12:28 AM
@TedShifrin I'm taking a seminar on abelian varieties. Of course, we're working over general base fields, but if I want to get some intuition from the complex case, what would you recommend?

Hmm, do you know about classical Jacobians of curves?

I've read about the Jacobian and the Abel-Jacobi map via integration of cycles
but not much

Mumford has a classic book. Lots of stuff, too, in Griffiths/Harris (e.g., conditions for a torus to be algebraic) and even Griffiths’ little book on curves. Also Gunning has stuff in the Princeton series.

the only AG book of Mumford I know is the red book

You should learn what the famous Torelli problem is, too.
He has a paperback on jacobians.

12:40 AM
there used to be a restaurant on hwy 29 on the way to napa called Tonelli's. i always wanted to open up Fubini's next door and run them into the ground.

I think we talked about Torelli's theorem over general algebraically closed field in the seminar premeeting

Do you know about Kodaira embedding? So for tori it’s the Hodge-Riemann bilinear relations that determine when it’s algebraic/projective,
There was a hwy 29 in Athens, too.

Is Kodaira embedding the theorem that closed submanifolds of $\Bbb{CP}^n$ are algebraic?

That’s Chow.

12:44 AM
Kodaira tells you a necessary/sufficient condition to get a projective embedding.

I see

Very ample line bundle ….
Curvature condition ….
So meandering through these things will give context.

I know (very) amble line bundles, but I don't know how this relates to curvature

For that look at G/H or your favorite German text.
Huybrechts, I mean.

thanks, I will have a look
I think Huybrechts wrote in English?
Though he's a professor at Bonn I believe

12:47 AM
Right.
I meant German prof.

I suppose it's nontrivial to show how the Jacobian defined as $H^0(X,\omega)^\vee/H_1(X,\Bbb Z)$ relates to $\mathrm{Pic}^0(X)$?

Not very nontrivial.

1:04 AM
okay I think I see how one might get a map. We have the Abel-Jacobi map $X \to J(X)$ (assuming we fixed a base point). If we think of $\mathrm{Pic}^0(X)$ in terms of Weil divisors, then because the group of Weil divisors is freely generated by the points of $X$ and because $J(X)$ is an abelian group, the Abel-Jacobi map extends to a map $\mathrm{Div}(X) \to J(X)$, which we can restrict to $\mathrm{Div}^0(X) \to J(X)$
I suppose the difficult parts are showing that this is surjective and the kernel consists of principal divisors
I was searching for some clever sequence of sheaves that gives the map, but it's easier to think about Weil divisors here

If I ever happen to contribute anything in mathematics, I think I'm going to name things formally with zoomer terminology solely because I think it would be humorous to see in titles of formal papers and their contents.
"Let a member of this set be formally defined as a bruh moment. If the value is real, then it is a real bruh moment."
I don't know why I think these things sometimes :)

@AMDG what about using emojis instead of letters for mathematical objects? "Let 😂 be 🙈-dimensional variety over Spec(😍) ... "

dear god

@LukasHeger Oooo, now that's awful. I like it.
"On the applications of the youjustgotprankt laughing_emoji theorem"

1:37 AM
@Ted thanks for the suggestions

1:52 AM
Hello
I see that lots of probabilistic proofs start with simple uniform distribution for their random variable at first. Any hint or clue please as to why they usually make that assumptions besides to start simple :/

Is anyone around to help me sort out an induction proof?

Can you post if pease?

How do I post images?

you need a minimum of reputation
like 75 or 100 i think

Ah dang, I only have 55.
I guess I'll latex it
Here is the problem: Suppose that one has proven the proposition that if $A \subseteq B$ and $C \subseteq D$, then $A \cup C \subseteq B \subseteq D$. Prove that for any integer $n \geq 2$ that if sets $A_1, A_2,...,A_n$ and $B_1, B_2,...B_n$ are sets that satisfy $A_j \subseteq B_j$ for $j = 1, 2, ..., n$ then $$\bigcup_{j=1}^n A_j\subseteq \bigcup_{j=1}^n B_j."$$
I'll post what I have so far.

2:02 AM
@shintuku. Hi?
@shintuku. I did not know about reputation, so I am not sure if I cant give hints too?

reputation points

yep
Can't we give hints though ?

it's a system, not your actual reputation

Sure\
@UnderMathUate. Can not you post it on main forum please?
@UnderMathUate. Post it and show any attempt you have done. I will also try to help

Base case: Assume $A_1 \subseteq B_1$ and $A_2 \subseteq B_2$. Let $x \in \bigcup_{j=1}^2 A_j$. Then $x \in A_1 \cup A_2$. As $A_1 \subseteq B_1$ and $A_2 \subseteq B_2$, so $x \in B_1 \cup B_2$. Therefore $\bigcup_{j=1}^2 A_j \subseteq \bigcup_{j=1}^2 B_j$. So $P(2)$ holds.

2:06 AM
it is not forbidden to give hints when you have low reputation points

@Avra I did, but I wanted to be able to discuss a little more openly since no one was really responding. I can post the link?
Above is what I have as the base case. Not sure if it makes sense.

I see the link no need

@Avra in other words, users that do not have at least 75 or 100 reputation points don't see the little "upload" button at the bottom of the chat room

@UnderMathUate. Your base case is right.

2:11 AM
Awesome! Ok, I'll do the step for $P(k)$ and post it.

I will post my answer there
@shintuku. tnx

@Avra Ok, I see it! So, in assuming $P(k)$, I can automatically say that $\bigcup_{j=1}^k A_j \subseteq \bigcup_{j=1}^k B_j$ without having to prove it like the base case?
And then just show how this assumption leads to $P(k+1)$, like in your answer?

@UnderMathUate. Nope. You don't have to prove assumption
This is why its called assumption

Oh right, of course.

So you take it for granted. This is called the induction hypothesis

2:22 AM
@Avra Ok, I think I get it now. I'm still getting used to this kind of reasoning, lol.
I'll accept your answer, I think I understand it all now. I really appreciate the help!

That's fine :) I had similar issues. Keep pushing and always show your attempt as folks here apprecite this behaviour

Ok! :)

If anything is unclear ask me there and I will answer you :()

Ok, no problem. I'm going to write it all out together now. I'll let you know if something seems off.

Good luck

2 hours later…
4:45 AM
the idea of using inner product spaces as a way to determine how well something approximates something else is a recurrent theme no?
the inner product giving a sort of measure of how far away (with respect to something) an element of the space is from our desired element

that's definitely part of it, yes. the same theme animates normed spaces that are not inner product spaces.

very cool!

a metric space provides a measure of distance.
an inner product provides a lot more than a distance. it provides some measure of 'alignment' as well.

people were talking about approximation in the sup norm the other day. that is a good example of a norm that is not induced by an inner product but is still of high interest.
what that guy said.

inner product spaces are for applied folks who like simple computations.

4:50 AM
a good deal of work in normed spaces involves approximate replacements for stuff that you get from an inner product. 'almost orthogonality' is a big thing in harmonic analysis for example.
the main justifications for least squares models, (1) it's a button in excel, (2) it makes computation easier.

very cool stuff

inner product spaces also provide a nice model of conditional expectation if you are into that sort of thing.

my daughter has stopped complaining about her leg. her wobbly walk is getting a little smoother but she is not pointing her foot in the right direction. orthopedist says it's not a problem unless it lasts for weeks.

excel is a degenerate piece of rubbish that is probably responsible for many scientific errors.
3
glad to hear things are improving.

between excel and java.

4:52 AM
my son returns tomorrow, so i am in a bad mood.

copper, as in orthogonality indicating impossibility or low likelihood?

we keep having weird conversations about a dead bird she saw at the duck pond.
i visited home i think three times in my first semester. i was a similar distance away (~ 40 miles). i said nothing.

well, the conditional expectation can be interpreted as a projection of sorts.
well, i have communicated ~5mins with him today. but that is better than nothing.
he went to church (his friends go, he is not religious) and then they went for food which apparently takes the entire day.

my dad didn't like me visiting even those three times. he had a weirdly idealized conception of college as something you go away to and never come back from. he was also in the process of being divorced by my mom which may have affected his disposition.

wow, that would have killed me. and my dad and myself had a fairly volatile (in all senses) relationship.

4:56 AM
he went to church. that's rich. i only go to church for funerals.

my kids have no idea what a proper dysfunctional family is like. they think my wife & myself having a disagreement is major event.

my dad had this idea that i was some kind of lothario because when i'd visit home for the weekend i'd stay at female friends houses. it was purely to avoid the awkwardness of being around him.
i think my daughter will have a somewhat better experience.

i think my kids have no idea how good they have it.

my parents never took ME to two different duck ponds. let alone every week.

5:12 AM
i used to let mine feed the ducks at jewel in direct contravention of posted signs.
jewel is almost completely gone. very sad.

are you guys getting any of this rain?

i used to bring mine to oakland zoo one a week. including the rain ride and an icecream.
a marginal amount of rainfall.
enough that i was concerned about my son's driving around.

we used to go to the OC zoo before the pandemic. OC is too full of wingnuts to go back during the pandemic.

i like zoos. i feel at home there :-)

we don't feed the ducks, but they tend to expect it. i took a great video this weekend of my daughter yelling as a gang of four ducks approached her. "agggh! i don't want them to come close to me!" and hiding behind me.

5:17 AM
my mom lived in fear of a goose or swan breaking our arms. no idea how she got that into her hear. she was a logical person, but had a few oddities. she thought one would die of cramps if you went swimming in the hour after eating.
i did an experiment and survived, albeit my dinner contributed to the general atlantic flora & fauna.

geese and swans do not f around. but my daughter seems cool with them. these ducks were coming too close.
some kind of authority figure should have intervened instead of making a video of it

have had a few standoffs with geese, but they can't do any real damage to an adult at least.

6 hours later…
11:02 AM
If I have a domain $\Omega$, and an open cover $\{U_{\alpha} \}$ of the domain (by open sets of $\Omega$), and functions $f_{\alpha \beta} \in C^{\infty}(U_{\alpha} \cap U_{\beta})$ and a partition of unity for $\Omega$ with respect to the open cover, $\{\psi_{\alpha} \}$, why is $g_{\alpha} = \sum_{\beta} f_{\alpha \beta} \psi_{\beta}$ necessarily smooth on $U_{\alpha}$?
here by $\psi_{\beta}$ it means $\psi_{\beta} \restriction_{U_{\alpha} \cap U_{\beta}}$
also $\Omega \subset \mathbb{C}$

11:16 AM
oh, is it because if $B(z_0,\epsilon_0) \subset U_{\alpha}$ is precompact, then $g_{\alpha} \restriction B(z_0; \epsilon_0) = \sum_{\beta ; supp(\psi_{\beta}) \cap B(z_0 ; \epsilon_0) \neq \emptyset} (f_{\alpha \beta} \psi_{\beta} \restriction_{U_{\alpha} \cap U_{\beta}}) \restriction B(z_0 , \epsilon_0) \cap (U_{\alpha} \cap U_{\beta})$
and the RHS is a finite sum (because $\{supp(\psi_{\beta}) \}$ is locally finite), and each summand is the restriction of a smooth function on an open set ($U_{\alpha} \cap U_{\beta}$) to a smaller open set

4 hours later…
2:54 PM
hello
I am confused why can you apply different limits when solving equation by separation of variable...

continuing my theme of "horrid elementary integrals"
is there a relatively 'slick' way to compute $\displaystyle \int_0^1 x^k \ln\left(\frac{1+x}{1-x}\right)\,dx$?
looking at it, it seems like i need to split between even and odd cases. i'm interested in odd $k$.

3:25 PM
sorry

hmm, this is suggestive: the values for odd $k$ are the same as the coefficients in the Taylor series of arctanh(x)^2
okay, so that seems to be coming from the following identity: $$\int_0^1 \frac{x}{1-y^2 x^2}\ln\left(\frac{1+x}{1-x}\right)\,dx = \frac{1}{4y^2}\ln\left(\frac{1+y}{1-y}\right)$$ which is valid for $-1<y<1$ (if you eliminate the removable singularity at $y=0$)
now expand both sides in powers of $y$ and match term-by-term
ah, drat, i transcribed that wrong
RHS should've been $\frac{1}{4y^2}\left[\ln\left(\frac{1+y}{1-y}\right)\right]^2=y^{-2}\tanh^{-1}(y)^2$
which is simultaneously tidy and gross

3:47 PM
Tidy and Gross would be a good name for a band
ideally a duo

hmm. i have no earthy clue why that integral works.

@leslietownes Sounds like the description of a public bathroom.

math.stackexchange.com/a/4276687/668308 I can't understand the given hint.

Hello,

If we have a list of jobs ordered according to their finish times (nondecreasing sort), then is it true please that we can find all jobs that are not conflicting with current job based on their finishing times in O(nlogn) please
The point is the jobs already sorted according their their finish time, so no clue why we need binary search to search all elements in O(nlogn) time where $n$ is the number of jobs.

4:25 PM
AMDG around here people usually skimp on the tidy and emphasize the gross
it could also be a good name for a joke law firm

5:16 PM
Les Tidy and L Otto Gross, LLP
@Semiclassical $\frac12\log\left(\frac{1+x}{1-x}\right)=\tanh^{-1}(x)$ I believe
@Semiclassical Ah, I see you realized this.
But it does suggest the substitution $\tanh(u)=y$
don't know if that helps

@leslietownes That is grossly unsettling.

6:19 PM
@leslietownes Sort of reminds me of The Odd Couple!

6:34 PM
these integrals just seem inherently gross to me

7:30 PM
Hello.

0

Problem: Suppose $c_1,c_2:(-\epsilon,\epsilon)\rightarrow M$ are smooth curves with $c_1(0)=c_2(0)=p$ , where $M$ is a smooth manifold. Then, $c_1'(0)=c_2'(0)$ if and only if there exists a chart $(U,\phi)\owns p$ such that $(\phi\circ c_1)'(0)=(\phi\circ c_2)'(0)$ My attempt: Forward direction: ...

1 hour later…
8:52 PM
hello, what it means "uniformaly" in this condition: $\lim\limits_{t\to0}\dfrac{f(x,t)}{t\phi(x,t)}=0\,\text{uniformly}\, x\in \mathbb{R}^N$

I would interpret as $\lim_{t \to 0} \sup_x | \cdots | = 0$.

3 hours later…
11:44 PM
hrmp who's a probabilist in the crowd?
I don't see a proof to Levy's inequality

is that in feller?

We have no probabilists, only aspiring ones like you.

maybe there is no proof. maybe it's levy's fraud upon the public.

Never hoyd of it.

11:48 PM
@LeakyNun to no avail
tried that, the cambridge website crashed. can you go yell at somebody over there

i'll try yelling at the air

Maybe Leaky has a key to the library.

@TedShifrin might you be able to help me with this question if you have time

i thought feller had some version of that in his treatise, which was published in english and ought to have infringing copies available. could be wrong.
i had a print set lost by the USPS. i have a long standing grudge against the postal service.

Leaky, I know nothing about étale cohomology.
Make DeJoy pay, leslie.

11:51 PM
i should. i don't think the statute has run on my claims.

court of federal claims, here i come. no recipe for success like suing the government in a no-right-to-jury forum, where a representative of the government without life tenure decides how much the government owes you.

Varadhan has proves what he calls Levy's inequality, but its not the traditional statement

i don't have a link, but this guy: en.wikipedia.org/wiki/William_Feller

@JoeShmo you can try asking for reference-request on math.SE

11:53 PM
it might be in volume 2

(i have access to the cambridge core website, but there's only 1 page)

yeah I think I saw it then
and it doesn't look like its the relevant stuff

yeah looks more like a review to me

Maybe @Balarka has encountered Levy.

varadhan proves $\mathbb{P}\{|X_i + \ldots X_n| \ge \frac{\ell}{2}\}$ for all $1 \le i \le n \implies \mathbb{P}\{T_n \ge \ell\} \le \frac{\delta}{1 - \delta}$ where $T_j = \sup |X_1 + \ldots + X_j|$

11:56 PM
my wife has a handful of real probability books although i think they are banished to the storage space above the garage.
her fake probability books are on a shelf in the living room.

excuse me, $T_n = \sup_{1 \le j \le n} |X_1 + \ldots + X_j|$
also, the condition is $ℙ\{|X_i+…X_n|≥\frac{ℓ}{2}\} \le \delta$