12:11 AM
I'm looking for a mathematical resource that discusses the following picture which talks about the normal line between two vectors in 3D

@user2103480 I posted my circle-squaring a while back.

@robjohn very clever

@CroCo the direction is unique, not the line segment.

@robjohn true. Can I imagine this vector to be if we slide one axis to the other, it will be the intersection point?
of course in the picture they will never intersect
@robjohn I think the direction and the distance not merely the direction, I guess.

Or perhaps I am wrong... the direction is the cross product, and if you slide a line with that direction along one line, it intersect the other line only at one point.
So it is unique. Sorry about that.

12:20 AM
where this kind of stuff are discussed in math books? Any suggestions?

@robjohn am I misremembering or did you do something in the direction of electrical engineering/signal processing in your academic life?

they half-ass a lot of that kind of stuff in vector calculus books. but it also seems to sort of be taken for granted in those materials so people can do stokes' theorem or whatever. i don't know of a good treatment.

If you did, I'd have a question on some physics-math-probability-ish thing, namely the derivative of poisson processes/spike trains

@user2103480 other than Fourier/Harmonic Analysis, not really

that may be enough
so the definition is fairly simple, I have a poisson process with jumps at time $...< t_{-1} < t_0 < t_1 < ...$
with limits going to -inf and inf in the respective direction
and I take the "derivative" of this thing which is just a sum of deltas $\frac{\mathrm d N_t}{\mathrm d t} = \sum_{n \in \Bbb Z} \delta(t-t_n)$

12:26 AM
mathrm d. you woke up today and chose violence.
but don't mind me

Okay, I think I got it myself. I got confused since one then just calls this a random variable but I suppose it's actually reasonable if one unpacks the definition. It's still super weird, we take the expectation of this and get a function $$r(t) = \sum_{n \in \Bbb Z} \Bbb E[\delta(t-t_n)]$$

here's a poisson process with jumps at time $t_[-1} < t_0 < t_1 < \dots$
i'll let myself out

soothing
Okay it's still not reasonable these are functionals wth is that expected value and the prof just states $\lim_{h \rightarrow 0} \Bbb P(\text{spike in }(t,t+h))/h = r(t)$

don't listen to the lies

12:42 AM
Probably equivalent to $$\frac{\mathrm d}{\mathrm d t} \Bbb E[N_t] = \frac{\mathrm d}{\mathrm d t}\Bbb E[\int_0^t X_s \, \mathrm d s] = \Bbb E[\frac{\mathrm d}{\mathrm d t} \underbrace{\int_0^t X_s \, \mathrm d s}] = \Bbb E[X_t] = r(t)$$
Still so weird
@leslietownes I wish I could but I'll have to answer these physicists' questions in the orals

@geocalc33 number theory is important to me because at present my existence depends on it being a thing
being a barely mediocre student pays the bills

@leslietownes I have standards

the physicist on my committee, without trying to be a jerk about it, knew his place as the outside member
and didn't trouble me with his gobbledegook
he did ask something about some unbounded operator and i felt like mr burns in that episode of the simpsons. "... are there any real questions?"

hahaha I thought I could strengthen my intuition by taking physics-adjacent courses and it just left me with improper math knowledge
but for the other course I can at least now program again

the thing that always bugged me about physics-adjacent courses is they'd trot out some horribly complicated proof of the cauchy-schwarz inequality, like, "here is a PROOF!" and the next thing they're integrating over paths and computing residues of things where they aren't defined

12:47 AM
and write a dozen emails about problems with the data

@CroCo You mean two lines, not two vectors? The vector orthogonal to both is easy, but you mean lines, I'm sure.

tbf he updated the logs at 11pm on a friday which is service

what really got me about it is they seemed to understand what they were doing

yeah i have huge respect for that. physicists often just seem to know

@TedShifrin in the picture these lines represent axes so I guess no, they are vectors and their directions are arbitrary.

12:49 AM
gwarn

i went with a dark blue adidas tracksuit with white piping

got it?

Huh?

oops no

pfft

12:51 AM
got it

You’re wrong, of course.
You need points through which the lines pass, not just their directions.

vibing to this

@TedShifrin I don't care about the direction of the axes but I really want to know is how can I compute the orthogonal between them? if any

@EdwardEvans rightfully so

You have to know both the directions and where the lines are sitting. Otherwise the lines could intersect and the distance between them is 0..
Give me the exact question.

12:55 AM
@TedShifrin Remember the other day when you said you were a department head and spent time dealing with people issues?
What happens next? When that happens?

When what specifically happens?

In this case, when she goes to the department and complains, with a few lies thrown in, about me. Dept. head wants to have a meeting.
She elevated our hostility to the coordinator, I responded in kind. She elevated to the undergrad chair.

@TedShifrin there are three cases for two lines in 3D. One, they are intersect and hence form a plane. Second, they are parallel hence, there are infinite normal lines. Third, there is a unique normal line. My question algebraically, how can I deter these cases given two lines?

You should document your side of the story carefully and calmly present it.

12:58 AM
Yeah. Well, I had sent her an email once about the issue, including taking some of the blame (I'm not blameless), but she never responded. Today we had emails with open hostility, and she copied the coordinator, blaming me for mistakes and bad instructions that she gave. I responded in kind, and then she elevated to the department char. (Though I'm sure you don't care about that)

What is her rank?

I would suggest talking about these details privately in case you can ever get traced to this discussion. That's very unlikely, but it's self-preservation.

@TedShifrin I started doing that already. But she's lied about me. And the situation isn't quite what you think. I"m not a grad student, I'm a part time employee with a not-very-strong master's degree.

@TedShifrin I need to determine these cases because the normal line will be the axis X. Could you please suggest a book that talks about this stuff?

@TedShifrin She's on the teaching track.

12:59 AM
Ah.

Accordng to her department page, she's an "Assistant Teaching Professor".

Xi-1 in the normal line

mike's point is a good one. it's probably fine if you don't go into specifics. any specifics, document as needed and talk with people over the phone for advice regarding particulars. you should assume that in litigation (god forbid) everybody would be reading everything that you wrote down.

You need to have points on and direction vectors of the lines. With those data I can answer. Otherwise it's just nonsense.

1:05 AM
@TedShifrin, let's say we have Q=[2,2,0] and P=[0,0,1]. How I can construct the normal line?

Parallel if and only if direction vectors are parallel. For the others, take the cross product of the direction vectors and project any vector from one line to the other onto it.
Come on, man. Two points tell me nothing.
I need the direction vectors of the lines and the points.

one moment, let me draw them

This should be standard in a multivariable calc class.
@Jeff You may have acted rashly, but my advice knowing no more (and wanting not to know more) is to speak privately with the person who hired you.

@TedShifrin Yeah, I probably did.
I wasn't really hired by a person, but a committee.
I'm going to lose any chance of a full-time job there (I probably have already).
I'm in a full-blown panic. Anyway, thanks!

Hi, @TedShifrin

1:10 AM
Hi Karim

I solved my PhD problem
@TedShifrin Just have to write the details down now
But I got the general idea down

famous last words. but congratulations if true.

What is the normal line?
Between P and Q?

Yeah but no congrats until I publish it even though I don't believe in this stuff don't want to jinx it.

1:12 AM
looks like the line through (0,0,2) with direction vector (0,1,0) and through (2,2,0) with direction (0,0,1), is that right?

@Jeff Did they assign a teaching mentor or someone you can talk with? If not, talk to the head separately first.
I agree, @leslie.

@KarimMansour That took me 2-3 years

Up Z+, Right Y+, Outward X+

without doing any calculations i'm guessing it's the line through (0,-2,2) and (2,2,2).

The direction of the normal line is $(0,1,0)$.

1:13 AM
outward X. you are a maniac.

Yes, that's standard.

@TedShifrin how did you compute it?

is it? i mess all of this up. to me right is X+. but i'm left handed and always ruined these kinds of calculations.

Your vectors used what he said, @leslie.

@MikeMiller Yeah I have a lot of stuff to pin down but I got the general idea down

1:15 AM
@leslietownes it has nothing to do it with left handed though

Write it down now in as much detail as you can

no, it doesn't. it's my excuse. if i miss a sign, get the wrong orientation, am off by a factor of 2, it's because i'm left handed and never had a chance.

Every time you have a new detail pinned down open a new TeX / text file and write it down in as much detail as you can

What vector is orthogonal to both the x-axis and the z-axis?

You will forget details later

1:16 AM
I am slightly confused.... so, infinitesimally at some point p (=tangent space at p) we can always find locally inertial coordinates which put the metric in canonical diagonal form = diag(-1,1,1,1). And around the point p the first derivatives of the metric components g_ij vanish....
Question: do the first derivatives of the metric components vanish only at the point p or infinitesimally around the point p?
(If it is only the former, isn't this trivial as the metric in p is in canonical form?)

We all do
Does "infinitesimally around the point p" mean "in a small neighborhood of p"
Because then definitely not you've just asserted that this space is not curved

So, Leslie guessed everything even though he's spatially challenged :)

Yeah I am recording all my conversation with my supervisor and also writing our email exchange down in latex file just in case I forget something

@MikeMiller no, a small neighborhood would be locally. Infinitesimally means only in the tangent space at p

@TedShifrin The undergrad chair has already reached out to talk to me. I think I just have to wait for it to play out. The consequences will be the consequences.
@TedShifrin Thanks.

1:18 AM

What does it mean "for the first derivatives to vanish in the tangent space at p"?
Do you just mean... the second derivatives vanish?

So you want values of $s,t$ so that the vector from $(s,0,2)$ to $(2,2,t)$ is parallel to $(0,1,0)$.

@TedShifrin please see the picture above, is this lucid?

Riemann Normal Coordinates would be one example of such locally inertial coordinates.
@MikeMiller no, the second derivatives cannot made to all vanish. otherwise we would have a flat spacetime

I was fine with the first picture. Read everything I wrote and follow up and do it.

1:20 AM
Yes, I agree. So maybe you could tell me what "the first derivatives vanish in the tangent space at p" means.
You don't need to tell me examples. I know the theory. You need to explain your language/notation.

@Jeff You should be contrite but calmly back up your side with evidence. Talk with the undergrad chair. Perhaps a weak background isn’t enough for the job.
Good luck!

@TedShifrin, but is the direction only we need to describe the normal? Shouldn't we include the distance?
Also, I'm looking for a general case to compute the direction. Since this is trivial, we can compute it by inspection.

No. You can find the distance but it's extra work.
If you read everything I have typed, I told you how to do the general case.

@TedShifrin Thanks again.

@TedShifrin I missed this line. Sorry.
Thanks

1:26 AM
I saw that you were going to upload a picture of Carroll's textbook. It does not surprise me if a physics textbook (even one on GR) does not lucidly explain the math. I'm almost certainly not going to be able to read his notation.

Why are all these physics people swarming on us? :)

I don't even mean this in a hostile way (for once). Physicists think about manifolds in a very different way than I do, a way which I find obscures both the language, the notation, and the ideas for me. Blame my inflexibility if you want!

this is how Carroll introduces the concept of locally inertial coordinates:
It is always possible to put the metric into canonical form. In fact it is always possible to do so at some point $p\in M$, but in general it will only be possible at a single point, not in any neighborhood of p. Actually we can do slighlt better than this; it turns out that at any point p there exists a coordinate system $x^{\mu}$ in which $g_{\mu\nu}$ takes its canonical form and the first derivatives $\partial_{\sigma}g_{\mu\nu}$ all vanish:

sorry for the dump question, how can I upload a picture plus adding comment here?

Yes, that's geodesic coordinates centered at the point.

1:33 AM
The indices sicken my stomach! But I agree with this. To say what Ted is saying in different language (I may as well use the language I am most comfortable with in case it's of any use at all), that holds in the coordinates given by the exponential map.

I'm ok with indices :)

@TedShifrin it is tricky. I have upload the picture and then type. If it happens, someone type, I will miss it. lol

@TedShifrin yes. My question: do the first derivatives of the metric components vanish only at the point p or infinitesimally around the point p (I assume its the former reading this. But if it is only the former, isn't this trivial as the metric in p is in canonical form?)?

You're using $\eta$ where we use $\delta$.

I have to type then upload and vice versa, I can't do them in a single comment. weird.

1:36 AM
I feel like my point has not been made clear. You have not defined the notion of "the first derivatives of the metric components vanish infinitesimally around the point p". You said something along the lines of "infinitesimally around the point p means on the tangent space at p", but I would then interpret this as saying "the first derivatives of the first derivatives of the metric components vanish"
AKA, the second derivatives of the metric components vanish. But you agree with me that's not the case.
So I'm not sure what you're asking, because the language is not clear to me.

When I type, the upload button becomes grey

Only AT the point, unless your metric is flat.
Maybe just take time to work through everything I've told you, @Croco. Stop talking and work.

@TedShifrin, lol. You're right. Sometime we try to escape from our duties. See you later.

@TedShifrin But then why the statement about the first derivatives vanishing? isn't this kind of redundant/trivial (based on the fact that the metric in p is in diagonal canonical form, i.e. diag(-1,1,1,1)... and first derivatives of those diagonal elements are obviously zero...?

Forget metrics. Let me rephrase what you just said. "We know that f(0) = 1. Isn't it kind of redundant/trivial that the first derivative vanishes at zero, because the derivative of 1 is obviously zero?"
When I say $f(0) = 1$ that just tells me the value at the point. Derivatives need nearby values.
When they say $g_{\mu \nu}(p) = \eta_{\mu \nu}$ that's just a value at a point. Nothing about nearby values.
So the derivative statement is more information, just like knowing that $f(0) = 1$ and $f'(0) = 0$ is more information than just knowing that $f(0) = 1$.

1:42 AM
@MikeMiller not sure why you think "the first derivatives of the first derivatives of the metric components vanish"... the metric components g_ij are functions of the spacetime...and when taking the first derivative of the metric components g_ij we are only taking the first derivative. Not sure where the second derivative comes into play here?
@MikeMiller gotcha! This makes sense... (although the first derivative is only zero in p based on the definition)....

Sure, which is why I said $f'(0) = ...$, and not $f'(x) = ...$
I'm doing nothing more than using your own words to try to help make them precise. Notice that you have still not precised what you mean by "infinitesimally around the point p"; you told me something vague, which I tried to interpret.
You asked: "Do $\partial_\sigma g_{\mu \nu}$ vanish infinitesimally around the point p?" I asked you what that meant, and you said on the tangent space. So then what does "on the tangent space" mean?
To me, if I have a function $f: \Bbb R^n \to \Bbb R^m$, and you say "$f$ vanishes on the tangent space at $0$", I assume you mean (and this is really the only reasonable interpretation) that the Jacobian $df_0: T_0 \Bbb R^n \to T_{f(0)} \Bbb R^m$ is the zero map. AKA, all the derivatives of $f$ at zero are zero.
So you handed me $f = \partial_\sigma g_{\mu \nu}$, and I said "ok, so you're asking that its derivatives vanish too." Derivative of a derivative is a second derivative.

@eigenvalue Carroll is not that great of a book for such things like normal coordinates(in a mathematical sense). A better idea would be to try, Wald's General Relativity or Hawking-Ellis Large Scale structure of spacetime.

I hope the above helps make this discussion make sense (if not on the first read, after a little comparison to what you were reading earlier, or some other discussion). I have to go do some writing, though.

i liked sachs and wu, general relativity for mathematicians. which is not to say that i did anything useful with it. i just smiled while reading the book.

@MikeMiller In math locally means “there exists a neighborhood of p", so infinitesimally refers to the tangent space which is a point wise approach (versus a local one).
@MikeMiller yes, thanks. Will think about what you said... :-)
@SayanChattopadhyay @leslietownes Will check it out! I feel like this topic doesn't get enough attention in most books. Couldn't find any useful and detailed information about those coordinates.

1:54 AM
@eigenvalue You're saying stuff that makes little sense. These things are not functions on the tangent space. They define a zero bilinear or multilinear function on the tangent space because the derivatives vanish at $p$.

i also liked jost's riemannian geometry. it does not handle the negative aspects of the metrics that people deal with in relativity. but is good about the normal riemannian aspects of riemannian geometry

It's sophisticated, though, not for physics types.
@eigenvalue Riemann normal coords, geodesics, exponential map are all in literally every Riemannian geometry text.

@TedShifrin yes, the metric is a bilinear or multilinear function on the tangent space. But the metric components are functions on the space time?

Only in a coordinate system, of course. They change when you change coordinates. That's the whole point of tensors.

@TedShifrin The problem with riemannian normal coordinates is that a metric compatible connection is needed to define them. I don't have such a connection and was looking for locally inertial coordinates that always exist

2:09 AM
It's true for any connection that you can choose a local frame so that the connection in that frame vanishes at the point. But I'm leaving now.

what does it mean for two subspaces to be "mutual annihilators" with respect to some symmetric bilinear form
is that just a fancy way of saying orthogonal

that would be my first guess at an interpretation

yeah, has to be that
weird terminology though

is there some requirement that they span the space? "mutual" is making me wonder. but yeah, weird term.

2:28 AM
Orthogonal, not necessarily orthogonsl complements, I would say.

yeah, that's also my current understanding, though I'm just extrapolating from what I'm reading

2:46 AM
actually, each is the orthogonal space to the other (i.e. the set of all vectors orthogonal to it), but they're not necessarily complements, since they can intersect non-trivially
so maybe this is the right terminology after all lol

They can intersect if the form is indefinite or degenerate.

0

Let $\Omega : \Bbb{N} \to \Bbb{N}\cup 0$ be any completely additive arithmetic function, of which the usual definition of $\Omega$ in elementary number theory is an example. We have: a graded ring $R$ w.r.t. $\Omega$. Let $R = R_{\Omega}$ be this graded ring. $R$ is indeed a unital ring when we d...

What's up, my algebros.

3:05 AM
right, a subspace intersects its orthogonal iff the restriction to that subspace is degenerate (but that can happen even when the form on the total space is non-degenerate)

@Thorgott you're good at algebra. Any ideas on above link?

I wish I was good at algebra
but no, though I also don't have time to look in detail rn

Thx, anyway

4:12 AM
I am grinding slowly through Dummit & Foote (my abstract algebra is woefully lacking) and find some of the exercises interesting but the computation tedious. I was wondering if there are systems that will let me do matrix multiplications in fields of my choice other than the reals/complex? Sage?

a guy i knew used GAP for that

Thanks! I'll add that to my list.

I'm pretty sure Matgematica will do it, too, but I haven't done such things in ages.

it's not too late to turn back and just do matrices over the complex numbers
all of that other stuff is just people goofing off

5:09 AM
@leslietownes it passes the time!

I will have a look at Mathematica (I did a little project for Fatemen at Berkeley with it).
Thanks @TedShifrin.
I need something to do while listening to customer meetings and waiting for compiles.
Just kidding. I adore my customers.
Bummer, Sage barfed on my Ubuntu 20 install. I think I had issues with Sage before. Pity, the docs look nice.

6:01 AM
Hmm, would help if I installed the correct package.

6:14 AM
I am never sure whether an exercise is meant to illuminate something or if it just grind & find. For example, one little problem is to show that $D_8$ is isomorphic to the upper triangular members of $GL_3(\mathbb{F}_2)$. There are only 8 elements so a little grind finds an answer. But is that it? or am I missing something deeper?

6:27 AM
Well, do we know the number of isomorphism classes of groups of order 8
In particular, do we know there are two non-abelian ones?
Or orders of elements ?

Quaternions, I presume. But I guess what I am asking is if I am missing some deeper structure

Generators and relations?

I mean the $i,j,k$ ones.

Right. The quaternion group is fascinating because every subgroup is normal.

Well, for example, to do the simple problem above I just went through the matrices until I found two that match the behaviour of the generators of $D_8$.
But I just found them by grinding, no intellectual effort.

6:32 AM
Yes, I would start by thinking of orders.

I guess I am asking the Bob Geldof :Is that it?"
Oh, I see.
You can see how naive I am with groups, etc.

I taught undergrad algebra a bunch and wrote a book, but it was my weakest area by far.

Apparently Hamilton carved the quaternion symbols on a stone beside Broom Bridge on the Royal Canal outside Dublin. I have yet to visit...

Ah, shame on you.

From my early high school days I have found it impenetrable.

6:35 AM
I made it more geometric, hence all the teasing I get here.

I enjoy reading Coxeter, but I am sitting in the tour bus, not out actively exploring.

You might find Artin a lot more interesting than encyclopedic Dummit & Foote.

I will get it then. I tried Hungerford and a few others. D&F was the most accessible for right now.

Why are you doing this, anyway?

It is sort of a challenge :-). Beats watching Netflix :-).

6:38 AM
I do rings and fields before groups. They are more natural.
Quotients of commutative rings way easier than dealing with normal subgroups early on.

That makes sense. I have a lot of examples from engineering stuff, it is trying to tie it together.

Motivated by modular arithmetic and solving polynomials.
You know FFT, of course.

I am curious about results that I know in the reals/complex and how general they are.

Hmm.

Unfortunately I learn very much from the bottom up.

6:41 AM
Like?

Simple things like determinants, inverses, that sort of things.

Linear algebra all?

Jordan decompositions, etc.

All linear algebra, then?

Well, I suppose that would be what I am most comfortable with from a real/complex perspective.

6:43 AM
I've solved twin primes :| Just need to write a paper on the approach

wo

stay away from characteristic 2. that's all i can say. characteristic 2 is a childish and immature habit. the smart boy will "cut it out"

Jordan needs eigenvalues in your field. Rational canonical works for any field.

That is the sort of thing I am interested in.
Elementary, but I like elementary.

Artin integrates linear algebra throughout the book, i think that's more interesting for you.

6:45 AM
This is Artin's Algebra?

Yes. Michael.

Thanks!
Wow, it just started blasting rain here out of the blue!!!

We had it on Wed, I think.

our mockingbird just started up, which must mean it's almost 11pm.

:-)

6:47 AM
Question: how do I write / publish a paper not being in accademia

The question is whether it is deserving of publication.
Most papers are not.

Twin prime theorem.

Unlikley.

that's going to be tough. fields with some intersection with applications are easier.

Well, I did it. I think... but how do I get peer-reviewed since MSE would just downvote it to death

6:49 AM
So many people think they've done number theory. Most of it is well known or just wrong.

you mean you solved the twin prime conjecture ?

Yes
Purely algebraic approach
I just have to prove that a certain monoid is not finitely generated, and that would wrap up the proof

I am not a number theorist, but I would bet money it’s wrong .

With the freedom of coefficients I got going on in my graded ring, it seems likely that such a proof to finish it off is not far away

i forget if there is gatekeeping on the arxiv. math.GM seemed to be something of a wilderness. maybe it is different now. really hard to get people to peer review a paper in number theory. there are too many people in line, and anything touching on famously unsolved problems provokes skepticism. justifiably, i would say.

6:51 AM
the chances are low
but not 0

What if one of you in accademia helped me publish it?

People like Granville know all the failed approaches.

It is probably not worth the effort give the chances of it being correct.

I'm taking the twin prime conjecture back from analysts :)
It's an algebraic problem!

i'm no longer in academia. i would not accept a paper from me if i were. even if it were just boring stuff and not a famously unsolved problem.

6:53 AM
It is like nuclear fusion.

you have wrote the paper and you want to publish it?

there was a guy who used to mail me a schedule of prices for his proofs of the invariant subspace conjecture, riemann hypothesis, and a few other things too. i think he mostly wanted currency denominated in US dollars. those kinds of people ruin things a little bit for everybody.

It's somewhat elementary, but on the other hand involves a graded ring that is isomorphic to a ring of Dirchlet polynomials. But the way I've defined it, Dirichlet series don't enter the picture.

3 years of CS at NAU

6:53 AM
@StudySmarterNotHarder nice

I got all A's in my math and CS courses

How many years writing math proofs that are criticized?

However, 90% of my math ability is self-taught
0 years
I'm new to peer-reviewing
However many years studying

It is a social process.

6:55 AM
I can show you the graded ring post

the issue with peer review is it's unpaid work by people who are normally paid for work. it's a higher bar than upvotes or downvotes on a website.

it's highly readable and easy to digest

Are you willing to pay people for their time to read it?

Yep
100 doll hairs

This is totally opposite my knowledge and interest, so I am not going to weigh in further.

6:56 AM
That gets you a few minutes...

lol
I will start emailing Terry Tao and so on at Berkely. I'm in San Diego, so maybe I could visit their offices :D
I mean once I have written the paper using Overleaf or something

Tao is not at Berkeley. Super famous people are way busy.

I imagine they get a lot of similar stuff, people sending them their ideas.

I took a wild guess

it is difficult to get anybody to read unsolicited manuscripts. a site like math stackexchange might be a better bet, although the downvote storm is a definite possibility.

6:58 AM
Well, to totally rule out non-accademics from the picture is just plain wrong
It's quite too long for an MSE post, there's no way in H they would read through it

You may not like it, but that is a reality.

if it could be broken into pieces, maybe. this is not my area of expertise. in my former field, a lot of very difficult math could be broken into pieces, portions of which would be of interest to a wider audience than the sum total.

I will self-publish on my blog or something, then post links on MSE with an abstract + an overview maybe?

hiding the ball a bit and not saying 'twin prime conjecture' might even help, if i'm being honest. but i don't know.

Is arXiv closed to me?