first cooking attempt :P

Leaky, you've become British already?

Looks dry and boring :D

Now eat it and see if you survive

you see the responses in TNB are much nicer

LOL ... you're doing food with the Frenet frame?

that does not look bad

@LeakyNun 8/10 would not smash

but did you make sauce?

its looks kinda dry =p

Simple probability question that I believe to have the correct solution to... Given a string of 12 light bulbs, with 3 of the 12 randomly burned out, what are the odds that the 6th bulb in the string is the 3rd and final faulty bulb?

ya it is

@TedShifrin I got question about dihedral group

@Eric (if you're still around): Earlier in class Benson was talking about viewing representations as $\mathbb{C}[G]$ modules or something

@KasmirKhaan dihedral

@TedShifrin how does one see that rs=sr'

donno whwere the head come from there lol

(1/6)*(5 choose 2)/(12 choose 3)

@Kasmir: You can see it geometrically or else you just compute. For most people, that's part of how the dihedral group is defined.

I meant geometriclly how do you see it ?=P

But immediately before he was talking about everything being done in char 0, is it that even over a generic char 0 field you can somehow transfer things to $\mathbb{C}$ or is it just that he was thinking about $\mathbb{C}$ but aside from part 2 of Schur, he'd be alright with whatever

Draw the polygon corresponding to the group, number the vertices, and play around with what the elements of the group do to the polygon?

Okay ill do that , thanks tasty

As I discuss in the book, conjugating is like changing coordinates ... like the change of basis formula in linear algebra. But just do it. Conjugate a rotation by a reflection and draw it out.

I did not read that chapter on dihedral grup

ill look it up now

@KasmirKhaan the dihedral group is the group of operations that leave the regular polygon unchanged

under functional composition

Some people define it to be generated by $r,s$ with that rule.

Yes it is a rigid motion

but r and s here

I know that r is defined to be 2pi /n

but how is s defined?

if we have an n is odd , does it fix vertex 1 ?

and flip the others?

@Daminark $kG$-modules $V$ are the same as representations $G \to \text{GL}(V)$ for some $k$-vector space $V$ regardless of the characteristic of $k$, innit

It's reflection across the perpendicular bisector of an edge ... so it switches a pair of adjacent vertices. (In the even case, two such.)

Yes I know that , but we do not care which lines it bisect?

like taking any reflection ?

Lol I mean Benson's restriction to char 0 is just a general thing

Right, one is all you need. You can get all the rest by conjugating. :)

the book am reading does not do much on explaing this

Like, we're only doing finite groups, char 0 fields, usually $\mathbb{C}$

All righty , since many of the examples are on dihedral , i need to reread this chapter

good

only normies use positive characteristic

@TedShifrin I don't know!

I mean false

well youre a normie so your opinion doesnt count

But that's great. I've spent all night going in the direction of something that was wrong all along...

I'm sorry you're more normie than I am. Number theory is the dankest of all human endeavors

number theory? more like, normier theory

ROASTED

Only a insert whatever you're doing would call that a roast

@TedShifrin I'm not sure I follow. How is the triple integrals analogy supposed to help?

I am currently roasting you, so that makes two of us

ROASTED

Because you think through setting up limits on iterated integrals exactly the way I said.

Sorry, I'm working on letters of recommendation for a former student.

It's fine.

@Daminark Benson just thinks things that aren't $\mathbf{C}$ aren't as important

and like, he's not wrong

Okay wow Balarka I'm gonna write a diss track against you

ull only make him become more powerful

@Daminark is the rule that profs have to give quality grades for undergrads taking grad classes for all grad classes or just the first year ones

@Daminark Stop diss-trackting me, I'm trying to do some math here

I have no idea

"trying"

GODDAMMIT

i have to email luis, i totally forgot there's a rule of this type and he's not really doing graded coursework

The bike carrier on the light rail train I'm on somehow made my front tire leak

So now it's flat

oof

Not cool, Semiclassic.

I still have no idea.

And I'm eventually getting off the bus about 2 miles from home

and that person who answered is not getting back to me anymore. I think they're done with me.

F***

I'm never getting home...

Plus I dunno if it's just flat or if I'll need a new inner tube

Which, since I was counting on being able to bike to transit for the upcoming week, is pretty f***ing bad

This sucks

The good news is that you can go to a bike shop tomorrow, Semiclassic.

Usually, yes

But the reason I need the bike next week is because I've been staying with my parents out in the suburbs

And they're out of town on vacation

Oh ... Isn't there a car you can use to take the bike somewhere?

that assumes I have a license.

oh wow

Yuuuuup

Hmm, I guess you can use Uber/Lyft.

Yeah, or I might be able to get help from a neighbor

Better idea.

I was going to suggest that, but then I figured you'd tell me you were too antisocial to know any neighbors.

Lol

I mostly am a hermit, but I do know one of our neighbors

I'm mostly just pissed at it happening. It's a vertical rack, so I must have hooked the front wheel in a bad spot (too close to the valve)

And once I noticed that it was leaking it was too late

Rest in rip

Yup. Most times we have an accident, it's too late ...

Hmm. There's two gas stations at the corner I'll be getting off at; maybe I'll luck out and they'll have a pump

Well, you need at least a patch PLUS a pump?

Well, the optimistic scenario is that the hook was pushing the valve out alignment somehow rather than a leak per se

Oh, I see.

You might be right, actually. I can't quite visualize what could have happened.

Bike repair: A Geometric Approach

Nor I. I'm on the bus now and the bike is on the rack on the front, so I can't check it until I disembark

Demonark: You think you're so funny.

@EricSilva: Any thoughts?

Truly

@TedShifrin I am guessing the formula for $d\omega(X_1, \cdots, X_k)$ is relevant?

It always annoys me how the exterior derivative is coordinate-wise a massive mess

Maybe, but when you trivialize the tangent bundle, it would be nice to use parallel vector fields or 1-forms so that we know the latter have 0 derivative. If you just take a random trivialization, you're not going to have any bracket control.

Right

@Balarka: I guess it boils down to this: Even if the manifold is parallelizable, you can't necessarily cover it all with one coordinate patch.

But, even so, I'm not sure that I know how do get from the basis for $\Lambda^k V^*$ to the basis for $\Lambda^{k+1} V^*$ with that formula.

This interesting question is still not getting any answers, and it is countably many dimensions. I guess for uncountably many, it will be even trickier

I wonder where are the functional analysis people in this chat went

Rip

Demonark's puns sent them out of finite dimensions and into Hilbert space.

they will probably need to wander for a long time before finding their way back here

Meanwhile, Leaky, simpleart and I were somewhere in the wilderness, near some amorphous sets. We also brought some ordinals as rulers for a safe measure

Yes, lost in your set theory morass.

Lel

Btw, any interesting maths you guys up to recently?

Finite topological spaces

details?

Basically, turns out that in a meaningful sense, partially ordered sets, Alexandroff T_0 spaces, small categories, and ordered simplicial complexes are the same thing

hmm, that will mean all posets can be treated as if they are ordered simplical complexes, or that they are equivalent under some class?

My prof has yet to explain the details but I think it's basically got to do with model categories

I see

But yeah I'll be doing a reading course on the stuff

Pay attention @Dami, one day you read about small categories, the next one you become an homotopy type theorist

Why should I be careful of that?

because we will all ignore you if you become an HoTTist

you'll be the only person in your vicinity who is an HoTTist

I made pretty graphs math.stackexchange.com/questions/2432239

I'll still have Peter May, Dylan, Akhil

and you will do fucken HoTT all day

@BalarkaSen want to meet single HoTTist near you? Sorry there's none

Lel

Balarka, you've overstayed your bedtime again.

Truly an unexpected turn of events

A turn without an appropriate signal?

I wouldn't go that far...

You missed your turn?

@AlessandroCodenotti ahaha

Sadly

@Ted My bedtime has overstayed me, it seems

What does that even mean

I must continue driving until I circle the world

Virus is a language from outer space, don't let it decieve you

I'm too confused to be decieved

Clearly you are though. You spelled "deceived" wrong

I have decieved you to spelling deceived wrong

And Balarka forgot to put in an extraneous apostrophe.

It's spelled dvdcieee so we're both wrong

Pah, sintax

imagination dead imagine

ok imma stop using stream of consciousness in conversations now

Wait you were conscious during these convos?

Learn something new every day

It's everyday bro

with the semiconscious flow

200 day dreams in a month, never done before

50 pages of Joyce in 5 months, never done before

Semiconscious flow is a very interesting area of research @Balarka

No there is no maths in my recent dreams

Secret: rip