The Picard group is super beautiful, and I guess you could do it in special cases without the formalism of line bundles if you just read about Dedekind domains
Yes! I remember being very confused about the point at infinity in S-T
I am still not great with projective (algebraic) geometry, tbh. Hoping it's something that improves in the next few years
IMO, best use of time is: be able to to do every calc problem (including multivariable), every linear algebra problem, do every practice test available on the internet, and know the major theorems that tend to pop up (e.g., classification of f.g. abelian groups, Cauchy-Riemann equations, Great Picard)
I mean, I'm sure you actually know these theorems, but the point is, one doesn't (unfortunately) have to actually know anything deep about them, just their statements
Lately I've been thinking of asking on here every single exercise of Magnus et al's book on combinatorial group theory. There's a good few hundred of them and they're very detailed, but I'm doing a PhD in the area and I'd like a firm grasp of the subject. It'd take a long time and I've already asked a handful. What do you think?
I don't care how difficult I find them: if it's too easy for me at first sight, perhaps I'm fooling myself as to the rigour of my solution, the details of which I can always post in a proof-verification; if it's too difficult for me, then a question is in order anyway! I'd ask on meta about it but I want some preliminary reactions first.
@Shaun you should probably try the exercises yourself first. Maybe for the first few easy ones if it's good to ask around after you do them and make sure you have the right idea, but it seems to me like at some point you'd want to gain confidence in your own ability to answer questions and believe you have them right without relying too much on others
I have considered UCLA, and I think it's moderately likely I'll apply there. For the most part my criteria at the moment is that there's at minimum good representation in the general sphere of algebra, number theory, and hopefully topology
Which UCLA seems to fit from what I hear, though at some point I'll have to check the faculty interests page and ask around
@Daminark Don't worry: I'll definitely try each if them first. If I'm confident in my answer, perhaps a proof-verification would be appropriate anyway to iron out any kinks.
In (the?) subring test, we show (i) $a+b$, (ii) $-a$, and (iii) $a \cdot b$ and (iv) $1$; belong to the subset. How come we don't show that $0$ belongs to the subset? Is it because (i) and (ii) give this, and (iv) ensures that it's nonempty, so no need to show that?
So the ICM still have not uploaded any of the plenary lectures (hopefully they will be up soon). But at least University of Bonn has uploaded the talk by Scholze.
You know a talk is in pure math when the final part is called "applications" and is followed by a slide titled "no torsion in cohomology of Shimura varieties"
@MatheinBoulomenos Is the easiest way to see a local ring with nilpotent radical is complete to think of the completion as Cauchy sequences which are just eventually constant? In fact, sequences which are constant after some fixed n?
That seems to make it clear all the Cauchy sequences have unique limits.
sorry I am probably going to need to alter and or edit a post ive made with the assistance of another how do I properly reference the assistance of that person into my SE question/poost
Crazy Turing Machine is the same as Turing machine with one stripe , except of the fact that after each ten steps the head jumps back to the beginning of the stripe.
Lunatic Turing Machine is the same as Turing machine with one stripe , except of the fact that after each 10, 100, 1000, ... steps...
@OskarTegby But Scholze has written a very nice answer on MO about what perfectoid spaces are, which are the main ingredient in a lot of his work (winning him the Fields medal this year).
Also, his ICM talk starts with an overview of the main ideas
(the talk is on the uni-bonn youtube channel, since the ICM have not yet gotten around to uploading the things)
@rschwieb I think the easiest way seems to be $A/(0) \cong A$, so $\varprojlim A/\mathfrak m^n \cong A$, since the inverse system is eventually constant
@Mathein I am too poor to finance studying immediately in the winter semester so I will probably work for 6 months to get some money together and then just pray that I can get a scholarship from the DAAD (I have quite a strong application I think so hopefully this works out)
Hi, if I have a normalized vector, is there a quick and easy way to denormalize it? I am working with python right now and there I get only the normalized eigenvectors of a matrix...
@ÍgjøgnumMeg did you check that it is possible to start in the summer semester? there are subjects where you can only start in the winter semester (but it maybe applies more to bachelor students)
@Mathein yeah i checked that you can start in the summer semester (at least I think I did!) but scholarships can only be given from the winter semester, so I would have to finance one full semester by myself I think
@Mathein Also I guess that there is also a chance that I would not be accepted in the summer semester (although I don't see why this would happen if they already accepted me now)
@TobiasKildetoft what I was doing was taking let's say $(1-x)^2+y^2=1$ and $(1-x)^.5+y^.5=1$ and compositioning them (substituting in the second one for the first one for x and y) and the result is y=x
@MatheinBoulomenos Ugh, I feel really dense, but I have a hard time seeing that isomorphism based on what little I know about direct/inverse limits. I can tell the "tails" look like elements of $A$ since they're constant... but it seems like the initial segment can vary a lot. How does the tail being constant choke it down to just $A$?
Thinking in terms of categorical limits is completely foreign to me, but I'd like to hear what the reasoning sounds like.
@OskarTegby And I do felt like you need to consider the case where f(x) is positive in some values and negative in others. Perhaps there is an $f(x)$ that grew in just a way so that it get cancelled out by $x^{-n}$ thus getting it to zero
but actually that equation is a bit strange, since you are starting with a circle thus $r$ should be constant, then it makes no sense why $s$ has to vary with angle
I think I am not very sure, unless I actually need to deal with 2 sets of (x,y) independently
Pretty visual theorem, basically compactness ensure every net will hit the boundary of the set or some points in its interior outside, hence guarentee closure
Now that makes me curious, how does that spillover occur for spaces that are non T2...
Ah, T1 spaces allow intersecting neighbourhoods, thus a net within the compact set can spillover to these neighbourhoods outside and hence landed in the neighbourhood of points outside the boudnary, thus preventing closure
@LeakyNun Yes. I think I figured it out. If $(X,\tau)$ be the topological space then we define $\text{Cl}_\tau$ as described here (this is also probably the one you mentioned. Let $\text{Cl}$ be any other closure operator on $X$ giving the same topology $\tau$.
user131753
Since $\text{Cl}$ is a closure operator giving the same topology $\tau$, so $\text{Cl}(A)$ is $\tau$-closed. Also since it is a closure operator we have $$A\subseteq \text{Cl}(A)$$ for all $A\subseteq X$. Then from the definition of $\text{Cl}_\tau$ we get, $$\text{Cl}_\tau(A)\subseteq \text{Cl}(A)$$To prove the converse we simply note that, $$A\subseteq \text{Cl}_\tau(A)\implies \text{Cl}(A)\subseteq \text{Cl}(\text{Cl}_\tau(A))\implies \text{Cl}(A)\subseteq \text{Cl}_\tau(A)$$
I think "recover" is a standard term used in mathematics (or maybe more in category theory)
Given a closure operator Cl on a set X, I can form a topology (X,τ).
What I mean when I said you can recover the closure operator, is that you can produce a closure operator Cl from the topology (X,τ) that is isomorphic to the original closure operator
where "isomorphic" really means "equal"
so I'm describing an equivalence between { closure operators on X } and { topologies on X } by giving a function in both ways
you already have the function in ---> way
when I said "recover", I'm referring to a function in <--- way
user131753
I see. Now it is much clearer.
user131753
I think I have essentially worked out the same details above.
maybe you should think about why "isomorphic" means "equal"
user131753
@LeakyNun Maybe it has something to do with the skeleton of the category $\mathbf{TCSp}$ but I have not gone that far in reading my category theory book.
Okay, I am going to do my best articulating my question. Suppose that $P = [p_{ij}] \in M_{n \times n} (\mathbb{C})$ is a positive semi-definite matrix. Then its off-diagonal entries satisfy $|p_{ij}| \le \sqrt{ p_{ii} p_{jj}}$, which says that the the entry $p_{ij}$ lies in the closed disc of ra...
How to show that multiplication of matrices, when defined, is associative and distributive over addition. Let $R$ be a ring, and the set of all $n \times m$ matrices over $R$.
Take the arc of the unit circle $(r=1)$, inside the square $[0,1]$ by $[0,1]$. Reflect this arc about $y=1-x$. Show or disprove that an approximation can be made arbitrarily precise to the reflected circle using the form $x^s+y^s=1$.
@Mike This question is probably too vague but can something interesting be said about topological spaces for which invariance of domain holds? (So spaces $X$ such that whenever $U\subset X^m$ and $V\subset X^n$ are open and homeomorphic we have $m=n$)
If $V$ is a normed linear space, and $W$ is a linear subspace with basis $B$, is $\overline{W}$ spanned by $\overline{B}$? Does $\overline{B}$ form a basis?
Under a discrete measure, Lp space, specifically, the part only located in the unit square, is not symmetric on either side of the function ϕ=1−x. So how exactly do you fill in the space with an infinite measure, say a probability measure in the unit square, such that the space becomes symmetric?
what is a measure, what is a discrete measure, what is a Lp space, are you saying an Lp space is a measure, what is an infinite measure, what is a probability measure, and when is a measure symmetric ?
Prove that $I^2/ \partial I^2$ is homeomorphic to $\mathbb{S}^2$
This was the idea I came up with to prove the above. I was thinking that I could prove the above by showing that $I^2/ \partial I^2$ was homeomorphic to the one-point compactification of $(0, 1) \times (0, 1)$ which itself is h...
How to show that multiplication of matrices, when defined, is associative and distributive over addition. Let $R$ be a ring, and the set of all $n \times m$ matrices over $R$.
Take the arc of the unit circle $(r=1)$, inside the square $[0,1]$ by $[0,1]$. Reflect this arc about $y=1-x$. Show or disprove that an approximation can be made arbitrarily precise to the reflected circle using the form $x^s+y^s=1$.
Prove that $I^2/ \partial I^2$ is homeomorphic to $\mathbb{S}^2$
This was the idea I came up with to prove the above. I was thinking that I could prove the above by showing that $I^2/ \partial I^2$ was homeomorphic to the one-point compactification of $(0, 1) \times (0, 1)$ which itself is h...