point-set-topology

[sleek thicket]

lmfao

Alright

Bold claim

[obtuse meteor]

they're linear spaces

everything is true for linear shit

[sleek thicket]

yeah but we're already allowing nonlinear functors

Like the tensor power or whatever

[obtuse meteor]

ye

hmmm

for this I want to like

recover

some stuff

is there a notable definition of differentiation for a diff-enriched functor?

[sleek thicket]

I do not know

[obtuse meteor]

and does that naturally give you a vect-enriched functor?

it should

[sleek thicket]

Also I've really been thinking about Mat

[obtuse meteor]

you should also have tangent categories or some real thing of that sort

ye Mat is good

[sleek thicket]

natural numbers and matrices of the appropriate size

[obtuse meteor]

mhm

skeletons

in your closet

[sleek thicket]

I see what you're thinking

I don't know any of it though

I was just trying to look at the algebra of it

since Mat -> Mat functors are a countable amount of algebra

I figured a simpler case would be for just continuous homomorphisms GLn -> GLk

[obtuse meteor]

ye

my enriched category theory is admittedly weak

[sleek thicket]

Ahhh sick there's a reference on Wikipedia

For what brof said

[obtuse meteor]

nice

hmm

I think you can pantomime the existence of a tangent category pretty easily right

to a category C we have the same objects and just replace each hom-manifold M with T M

[sleek thicket]

I think I agree that this is well defined

But it's not Vect enriched

it's a weird kind of bundle over C

[obtuse meteor]

Composition is an operation like M x N -> K and becomes T(M) x T(N) = T(M x N) -> T K

yeah

we want it to be like a Vect-Bundle over C

in some appropriate sense

so we get a clear map T C -> C which is identity on objects and the projection T M -> M on hom-manifolds

we wish for this to be a Vect-Bundle

this seems like it should be true

instead of like homeomorphism

we're going to want to replace it by a Top-equivalence of categories

hrm IDK if this gives us the right thing

what I kinda've want to say intuitively

is that a continuous functor locks you into very rigid actions on Vect / Mat, and we can't really vary too far away from that because we're working with linear data

[sleek thicket]

well the proof for lie groups seems unlikely to generalize to this case. It relies heavily on the exponential map

[obtuse meteor]

but this is just hogwash

[sleek thicket]

I also don't really understand your intuition here

[obtuse meteor]

you might be able to take the proof for lie groups and yeet it higher

like take the basic result for lie groups

[sleek thicket]

Sure

[obtuse meteor]

and then say bc this holds it holds in the categorified version

this happens (often) enough

I haven't written anything down yet so I'm just talking

let me thonk

can you send me the lie group ref?

[sleek thicket]

But we're not working with a categorified version of a lie group, since Mat isn't a groupoid. I don't even see how to generalize the lie group proof to the smooth monoid case

And sure

[obtuse meteor]

yeah that's fair

is it true that every continuous map Hom(V, V) -> Hom(V, V) is automatically smooth? This feels untrue, but maybe functoriality like gives us an upgrade

like R -> R it's certainly not true

and Hom(R, R) = R

[sleek thicket]

Yup

[obtuse meteor]

and for some reason functoriality doesn't feel strong enough to give us this

[sleek thicket]

so the right thing to look at is continuous monoid homomorphisms R -> R

which are smooth, I believe

[obtuse meteor]

yeah those are

[sleek thicket]

like the right analogue

[obtuse meteor]

are Top-enriched things going to do that?

[sleek thicket]

wym?

[obtuse meteor]

I don't think they will

the original question was are all Top-enriched functors on Vect also Diff-enriched

[sleek thicket]

Sorry I'm confused by what this means in response to the thing about monoid homs (R,*) -> (R,*)

[obtuse meteor]

oh yeah I agree that cts monoid homs will be smooth

I was going back to your original question

bc I believe I can break it by treating it in the more basic case somehow

[sleek thicket]

yeah so at least if you have a Top enriched functor F : Vect -> Vect we get a smooth map Hom(R, R) -> Hom(R, R)

[obtuse meteor]

yeah because you get a monoid hom by pre-composing with (R, *)

you might be able to upgrade it to a Vect hom by pre-composing with (R, *) right?

in the more general case

[sleek thicket]

more basic case like focusing on continuous monoid homs Hom(R^n, R^n) -> Hom(R^k, R^k)?

that's what I've been thinking about

[obtuse meteor]

mhm

sorry I am juggling a lot of data

[sleek thicket]

yeah haha no problem

oh yeah the reason this shows up is that you can lift a continuous/smooth endofunctor to an endofunctor of the category of continuous/smooth vector bundles on X

[obtuse meteor]

ye

makes sense why you'd care then

[sleek thicket]

so they've shown up in my bundles class and I got to talking to my prof about this stuff, and if smooth = continuous = arbitrary functor

yee

the second equality "continous = arbitrary functor" is false btw

as is linear = smooth

wait umm I have just had a Realization

I think the lie group thing actually suffices for the bundles application!

[obtuse meteor]

oh nice!

[sleek thicket]

for bundles we only ever apply this to isomorphisms!!!

the transition functions

[obtuse meteor]

ahhh aha yeah

[sleek thicket]

h*ck yeah

[obtuse meteor]

So I think I just had a nice realization

[sleek thicket]

: O

[obtuse meteor]

that might allow us to fix it in the general case

not like, linear != smooth but that's w.r.t the traditional vector space structure you place on Hom(F A, F B)

But what you can instead do is like

btw you can extend like a lot of the structure by considering that like ok given two vector spaces A and B and a fixed real number

fix a real number r, then multiplication by r is a morphism r : B -> B, By composing things and by functoriality you get a square like this, where we're considering them with the cts structure

now let me launch quiver

,texit % https://q.uiver.app/?q=WzAsNCxbMCwwLCJIb20oQSwgQikiXSxbMSwwLCJIb20oQSwgQikiXSxbMCwxLCJIb20oRiBBLCBGIEIpIl0sWzEsMSwiSG9tKEYgQSwgRiBCKSJdLFswLDEsInIgXFxjaXJjIl0sWzAsMl0sWzEsM10sWzIsMywiRihyKSBcXGNpcmMiLDJdXQ==
[\begin{tikzcd}
{Hom(A, B)} & {Hom(A, B)} \
{Hom(F A, F B)} & {Hom(F A, F B)}
\arrow["{r \circ}", from=1-1, to=1-2]
\arrow[from=1-1, to=2-1]
\arrow[from=1-2, to=2-2]
\arrow["{F(r) \circ}"', from=2-1, to=2-2]
\end{tikzcd}]

[gentle osprey]

Cohomologay

[sleek thicket]

oh lol actual quiver

let me see if I understand this

[obtuse meteor]

now you can give Hom(F A, F B) a vector space structure by taking the action of F(r) \circ as the multiplication

I think if you equip it with this particular vector space structure then you actually get a linear functor

[sleek thicket]

sorry, give me a sec to thonk on that diagram

[obtuse meteor]

no problem

[sleek thicket]

so when you write F(r) are you identifying R with Hom(R, R)?

[obtuse meteor]

yeah I'm saying r is a vector space homomorphism B -> B

[sleek thicket]

oh okay, yeah

[obtuse meteor]

but in particular it should be a scalar multiplication

the point being that like there's multiple ways to give Vect a Vect-enriched structure, and you can give the second Vect in F : Vect -> Vect

a different linear structure

[sleek thicket]

yes all right F(rT) = F(r) F(T)

is what this diagram says

[obtuse meteor]

ye

[sleek thicket]

hmm

I am concerned

[obtuse meteor]

a linear structure which is determined so that F will in fact be linear

[sleek thicket]

so I don't think this can work

[obtuse meteor]

I agree let me find problem

[sleek thicket]

there are nonadditive Fs

for one

[obtuse meteor]

mhm

hmmm

I think

the reason there are non-additive Fs

might actually come down to how you enrich Vect with Vect structure but I'm not sure

like ok there's a nice canonical way to say Hom(A, B) is a vector space by just doing everything coordinate-wise with B right

[sleek thicket]

yup

[obtuse meteor]

but you could do everything coordinate-wise with B but arbitrarily decide to multiply by two

but you'll get a Vect-isomorphism there too

so it should preserve additivity of functors

but maybe more complicated things won't have a Vect-isomorphism there

so like what I'm thinking is

if you have a morphism f : Hom(F A, F B) and a real number r

you know r gives you a thing beta : Hom(A, B) -> Hom(A, B) by acting pointwise

[sleek thicket]

I think I see where I think this breaks down. what if F(r + s) \neq F(r) + F(s)?

[obtuse meteor]

and you can use the appropriate F(beta) to extend it

for real numbers r and s?

that feels valid

[sleek thicket]

Yup

[obtuse meteor]

so our R loses its field compatibility

[sleek thicket]

or any morphisms of vector spaces

[obtuse meteor]

yeah yeah you can be non-additive

[sleek thicket]

and if F is already additive to start with you can show it's linear via continuity

[obtuse meteor]

ah I didn't know that

[sleek thicket]

yeah, it's additive so Q-linear, then take limits

is the outline

[obtuse meteor]

makes sense

same as in vector space case

and preserves identity

so we're gold

ok ok

hrmmm

this is an interesting problem

[sleek thicket]

right, we have a continuous additive map Hom(V, W) -> Hom(F(V), F(W)) for each V, W, and those are vector spaces

yeah, I think so too!

oh lol the continous => smooth for lie group homomorphisms problem is in ISM

i've probably done it

[obtuse meteor]

lol

[sleek thicket]

Lee has definitely done it

smh my head

so if Vect^* is the groupoid of isomorphisms of Vect then we know continuous => smooth for endofunctors there

because such a functor is just a bunch of continuous homomorphisms between lie groups, which are smooth

[obtuse meteor]

so we're going to be smooth on the subspace of isomorphisms

[sleek thicket]

yup

[obtuse meteor]

what we wish we could say is that like

[sleek thicket]

but this gives no information about going between n and m

[obtuse meteor]

yeah

this should tell us that going from n to n

we're safe

because any homomorphism

[sleek thicket]

i wish that too

but I didn't see why

[obtuse meteor]

is close to an isomorphism

[sleek thicket]

hmm

I do not see why

[obtuse meteor]

being singular is shaky

[sleek thicket]

consider the 1d case

[marsh forge]

the 0 map: am I joke to you

[obtuse meteor]

lol

[sleek thicket]

|x| is a smooth lie group homomorphism GL(1, R) -> GL(1, R)

yeah?

and is a continuous homomorphism M_1(R) -> M_1(R)

but is not smooth

[obtuse meteor]

this is absolutely sad

[sleek thicket]

lol

hmm I think I see how to prove continuous => smooth

for lie group homs I mean

yes, my guess was right

nice

[obtuse meteor]

I very much remember we proved in class once that like being singular is shaky

maybe I am misremembering something

bc like you can just shove epsilon along the diagonal for even the zero map

[sleek thicket]

yeah, GL is a zariski dense subset

big big open

[obtuse meteor]

big boi

[sleek thicket]

singular matrices are measure 0/codimension 1 subvariety of all matrices

since it's defined by the vanishing of a single polynomial

[obtuse meteor]

yeah

[sleek thicket]

but you can't check smoothness on a large set

[obtuse meteor]

yes

very sad

this is why continuity

is good and based

[sleek thicket]

any increasing function R -> R is differentiable ae, for example

you can't do this with continuity either lol

[obtuse meteor]

you can't

[sleek thicket]

you can check continuity on a cover

but not on a single dense open

[obtuse meteor]

and so can smoothness

[sleek thicket]

right

[obtuse meteor]

but often in topological arguments it's enough to consider a dense open

the fact that smoothness isn't a topological property

makes me sad

is what I'm communicating

[sleek thicket]

lol

[obtuse meteor]

lmao

the fact that you can be smooth everywhere except a single point

[sleek thicket]

it's a topological property because all topological spaces are smooth manifolds

[obtuse meteor]

is hella unbased

[sleek thicket]

> this is what brendans actually believe

[obtuse meteor]

lol

[sleek thicket]

there are exactly two kinds of topological spaces

> smooth manifolds
> affine schemes

[obtuse meteor]

🧠

you ever met the hawaiian earring

it is making you sad

[sleek thicket]

it is p bad

I would prefer that it would not

[obtuse meteor]

it doesn't have a covering space sham

or rather a simply connected one

but all covering spaces strive to be simply connected

[sleek thicket]

tfw no cs

[obtuse meteor]

😌

also this thread is making me realize that I need to not do math immediately after taking a nap

[sleek thicket]

lol

[obtuse meteor]

I've been like

in my bed and just not in things

also sham

aren't we both supposed to be writing letters

[sleek thicket]

...yes

i am also supposed to be doing my analysis homework

I sat down to write it and everything

but then got the ping from brof

look now i have a neat fact to tell lee tomorrow because I crave the approval of my professors 😌

and hey maybe it even gets into the book!

alright this was fun, time to go back to functions of bounded variation

[obtuse meteor]

@sleek thicket would you believe that we are dumb?

[sleek thicket]

Yes

[obtuse meteor]

send V to R

[sleek thicket]

Always

[obtuse meteor]

send T to |det(T)|

[sleek thicket]

oh no

[obtuse meteor]

I literally thought of this

2 hours ago

and said "no way this is functorial"

and stopped

[sleek thicket]

this makes sense to me, unfortunately

[obtuse meteor]

before checking

lmao

[sleek thicket]

wait no

This isn't well defined

[obtuse meteor]

det is defined without a basis

[sleek thicket]

the determinant only makes sense for endomorphisms

[obtuse meteor]

oh shit

you are smart

indeed

sarahz and I

don't know linear algebra

wait wait wait

We can fix this

take operator norm

and only do it over Mat

[sleek thicket]

wym operator norm? The definitions I know aren't multiplicative

Only submultiplicative

[obtuse meteor]

oof that sucks that's true

me

[tough imp]

tf is submultiplicative

[obtuse meteor]

literally having shown operator norm is submultiplicative

me

[sleek thicket]

what do you think Alex

[obtuse meteor]

dumb

[tough imp]

Idk lol

[sleek thicket]

|AB| <= |A| |B|

[obtuse meteor]

I feel like there should be a way to fix this

[tough imp]

oh

[sleek thicket]

This is about the continuous functors thing

[tough imp]

I forgot that you have access to this forbidden <=

[sleek thicket]

It's sufficient to consider continuous functors for bundles!

[tough imp]

I mean I figured haha

so you're trying to like

show continuous => smooth?

[sleek thicket]

Yup

[tough imp]

Sham is doing stuff that's like new

novel

interesting questions

[sleek thicket]

probably not lol

[obtuse meteor]

okay hm

[tough imp]

I spent 8 hours writing up abelian category bs

[obtuse meteor]

if you have A an n x m matrix

[tough imp]

🥴

[obtuse meteor]

is |AA^T| multiplicative

[sleek thicket]

chm why don't you think about manifolds as locally ringed spaced

*spaces

[obtuse meteor]

I don't think so

this is cringe

[sleek thicket]

This is what I was doing last year

It isn't Faye

[obtuse meteor]

there has to be a way

I refuse to believe

[sleek thicket]

lol

[tough imp]

Sham

Math overflow time

jk

I think this is something that would be MO level tho

[sleek thicket]

I have

[tough imp]

probably

[sleek thicket]

Googled

[obtuse meteor]

this is so cursed

[sleek thicket]

There's a conjecture by that one super active dude that continuous functors are like a direct sum of "schur functors"

And so smooth

[obtuse meteor]

brendon

why did you like

[sleek thicket]

lol

[obtuse meteor]

curse me with this question

[tough imp]

what is the super active dude

[obtuse meteor]

I hate it

[sleek thicket]

Lmfao

[tough imp]

lmao

[sleek thicket]

I forget his name

[tough imp]

-_-

lmao

I mean

I know a few very active ppl

but they all do AG

[sleek thicket]

Qiaochu Yuan

[tough imp]

oh lmao

I thought maybe him but

went "naaaah"

[sleek thicket]

https://math.stackexchange.com/a/3887358/408287

this is the answer that made me realize my complex counterexample gives a real one

yeah this question has been asked before

https://math.stackexchange.com/q/3016674/408287

but no resolution

if only smooth monoids had an exponential map....

hmmmm apparently lie monoids still have lie algebras..........

oh fuck jonathan beardsley is on the homotopy theory chat

i iwsh i had not emailed him

because it would be a dick move to ask him in the chat now

instead of waiting for a reply

[shut moat]

you know significantly more math than me so if you couldn't figure it out I'll just move on ig 😂

[gritty widget]

Is there anywhere I can read about group objects

Or group theory on group objects in general categories

I’m just interested in seeing how much from group theory we can carry over to group objects

[sleek thicket]

lol nvm he replied to my email this morning

[sleek thicket]

I didn't think about it very hard!! Keep at it

[marsh forge]

I think that the reason for group objects is generally a theory of group (object) actions which comes up in a lot of places

e.g. the right notion of a G-Space X is a topological group G w continuous GxX->X action

I dont think that the majority of big group theory ideas apply in this generality

[gritty widget]

Actually why are group actions interesting

I’ve learned about plenty of examples of things with group actions like principal g bundles

But I couldn’t succinctly say why group actions are interesting

[marsh forge]

group actions like

prescribe a set of important symmetries

this is how i think abt it

for example take S^1

this is clearly a very symmetric object

but it has a nice Z/2 action which just fixes the top and bottom and flips everything else along that axis

this is a Z/2-space version of S^1

In some sense, we are saying that while S^1 has lots of symmetries, this specific symmetry is the one we want to care about

this data gets encoded in morphisms as well: an equivariant morphism here must respect this flipping Z/2 action

but can ignore other symmetries

(another obvious way of saying this is that a group action is basically a map G->Aut(X) which is basically just a subgroup of Aut(X) so a G-Space X is really just X and a subgroup of Aut(X) described by G)

[long coyote]

Let $F$ be any family of subsets of a space $X$. Show that there is the smallest
topology $\tau_F$ on X containing $F$. i have googled the question, but i couldn't understand the meaning of smallest topology

[gentle osprey]

亜城木 夢叶

[marsh forge]

so you can think of topologies as subsets of PX the powerset of X

a smallest topology with property P is a topology that is a subset of every other topology w property P

[long coyote]

do you mind give me an example

[tough imp]

The topology {X,empty set} is the smallest topology that is a topology

[long coyote]

ok, now i see, thank you so much

[sweet wing]

usual construction for like "smallest/lower bound" is jus intersect everything and hope it works

[tight agate]

Does anyone know if there are expository articles on Adams' original proof of the hopf invariant one problem?

the secondary cohomology operations one

not the k-theory proof

[karmic smelt]

it's good channel for differential equation

?

[river granite]

depending on what exactly your question is about, #multivariable-calculus, #advanced-analysis or #dynamical-systems

might be better fits

[gritty widget]

inb4 they start talking about stable manifolds

[river granite]

that'd be fun at least

[summer jolt]

Suppose $p \in U \cap V$ (charts on manifold) and let $T_pM$ be the tangent space at $p$. A tangent vector at $p$ can be written in two different basis:
$$
v_p = v_p(x^i)\left(\frac{\partial}{\partial x^i}\right)_p;
v_p = v_p(y^i)\left(\frac{\partial}{\partial y^i} \right)_p;
$$

where the $x^i$ are the local coordinate functions at $p$. Now my book claims that from here it follows that:
$$\left(\frac{\partial}{\partial y^i}\right)_p = \left(\frac{\partial} {\partial y^i} \right)_p (x^j)\left(\frac{\partial}{\partial x^i}\right)_p$$

However, I'm struggling to see why would this be the case. Does someone have any idea?

[gentle osprey]

snypehype46

[gentle osprey]

TTerra

[gritty widget]

(i'm going to assume that you meant to write d/dx^j in your second displayed equation)

@summer jolt

it's just linear algebra

if you're familiar with differential forms, it's the same thing (literally) as applying dx^j to both sides of the second displayed equation

[gentle osprey]

TTerra

[ivory dragon]

[gritty widget]

what

[tough imp]

[river granite]

[gritty widget]

im just partaking in some physicist's diffgeo

[shut moat]

[gritty widget]

my free speech is being oppressed

[gentle osprey]

shamrocK^n(X)

[small obsidian]

@spring geode

We can do here if you want

The easier rule to do is to show that the intersection of any two open sets is also open

[spring geode]

yes sorry

i was helping a guy

how do i show that rule

[marsh forge]

does anyone know of a good spectral sequence computation that converges on page 3

i built a small visualizer for multi-page SS and want to work out an example

[small obsidian]

I'd say just take any two. What's:
(a, inf) ∩ (b, inf)?

[spring geode]

umm

inf?

no wait

[small obsidian]

It of course depends on which is larger, a or b

[spring geode]

yeah

[small obsidian]

Just pretend b is larger wlog. Then that intersection is (b, inf)

Which is an open set!

[spring geode]

[small obsidian]

So any intersection of two is an open set and we win again

[spring geode]

pretty cool

👍

[small obsidian]

But the generalized union is a bit tougher

[spring geode]

what bout (a, inf) union (b, inf)

ye i see how this is tougher

😦

[small obsidian]

The problem is that we need the infinite union to be an open set as well

So like we can take finite unions pretty easy. (a, inf) U (b, inf) is (a, inf) if a is smaller

And (a, inf) is open. So, finite unions are open

[spring geode]

oh

i see

there's 1 more part of the question tho

"In this topology, what is the closure of a set A subset R?"

o frogot what closure of a set is

[small obsidian]

Just include all of the limit points of A

The closure of (a, inf) is [a, inf)

[gritty widget]

smh not including infinity

[obtuse meteor]

another way to say closure is "smallest closed set containing A"

[marsh forge]

I think this is wrong

the closure is not the closure in R

its bigger

@spring geode @small obsidian if the topology is the one consisting only of sets (a,inf) then the closure of (b,inf) is not [b,inf)

you really need to use the defn of a limit point here

[obtuse meteor]

yeah I didn't see this context

[small obsidian]

Oh yeah, the open sets are just (a, inf)

[marsh forge]

dont spoil it

this is an essential part of the exercise

[obtuse meteor]

ye

I would use the smallest closed set defn

[marsh forge]

er

yeah i suppose that works

I think the direct approach is easier

characterize the limit points of (a,inf)

[obtuse meteor]

to me that is the direct approach lol

[small obsidian]

Oh haha, yeah I assumed an incorrect answer here

[marsh forge]

closure has an explicit construction

[obtuse meteor]

you can take intersection of all closed sets containing

[small obsidian]

Notably, [a, inf) isn't even closed so I should have realized I was wrong

[marsh forge]

im aware

[obtuse meteor]

ye I know you are max I was just like

commenting for the person doing the problem

[marsh forge]

(proving this coho's way requires you to characterize the closed sets which is not hard)

[digital glacier]

Does anyone here know where the geometric langlands has gotten to lately?

[cedar pebble]

@digital glacier Yes, it's been quite active lately. I think the biggest things happening right now are related to the recent paper of Arinkin-Gaitsgory-Kazhdan-Raskin-Rozenblyum-Varshavsky (https://arxiv.org/abs/2010.01906) and work of Fargues-Scholze on the geometrization of the local Langlands correspondence for p-adic fields

[gritty widget]

that arinkin guy has got a lot of last names huh

[cedar pebble]

😛

nah this is just a really long paper with a lot of big name authors and it's one of the most precise forms of categorical geometric Langlands that has been given so far

Since then there have already been two more papers by the same authors that prove some of the claims in the main paper

very exciting stuff

The work of Fargues-Scholze is the most exciting to me at the moment, they are really having to develop some new fountational machinery (condensed mathematics) to deal with some of the issues surrounding geometrizing the local Langlands correspondence for p-adic fields

There are some speculative ideas about how to geometrize the global Langlands correspondence for number fields but this is still extremely mysterious and a long way away form being realized

(nobody has a good idea of what "global shtukas" and their moduli are supposed to be)

[willow spear]

what doe sit mean iof something is open in topology

im still trying to wrap my head arond this lol

[marsh forge]

i think you should focus on the metric space inuition

points that share a lot of open sets are, in some sense, 'close'

[willow spear]

so what is teh relation between metric psaces and to[pology

[marsh forge]

a metric on a set induces a topology

the basis is given by the balls of all radii

[willow spear]

so if im new to point set topology, what should i start learning

is hatcher's notes a good place?

[marsh forge]

I normally think that a little analysis can be a gentle introduction

but yeah, hatchers notes works

[willow spear]

is that aalone fine, do I need to supllment it with something

topology is hard to wrap ur midn around first lol

[marsh forge]

uh if you find it difficult you can supplement w like janich

or munkres

[cedar pebble]

@willow spear so strictly speaking, there is some choice involved in which subsets are called open in a given topological space

between two extremes, given a set X you can consider the discrete topology on X where every subset of X is considered to be open, or you can consider the codiscrete topology on X where only the empty subset and X itself is considered to be open

a topological space, after all, is a set X along with a choice of subsets of X which are considered to be open, which are required to satisfy some closure properties (the empty subset and X itself are required to be open, arbitrary unions of open subsets are required to be open, and finite intersections of open subsets are required to be open)

Now if you have a metric space X, so you have a set along with a notion of distance between any two points, then you can obtain a topological space from this: the open balls around each point generate a topology

(that is, every open subset can be written as an arbitrary union of open balls around different points)

But such topological spaces are quite special: these are the metrizable topological spaces, and they satisfy certain very nice properties that not all topological spaces satisfy

(if you're looking for a place to read about this, Munkres' "topology: a first course" is good choice, as is Lee's "introduction to topological manifolds")

[west brook]

What is your favorite polytope?

[marsh forge]

circle

[west brook]

Circles aren’t polytopes.

[marsh forge]

they are in this channel

[west brook]

So?

[marsh forge]

no, i mean in this channel, circles are polytopes

[west brook]

No, they aren’t.

[marsh forge]

Yes, they are.

[west brook]

https://polytope.miraheze.org/wiki/Polytope
None of these definitions admit circles.

[gritty widget]

[marsh forge]

see this circle is a polytope

[west brook]

That is a polytope but not a circle.

[gritty widget]

that's a circle.

[marsh forge]

It's a circle in this channel.

[west brook]

Why?

[marsh forge]

its homeomorphic to a circle

this is a topology channel

[west brook]

It’s also a geometry channel.

[marsh forge]

thats not geometry i dont see any rings

well i see one ring, the circle

[gritty widget]

brb posting in #old-suggestions "rename #point-set-topology to # topology"

[marsh forge]

i support this motion

[west brook]

So then where can polytopes be discussed? They’re too advanced for #geometry-and-trigonometry.

[marsh forge]

are they

i was mostly memeing

[west brook]

Yes

[marsh forge]

i expected you to get the joke

you're allowed to discuss polytopes

[chrome dew]

woah aren't you a bit old to be making juvenile jokes 😏

[marsh forge]

?

[gritty widget]

[west brook]

Is anyone here familiar with the operation of blending?

[cedar pebble]

favorite polytope is the K5 associahedron polytope

I have a zome model of it in my apartment

[sleek thicket]

Max that's clearly not a circle

You're mixing up simplices and spheres

very different objects smh

i am also an associahedron fan

[obtuse meteor]

what is the associahedron for again?

[marsh forge]

its clearly a simplicial circle

[sleek thicket]

I think of it as an operad thing

[obtuse meteor]

I should at some point learn about operads

[sleek thicket]

So like, say you have a topological space X

Then we have paths in X

Eg say you have three paths p, q, r

[obtuse meteor]

ye

and you have like the associahedron

[sleek thicket]

how can we compose p, q, r ?

[obtuse meteor]

which tells you all the homotopies

between different compositions

is that the idea there?

[sleek thicket]

Wait I just wanted two paths lol

Anyways yeah

[obtuse meteor]

and this should take place in the loop space

[sleek thicket]

yup

Each point in the nth associahedron parameterizes an n ary operation on the loop space

[obtuse meteor]

so in Ω(X) some sort of simplex

each point of which is a possible composition

[sleek thicket]

So the assosciahedra assemble into an operad and this shows Ω(X) is an algebra over that operad

Hmm, I'm not sure what you mean

By that last statement

[obtuse meteor]

so like, we're take Ω(X), and what we can do if we have three points p, q, r is have two vertices p(qr) (pq)r, representing the two associatifdsaflsa things

and then we have a line between these points, a simplex

[sleek thicket]

right

[obtuse meteor]

and this simplex lives in Ω(X) and exhibits the homotopy

[sleek thicket]

So we actually have three vertices

[obtuse meteor]

maybe (pqr) where you weight each of them equally?

with 1/3

[sleek thicket]

No sorry I think I'm wrong

[obtuse meteor]

hmm, no I think it should be three vertices

[sleek thicket]

I was thinking of doing p and r for 1/3 time

[obtuse meteor]

because like you're splitting the unit interval into three pieces

x1 x2 x3

and x1 + x2 + x3 = 1

p gets x1 time q gets x2 time r gets x3 time

it gives you the interior of a 2-simplex with three vertices yeah

[sleek thicket]

I am confused and I'm thinking about why

This feels off to me but I can't pin down why

I think it should be a line segment

Where the midpoint is to traverse all for 1/3 of the time

[obtuse meteor]

yeah that also makes sense

[sleek thicket]

Yeah I'm not sure, sorry

[obtuse meteor]

it's okay!

[sleek thicket]

I don't know this stuff super well, my work with operads was mostly like, algebraic

But yeah the idea is that they form an operad parameterizing ways to compose paths

An operad being a sequence of things X1, X2,... with some composition operators where Xn is supposed to represent n ary operations on something

Analogous to how a group represents automorphisms of something

[obtuse meteor]

hmm

[sleek thicket]

And algebras over an operad actually realize those as concrete n ary operations, in the same way a G-set represents the elements of a group as actual automorphisms

anyways yeah, Ω(X) is an algebra over the assosciahedron operad

Is what I was trying to say

[obtuse meteor]

sarah tells me 1 object multicategories lmao

[sleek thicket]

this is also true

[obtuse meteor]

which means I can draw the axioms easily

[sleek thicket]

I mean like, you have n ary operations

but only on one thing

[obtuse meteor]

yeah it makes sense ^^

can say more?

[sleek thicket]

Oh me?

About anything in particular?

[obtuse meteor]

just operad stuff. It's interesting

[sleek thicket]

It's cool! So like, one example of an operad is the one with a single point in each degree

[tidal cedar]

There's a good PDF on the associahedron that Ultra posted a while back

[sleek thicket]

Then an algebra over this operad in Set is a set X with a map {pt} -> Hom(X^n, X) for each n

And these maps have to respect composition

What kind of a thing is this?

Also I have a meeting at 6 so I can't go into anything super deep

[tidal cedar]

operads-talk.pdf

Ah here's the Ultra one

[obtuse meteor]

that's alright

this is neat

I'll take a look at that pdf

[sleek thicket]

I used this for my first introduction https://www.semanticscholar.org/paper/AN-INTRODUCTION-TO-OPERAD-THEORY-SAMCHUCK-SCHNARCH/18263aba4731dd91725379b8b7755d0ff89136e7

I liked it

Math3ma also has good high level blog posts on operads

for me they came up in the context of braid groups, since braided monoidal categories are algebras over a certain operad in Cat

[tidal cedar]

I have yet to really apply them but they seem very cool, coloured operads also apparently give you "types" for the operations

[sleek thicket]

We wanted to find an analogous notion of braided monoidal category for the singular braid monoid, although it turns out operads can't work here

But we ended up with something called a PROB that does work

[tidal cedar]

E_k operad?

[west brook]

My favorite polytope is the 30-naq.

[obtuse meteor]

hmm now I'm thinking about braided monoidal categories

jumping to #category-theory to make the geometers not get mad at me

[sleek thicket]

I'm not sure what you're asking fiona? I don't think braided monoidal categories are algebras over the Ek operad

Iirc it's like the fundamental groupoid of the E2 operad actually

@obtuse meteor braided monoidal categories are knot theory, so basically low dim topology :^)

[tidal cedar]

Ohh

[sleek thicket]

For a reference check "Homotopy of Operads & Grothendieck-Teichmüller Groups" by Fresse

[tidal cedar]

Hmm

[obtuse meteor]

lol

[tidal cedar]

NLab, if it is to be trusted, says the braided monoidal categories are categories that are algebras over the little 2-cubes operad

[sleek thicket]

Hmm, I might be wrong then

although nlab might be saying "lol topological operads are the same as operads in groupoid" or something

Like you gotta make sense of an algebra in Cat over a topological operad

[tidal cedar]

I think that's what NLab is saying

[sleek thicket]

And the way I would do that is to take the fundamental groupoid

I think nlab's statement is confusing

anyways, i gtg

fresse is very scary imo but should answer questions about this stuff

[tidal cedar]

Thanks!

[cedar pebble]

Oh god this is one of my favorite books

[cedar pebble]

This is close, braided monoidal categories are braid theory. Given an object V of a braided monoidal category C and a (parenthesized) braid B on n strands you can build a morphism V^{\otimes n}->V^{\otimes n} in C. Moreover the axioms of braided monoidal categories are more or less the minimal set of axioms that make this morphism only depend on the isotopy class of B. That is, this gives you invariants of (parenthesized) braids.

When you want to do knot theory you need a bit of extra structure. If you think about what kind of things you're allowed to do in a braided monoidal category, the braiding and the associator are what allow you to build parenthesized braids

But if you want to build knots, you also need to be able to create and annihilate strands; this is precisely what you're able to do in a rigid braided monoidal category where you have a notion of duality

(there are other refinements of this as well; if you care about framed knots then you need to work in something like a ribbon category, so you have extra structure that lets you twist the framing around strands)

Incidentally I think this is one of the clearest ways to understand how quantum knot invariants are defined. Whatever a quantum group is, such a thing gives you a rigid braided monoidal category Rep(U_q(g)) which is linear over C[[\hbar]]. Given a knot K the rigid braided monoidal structure in Rep(U_q(g)) gives you a linear morphism from the unit object to the unit object, that is a linear morphism C[[\hbar]]->C[[\hbar]]. But this is just given by multiplication by a formal power series in C[[\hbar]], and that's your quantum knot invariant (if you do this for g=sl_2 you essentially get the classical Jones polynomial of knots)

[cedar pebble]

Yes, this is correct. Since this only matters up to homotopy, braided monoidal categories are algebras over E_2 operads (so this means things like little 2-cubes, little 2-discs, or perhaps more applicably to the braid theory picture, the parenthesized braid operad)

There's a related notion called "infinitesimally braided monoidal categories" which are algebras over the so called parenthesized chord diagram operad

Drinfeld associators (these are one of the main topics of Fresse's book, and are very closely related to the Grothendieck-Teichmüller group) give you a way to turn infinitesimally braided monoidal categories into braided monoidal categories

(Drinfeld associators have various definitions, but the relevant definition here is that they are equivalences PaB^->PaC^ between a certain completion of the parenthesized braid operad and a certain completion of the parenthesized chord diagram operad. Then, more or less by definition, an infinitesimal braided monoidal category can be regarded as a "representation" of PaC^, and you can precompose this with a Drinfeld associator PaB^->PaC^ to obtain a "representation" of PaB^, regarded as a braided monoidal category!)

(actually the main result of Fresse is that the Grothendieck-Teichmüller group GT is the group of homotopy automorphisms of PaB^, and that the "graded" Grothendieck-Teichmüller group GRT is the group of homotopy automorphisms of PaC^. So this immediately gives the structure on a GT-GRT-torsor on the set of Drinfeld associators)

okay enough chat spam for now I love this stuff to death though

[sleek thicket]

yeah this is how I got into this stuff, my reu this summer was trying to generalize this process to find singular braid/knot invariants

[cedar pebble]

Oh that's right! I remember seeing you post about this at some point on twitter

Did anything ever come of that project?

[sleek thicket]

Beardsley gave a talk at an operads conference and we're working on a paper

I'd talk about it but I'm in class

[cedar pebble]

Maybe another time then, I'd love to hear about it someday. Glad to hear that you're both making progress.

[obtuse meteor]

@cedar pebble me and sarahz were doing knots in homotopy.io for fun a few days ago, you can get some really pretty pictures

https://estrogen.fun/i/auwl.png

white and black are dual objects here

[cedar pebble]

Oh that's cool!

Is there anything like this that supports adding parenthesizations to the strands (that is, keeping track of the associators, not just the braiding and duals?)

[obtuse meteor]

hmmm. I'm not sure

homotopy.io keeps track of every slice

so I think internally it's keeping track of the associators

this is a projection of the full 3D model

[cedar pebble]

(there's a certain knot invariant which requires you to keep track of this, I'm sure someone has written software to handle this but I'm not sure)

[obtuse meteor]

to a slice omitting any 0-cells

it's an unfortunate fact that in order to do braiding in homotopy.io you need to make your objects endo 2-cells on the identity of the identity of a 0-cell C

[cedar pebble]

aaaaaa

[obtuse meteor]

https://estrogen.fun/i/bfkq.png

Also, related thingy I was talking about in #category-theory

[cedar pebble]

mhm!

[obtuse meteor]

I think there might be a nice notion of "braided" (may be slightly weaker than the usual def) for bicategories

[cedar pebble]

yea I was thinking about this

[obtuse meteor]

namely you can require that all left and right adjoints agree

since this is one of the things that braiding gives you in the delooping

I don't think that's an if and only if in the case of braided monoidal though

[cedar pebble]

right I know there are versions of braidings for bicategories

[obtuse meteor]

so it's a weak generalization

[cedar pebble]

with an extra step in the middle between braided and symmetric

[obtuse meteor]

I've never heard of any

ah interesting

this has to do with the like

k-tuply monoidal categories

stuff

[cedar pebble]

yup!

people have worked out coherence theorems in this setting as well

[obtuse meteor]

is it true that if you're a single object k-tuply monoidal n-category then somehow you're delooping a (k-1)-tuply monoidal (n-1)-category

up to whether or not these are the right indexes lol

[cedar pebble]

Something like this yea. For instance a concise definition of a monoidal n-category is an (n+1)-category with one object

[obtuse meteor]

yeah

[cedar pebble]

iirc the k-tuply monoidal structure passes through this delooping procedure

[obtuse meteor]

ah I think it should be like

(k+1)-tuply monoidal (n - 1)-category

[cedar pebble]

just that at each step there will always be one more stage before the stabilization

[obtuse meteor]

you shift up your coherence

because you lose like

cells to not be coherent in

[cedar pebble]

yea something like this

been a while since I did this yoga

[obtuse meteor]

Lol

[coral pawn]

I posted this in algebra channel too, so let me know if this counts as spam and I'll remove it from here

[wanton marsh]

what's the topology on C_E and C_F ?

in case K is not C

[cedar pebble]

you usually want to be thinking about the Zariski topology

The correct replacement for finite covering spaces when you're working over arbitrary fields (or even over arbitrary commutative rings) are finite etale morphisms

So the correct picture here is that the curves are schemes (or if you like, varieties) with the Zariski topology and the covering map is a finite etale morphism between them

(just as the fundamental group classifies finite covering spaces of topological spaces, there is something called the etale fundamental group that classifies finite etale covers of schemes; taking the usual topological fundamental group for the Zariski topology is the wrong thing to do here)

(incidentally if X is an integral normal scheme (the curves you're thinking of are examples of such) then the etale fundamental group of X is isomorphic to the Galois group of a maximal unramified extension of the function field of X. So we're back to the beginning of this dictionary: finite etale covers of algebraic curves (which you can think of as finite covering spaces in the usual sense) correspond to finite unramified extensions of the function field)

[sleek thicket]

The frame bundle, discuss

[cedar pebble]

I think my favorite fact about framings is that S^n admits a framing iff n=0,1,3,7 (corresponding to the four normed division algebras)

[sleek thicket]

yeah that result is awesome

I would like to learn k theory in the near future

And I've heard that there's a nice proof using that

[cedar pebble]

(also stable homotopy groups of spheres but I didn't want to get too pretentious lol)

oh cool, I've been totally in the dark about proofs of this

[long coyote]

Show that $x$ lies in the closure of $S\subset \mathbb{R}l$ if and only if there is a sequence ${x_n}{n\in\mathbb{N}}\subset S$ such that $x_n\geq x$ and $\lim_{nto\infty}x_n=x \in\mathbb{R}$.

since $\lim_{n\to\infty}x_n=x\in\mathbb{R}$ where ${x_n}\cup{x}\subset S$ and $x_n\geq x$ for all $n$, we have $B_{\epsilon}(x)\cap S\neq\phi$ for all $\epsilon>0$. That means $x\in S$; thus, $x\in\overline{S}$.

how to do the other direction

[gentle osprey]

亜城木 夢叶

[sleek thicket]

I don't understand your proof for the first direction

It's not true that any element with a sequence in S converging down to it is in S, that would mean all sets are closed

Is this the lower limit topology?

[long coyote]

yes

but $x_n\geq x$

[sleek thicket]

I don't understand how you say ${x_n} \cup {x} \subset S$

[gentle osprey]

shamroc$\overline{k}$

[long coyote]

that is not right, my bad

[sleek thicket]

also this is the Picard group right?

my ag is weak

can someone tell me why this matter, either for manifolds or (at a high level) in AG?

[cedar pebble]

Right this is very closely related to the Picard group in AG

the group of isomorphism classes of complex line bundles on M is isomorphic to H^2(M,Z)

The way to see this is the exponential exact sequence: complex line bundles on M are C*-principle bundles on M

which are classified by H^1(M,O*_M) (this is exactly the topological version of the Picard group, which is verbatim the same group if M is replaced with a scheme)

we have the exponential exact sequence 0->Z->O_M->O*_M->0

so you get a corresponding long exact sequence in cohomology

$H^1(M,\mathcal{O}_M)\rightarrow H^1(M,\mathcal{O}^\times_M)\rightarrow H^2(M,\mathbb{Z})\rightarrow H^2(M,\mathcal{O}_M)$

[gentle osprey]

nGroupoid

[cedar pebble]

the connecting homomorphism in the middle is an isomorphism since H^1(M,O_M)=H^2(M,O_M)=0

Another way to think about this is the following: the classifying space for complex line bundles is the space BU(1). But this is equivalently K(Z,2) (both are actually modeled by CP^\infty). Isomorphism classes of complex line bundles on M correspond to homotopy classes of maps M->BU(1), that is homotopy classes of maps M->K(Z,2), that is classes in H^2(M,Z)

Now we can say a similar thing for real line bundles: the classifying space for real line bundles is the space BO(1)=B(Z/2Z). But this is equivalently K(Z/2Z,1) (both are actually modeled by RP^\infty). Isomorphism classes of real line bundles on M correspond to homotopy classes of maps M->BO(1)=B(Z/2Z), that is homotopy classes of maps M->K(Z/2Z,1), that is classes in H^1(M,Z/2Z)

With this in mind, V^1_R(M) and V^1_C(M) (in the above notation) are Abelian groups with respect to tensor products of line bundles, the unit element is the trivial line bundle, the inverse is the dual line bundle

[gritty widget]

is this lee's new book

[cedar pebble]

wait Lee has a new book?

[sleek thicket]

Oh yes lol

Not sure if I'm allowed to post screenshots

[cedar pebble]

woah it's that new

[sleek thicket]

And yeah NG I'm in his bundles course

It's a work in progress

[cedar pebble]

nice!

wait btw are you in Jarod's moduli course?

[sleek thicket]

Nope

[cedar pebble]

I don't think I can keep attending now that my classes are starting but I enjoyed the first few

[sleek thicket]

Yeah he is much more ag brain than me

I sort of gave up in our AG class last year

After not being scheme brained enough

[cedar pebble]

oh hey Chmonkey

[sleek thicket]

And did a Hatcher thing instead

[cedar pebble]

nice

being Hatcher brained is also respectable

[rugged swan]

woke

[sleek thicket]

Someday I will learn algebraic geometry

[tough imp]

lol

Chmonkey brain

[rugged swan]

🐒

[sleek thicket]

[tough imp]

[sleek thicket]

I hate this emote

[tough imp]

F U

[cedar pebble]

[rugged swan]

[sleek thicket]

No I meant

[tough imp]

Oh

[sleek thicket]

I like

[rugged swan]

the only emote we should post

[cedar pebble]

I think Chmonkey actually did more of an honest job doing Hartshorne exercises than I have and it terrifies me

[tough imp]

lmao

[sleek thicket]

He is very diligent

[tough imp]

I mean

I want to do cool stuff now

not just spend my days going

"ugh, gotta do II.6.4 now"

or some shit

There's too much cool shit

and idk how anyone learns all the cool shit

[cedar pebble]

every day I regret doing cool shit too early

[tough imp]

because when you learn more

the more you learn exists

[rugged swan]

I don't like Hartshorne. I wonder how did you manage to read it, it has no examples

[tough imp]

The rate at which I do stuff is slower than the rate at which I learn about cool stuff I want to learn

Idk

I don't like it either

but I've spent too many hours with it to like

not say I like it

[rugged swan]

haha

[cedar pebble]

yea it's not a friendly book

there are much better books if you want examples

but the exercises from it are very very good

[tough imp]

The thing I don't like is

his examples usually involve like intuitively working with these things as if they're varieties

but then there's no translation to how to do it rigorously with schemes

[cedar pebble]

aaaaaa

[rugged swan]

I've only read the begining and it's still frustrating to have a definition of morphism which comes from nowhere

[cedar pebble]

there's a pretty famous etale cohomology book I worked through a few summers ago that has precisely 7 exercises

[tough imp]

lol

[cedar pebble]

exercise 1 is on page 150 or so

[tough imp]

" a few summers ago"

[rugged swan]

the only holy book is ega

[tough imp]

You mean you didn't learn etale cohomology through DLitt's class?

[rugged swan]

[tough imp]

EGA is

I've tried to read sections like

twice

[cedar pebble]

no I think Litt's course was the third time I learned etale cohomology

[tough imp]

My thing is anytime I look at EGA

[rugged swan]

[cedar pebble]

which was kinda irritating because as a third pass I just wanted to like

[tough imp]

I go see how far back the tree of lemmas I need is

and it's like 6 deep

and I jsut say nah

[rugged swan]

xD

[cedar pebble]

actually see and understand all the gory details of the proofs of the big theorems

which the course didn't really do

(and a course really shouldn't do, to be fair)

[rugged swan]

but it's well written tho

[cedar pebble]

(nobody should be subjected to that haha)

[tough imp]

big theorem as in like

Weil conjectures?

[cedar pebble]

oh no I'm quite happy with the proofs of Weil I and Weil II actually

I mean more like

proper base change, smooth base change, Kunneth, Poincare duality, cycle class maps...

the proofs are just really irritating

[tough imp]

I don't know what any of that is 👍

[cedar pebble]

the base change theorems are just some technical tools you need
Kunneth is much like the Kunneth formula in topology that relates the cohomology of a product of spaces to the cohomology of the individual spaces
Poincare duality is much like Poincare duality in topology e.g. for smooth compact manifolds of dimension d that relates H^k to H^{d-k}
Cycle class maps is much like the fact in topology that a submanifold defines a cohomology class...

[sleek thicket]

(poincare duality and the kunneth formula are things that show up for singular cohomology on manifolds, presumably NG is talking about some kind of generalizations to schemes and etale coho)

[cedar pebble]

yea much of the point for \ell-adic cohomology is you get a cohomology theory for varieties say over finite fields that behaves like and satisfies essentially all the same kinds of theorems that singular cohomology for manifolds does

[tough imp]

How much work is needed to show that a scheme is a sheaf on the etale site?

Like, showing you can glue maps from an etale cover

I glanced at Vistoli's descent notes which I want to read at some piont

but idk if it's the proper time to due to everything I have on my plate already

[cedar pebble]

it's a little annoying to prove

iirc Daniel's course had a proof of it

[rugged swan]

ega has a proof

[tough imp]

blech

[cedar pebble]

actually Daniel proved fppf descent which is stronger

[tough imp]

Does it require a lot of machinery to build up tho?

[cedar pebble]

not really

[tough imp]

If it's just an isolated hard proof

then I'm fine just working with it

I just don't want to have to go through 40 pages of building stuff up to prove it

[cedar pebble]

e.g. there's a pretty short proof of this at the beginning of these notes: http://pub.math.leidenuniv.nl/~zomervruchtw/docs/etdesc.pdf

[honest narwhal]

Going through those proofs in full detail is a tall order

[cedar pebble]

it's essentially reduction to the affine case and then a really clever homological algebra trick

this is essentially the same strategy that appears in lectures 4 and 5 of Daniel's course

(and the machinery you need is all in lecture 3. So it's not actually a dreadful amount of background)

[sleek thicket]

quote from my AG prof will never stop being relevant https://twitter.com/grassmannian/status/1191103487729233920?s=19

Brendan TKOIBP (@grassmannian)

“Theorem: Cool AG thing or whatever
Proof: Reduce, reduce again, reduce until it's some dumb algebra fact”

Twitter · Nov 3, 2019, 9:22 PM

[cedar pebble]

LITERALLY

[sleek thicket]

just go to affines ezclap

[cedar pebble]

when I took a course on moduli last year

did some complicated reduction to the affine case

professor stared at it

got frustrated

[gritty widget]

dox

[cedar pebble]

grumbled

[sleek thicket]

lol

[cedar pebble]

"idk just look at Atiyah MacDonald"

"it's probably in there somewhere lol anyways moving on"

[sleek thicket]

@gritty widget you're gonna find out who I am anyways when we both get the fields institute thing

and crush the symmetry conjecture this summer

[gritty widget]

im going to write all the emails i need to tonight

currently working out a shitty residue integral

can't wait to put together a completely blank cv

[sleek thicket]

lmfao

I submitted my app last night

Or maybe two nights ago

One prof has replied saying he submitted the email already

[cedar pebble]

looking forward to postdoc applications someday when everyone's "conferences attended" section is INSANELY inflated

[sleek thicket]

[gritty widget]

i need to do the cv bit and the references bit

[cedar pebble]

"I went to 20 conferences in 2020"

[gritty widget]

good luck, shamrock!

[cedar pebble]

"how many were on zoom that you didn't have to apply for"

"uh"

[sleek thicket]

You too!

also I realized both of my letter writers are named john

[cedar pebble]

when I applied for NSF two of my letter writers were Dan(iel)

[sleek thicket]

lol

[cedar pebble]

another friend that applied to NSF also had two Dans

incredible world

[honest narwhal]

Now I'm sad that is sully rather than Dan

[cedar pebble]

(it was grad apps Jeff )

oh my god

[honest narwhal]

I was gonna make a joke and it doesn't work anymore

Oh lawd

[cedar pebble]

[gritty widget]

i have a dan emote server

[sleek thicket]

incredibly based

Put that on your cv!

[honest narwhal]

I was n people away from having someone named Daniel as an advisor as well lol

[gritty widget]

cv: graduated highschool, doing undergrad, have a dedicated dan emote spam server on discord

literally guaranteed

[sleek thicket]

And lowmath honorable?

talk yourself up dude

[honest narwhal]

Lol speaking of CV I should put a resume together for an internship application I'm thinking of doing

[gritty widget]

"spend free time tutoring students in extremely advanced topics such as the product rule"

[sleek thicket]

I managed to restrain myself from putting a link to my Twitter on there. Not putting it on there because I think it helps my chances of getting in but because the person reading apps might follow me

[gritty widget]

Couple weeks into calculus 1 now, doing well, already past the chain rule and beyond. Quotient rule was a joke. Product rule remains my specialty.

I ask my professor his thoughts on quantum mechanics and partial derivatives. He's impressed i know about the subject. We converse after class for some time, sharing mathematical insights; i can keep up. He tells me of great things ahead like series and laplacians. I tell him i already read about series on wikipedia. He is yet again impressed at my enthusiasm. What a joy it is to have your professor visibly brighten when he learns of your talents.

And now I sit here wondering what it must be like to be a brainlet, unable to engage your professor as an intellectual peer.

All of the deep conversations you people must miss out on because you aren't able to overcome the intellectual IQ barrier that stands in the way of your academic success... it's so sad.

My professor and I know each other on first name basis now, but i call him Dr. out of respect.

And yet here you brainlets sit, probably havent even made eye contact with yours out of fear that they will gauge your brainlet IQ levels.

A true shame, but just know it is because i was born special that i am special. I can't help being a genius, nor can my professor.

Two of a kind is two flocks in a bush.

[honest narwhal]

I'm just imagining a disaster scenario in which you did mention server_which_shall_not_be_named and Vakil reviews your app

[sleek thicket]

Lmfao

[cedar pebble]

man I really regret that place in an profound way

it is what it is I guess

[honest narwhal]

Throwback to when it was a server to hyperventilate about grad apps

[cedar pebble]

oh man

those were the days

just giant walls of pink wojaks

[honest narwhal]

Yeahhhhhhhhh

January 28th to February 6th, 2019

[cedar pebble]

oh man I remember your grad apps screw up

[honest narwhal]

Arguably the worst period of my life

[cedar pebble]

turned out alright though

but holy wow that was a time

[honest narwhal]

Yeah I think I made 4 different mistakes at different times lol

[sleek thicket]

everyone send good vibes to @tough imp, he's sending a long shot app to Columbia this year

[honest narwhal]

I sent my friend's CV to Columbia instead of my own

[sleek thicket]

Lmfaooo

[honest narwhal]

I titled the UChicago SOP "MIT Statement of Purpose"

[cedar pebble]

Good luck Chmonkey!!

[honest narwhal]

Misspelled "Andy Putnam" on my Notre Dame SOP

[cedar pebble]

oh nooo I forgot about that one

[honest narwhal]

Proceeded to send that SOP to Wisconsin, Duke, Penn, and Minneosta

[sleek thicket]

Bro

How did you get in

[cedar pebble]

man Wisconsin really saved your ass on that one

[sleek thicket]

Anywhere

[honest narwhal]

I flexed

Really fucking hard

[sleek thicket]

Lmfao

[cedar pebble]

Wisconsin was nice enough to email and be like

[honest narwhal]

And Madison was just intimidated

[cedar pebble]

"hey"

"I think you might have"

[honest narwhal]

Oh no they didn't

[cedar pebble]

oh they didn't?

oh jeez

[honest narwhal]

Negi was like

Yo why do you even give a shit about Duke anymore?

We know Dick Hain is very not based

[cedar pebble]

don't remind me about Duke

[honest narwhal]

Who else there would you even have as an advisor?

This was the day Duke acceptances were coming out

And I was like

Yeah I can't think of anybody else at Duke tbh

Lemme check my SOP

[cedar pebble]

Leslie Saper?

If you're into L^2 cohomology lol

[honest narwhal]

And then I see "I would be interested in working with Andy Putnam..." and I'm like hooooooooollllllllllllllddddddddd up

[sleek thicket]

My mom just said "if you choose to do grad school at Stanford"

[cedar pebble]

oh god that was a REALLY close save then

[sleek thicket]

Relevant to this convo

[honest narwhal]

Yeah lol

January 28th is when this happened

And then I checked the other SOPs to see which ones got fucked

[cedar pebble]

aaAAAAAA

[honest narwhal]

And asked each place to switch

Also I forgot to send my transcript to Rutgers that happened too lol

[cedar pebble]

get into Notre Dame 4 times