for any twist. It's not clean!
Easiest approach I had was the splitting principle.
@MikeM: As you know, I don't think about SW. But, yeah, for Chern classes there's a well-known formula for tensoring with a line bundle, and the splitting principle is the easiest way to see it.
@Alessandro: I'd have to look more carefully to figure out what the background is. Certainly $L^p$ and measure theory. Dunno if there's much PDE, for example.
@MikeM: I've certainly had to use the Chern class formulas in several papers.
Much as I'd love you to learn some diff geo, @Alessandro, jumping into an advanced course without the "undergraduate" (ha!) background isn't going to be smart.
Well, I've certainly needed arbitrary rank bundles for this, @MikeM. It comes up naturally even for complex submanifolds of projective space, because of the Euler formula for the tangent bundle.
@Alessandro: That's what happens when you slum and ask me questions instead of Mathein. (Probably I was the appropriate person, but never mind ... ) :D
I'm considering many courses at the moment, but in the end it'll be models of set theory+another course+a seminar (+ a third advanced course with no psets at most)
@TedShifrin Well I wanted to hear impressions from someone who actually took the course, I guess Mathei knows such people
today I learnt the shocking fact that apart from green's theorem, stoke's theorem, and divergence, there is yet another special case of Stoke's theorem that everyone knows
and it helped me understand Stoke's theorem so much better
OK, @quallenjäger, so I would say nondecreasing (or the first time make sure to tell people that increasing means that). I don't see the point of "monotone" in addition.
Do you have any intuition behind the Serre's twisting sheaves we mentioned earlier? Like I get the definition, it's an easy one, but why did Serre care about them? Or why should I care about them?
@Alessandro: In just the setting of complex manifolds it's very natural and very important. It's looking at holomorphic $n$-forms on an $n$-dimensional complex manifold. It comes up crucially in algebraic geometry because of Serre duality for computing cohomology of bundles (and more generally ...).
The only application I saw involving them is how to get an injection $\Bbb Z\to\mathrm{Pic}(\mathrm{Proj}(A))$ where $A$ is a graded ring generated by $A_1$ over $A_0$ (polynomial ring with all variables of degree one for example)
@Alessandro: Nah, even for curves, the genus is the dimension of the space of global sections of this line bundle. When $g\ge 3$ and the curve is non-hyperelliptic, sections of this bundle give you what's called the canonical embedding of the curve into $\Bbb P^{g-1}$.
I can recommend more geometric reading for you if you're interested sometime, @Alessandro.
@CaptainAmerica: There's not much content to turning "if P, then Q" into $P\implies Q$.
@TedShifrin But what if I'm doing Spivak and and I get stuck and your like this is super easy stuff why don't you know this? And I'm like I never did intro proofs.
@TedShifrin I feel like these two years I've been focusing too much on formality and nukes and abstract nonsense and making definition that "type-checks" and forgot what mathematics is
Until 25 years ago, there were no "intro to proofs" classes. People just learned doing actual math. Like Spivak, like my course. Like abstract algebra. Then we started having people majoring in math who couldn't think at all, and so we created the "intro to proofs" course to help them be successful.
@CaptainAmerica: I really want you to make it a point to get to actual calculus. You still haven't gotten out of Chapter 1 and it's halfway to February.
The beautiful stuff is chapters 5-8 and then derivatives finally are 9-10-11.
Speaking of being formal, let's say I'm learning differential geometry, I feel if I'm not being completely rigorous and formal about what I'm doing then I'm not actually doing math. As an example I feel I've spent so much time trying to nail down precisely what authors mean by coordinates when I could be progressing quicker and learning other things like submersions, immersions etc.
Informal logic? Lots of politicians and "real-life" people have no idea about logic or how to reason in a valid manner.
@Perturb: I think you've lost the forest for the trees (or vice versa). You have spent weeks on something that should be a minute. I truly can't understand how this is happening.
What I mean is that I don't care whether ZFC is true in some platonic or philosophical sense, I just see it as an interesting first order theory that's worth studying
I must prove that an hypersurface $M$ on $\mathbb{R}^{n+1}$ that is Einstein and compact can be only the $n-$dimensional sphere when $n>2$
The Einstein condition we permits to say that scalar curvature of $M$ is costant because $n>2$.
The fact that it is an hypersfurface of $\mathbb{R}^{n+1}$ c...
