#point-set-topology | Mathematics | Page 210

fading vale Mar 8, 2021, 7:20 AM

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catKing

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shamrock i saw the equivalence of COV_B and TRA_B

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when B is transport local

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scary tbh

viral atlas Mar 8, 2021, 7:21 AM

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Ahhhh, so like, within discrete metric a convergent sequence would eventually have the same point recurring in the sequence, and within the regular metric the corresponding sequence converges. However, there are more sequences in the regular metric converging to the same function value, while they all don't exactly have a pre-image in discrete metric? Idk I'm just rambling

fading vale Mar 8, 2021, 7:21 AM

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that sounds right

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i dont know analysis

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but that sounds right

sleek thicket Mar 8, 2021, 7:22 AM

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oof

sleek thicket Mar 8, 2021, 7:22 AM

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viral atlas Ahhhh, so like, within discrete metric a convergent sequence would eventually ha...

This is exactly correct

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if you like analysis, there's lots of examples on function spaces

fading vale Mar 8, 2021, 7:23 AM

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moth AU where they know literally any analysis

sleek thicket Mar 8, 2021, 7:23 AM

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On the set of continuous functions [0,1] -> R there's two natural metrics

fading vale Mar 8, 2021, 7:23 AM

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coming in 15 business days or so

viral atlas Mar 8, 2021, 7:23 AM

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hype

sleek thicket Mar 8, 2021, 7:23 AM

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$d(f, g) = \sup_{x\in[0,1]} |f(x) - g(x)|$

gentle ospreyBOT Mar 8, 2021, 7:23 AM

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!!Shamrock!!**

viral atlas Mar 8, 2021, 7:23 AM

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sleek thicket On the set of continuous functions [0,1] -> R there's two natural metrics

Euclidean, sup-norm, taxi-cab are the ones I know of

sleek thicket Mar 8, 2021, 7:23 AM

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$d'(f, g) = \int_0^1 |f(x) - g(x)|, \mathrm{d}x$

gentle ospreyBOT Mar 8, 2021, 7:23 AM

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!!Shamrock!!**

sleek thicket Mar 8, 2021, 7:24 AM

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ah yeah this is sup norm vs Euclidean

viral atlas Mar 8, 2021, 7:24 AM

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Oh, these are metrics on function spaces

sleek thicket Mar 8, 2021, 7:24 AM

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Although I think taxi cab and Euclidean are equivalent?

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yes, sorry

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but does that make sense?

viral atlas Mar 8, 2021, 7:24 AM

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Taxicab is generally greater than Euclidean

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It does, thanks!

sleek thicket Mar 8, 2021, 7:24 AM

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Exercise: prove that these do not generate the same topology

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Find a sequence converging in one but not the other

viral atlas Mar 8, 2021, 7:24 AM

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catThink

fading vale Mar 8, 2021, 7:24 AM

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cring

sleek thicket Mar 8, 2021, 7:25 AM

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and show that one is finer than the other

viral atlas Mar 8, 2021, 7:25 AM

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Nice

clear storm Mar 8, 2021, 7:25 AM

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does anyone here know about fundamental groupoids

viral atlas Mar 8, 2021, 7:25 AM

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I'll try this!

sleek thicket Mar 8, 2021, 7:25 AM

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(all open sets open in one are open in the other)

sleek thicket Mar 8, 2021, 7:25 AM

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clear storm does anyone here know about fundamental groupoids

hyperhonk

fading vale Mar 8, 2021, 7:25 AM

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@clear storm i know some

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lmao sham knows more

sleek thicket Mar 8, 2021, 7:25 AM

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You've been learning about them a lot

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Recently

fading vale Mar 8, 2021, 7:25 AM

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that is true

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i have learned terrible things

viral atlas Mar 8, 2021, 7:25 AM

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sleek thicket (all open sets open in one are open in the other)

So a sequence converging in one metric converges in the other, but the converse fails to hold?

sleek thicket Mar 8, 2021, 7:26 AM

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Exactly

fading vale Mar 8, 2021, 7:26 AM

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like the equivalence of functors from Pi(B) -> SET and Pi(b, b)-SET

viral atlas Mar 8, 2021, 7:26 AM

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Neat, I'll try this.

sleek thicket Mar 8, 2021, 7:26 AM

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also, equivalent means they generate the same topology

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Not that they're actually equal

sleek thicket Mar 8, 2021, 7:26 AM

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viral atlas Taxicab is generally greater than Euclidean

Replying to this, sorry

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You can get uniform bounds

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c Euclidean <= taxicab <= C Euclidean

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The two metrics I showed you are the sup (or "L^infinity") and L^1 metrics

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Very important in functional analysis

fading vale Mar 8, 2021, 7:27 AM

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shamrock cna you believe im actually going to learn multivariable

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its been like

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2 years

sleek thicket Mar 8, 2021, 7:27 AM

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I'm so excited for you

fading vale Mar 8, 2021, 7:28 AM

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my streak 😭

sleek thicket Mar 8, 2021, 7:28 AM

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That was my fave math class

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Before proof stuff

fading vale Mar 8, 2021, 7:28 AM

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im doing it for the difftop pill

sleek thicket Mar 8, 2021, 7:28 AM

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g e o m e t r y

fading vale Mar 8, 2021, 7:28 AM

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broosh

sleek thicket Mar 8, 2021, 7:28 AM

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hi sei, you can ignore our spam

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feel free to ask your question whenever

clear storm Mar 8, 2021, 7:32 AM

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@fading vale I am trying to show that given a circle, set of all paths between points on circle modulo endpoints preserving homotopy is homeomorphic to R.

viral atlas Mar 8, 2021, 7:32 AM

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Thanks for all the help, everyone.

clear storm Mar 8, 2021, 7:32 AM

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also

sleek thicket Mar 8, 2021, 7:32 AM

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clear storm <@!358385343629885440> I am trying to show that given a circle, set of all paths...

is this about universal covering spaces?

clear storm Mar 8, 2021, 7:32 AM

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the way topology is defined on the set of paths are very weird to me

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yes

fading vale Mar 8, 2021, 7:33 AM

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im not totally sure what you mean by this, the set of paths between two points x, y up to htpy?

clear storm Mar 8, 2021, 7:33 AM

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basically the end goal is showing that fundamental groupoid of circle is homeomorphic to the infinite cilinder

sleek thicket Mar 8, 2021, 7:33 AM

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moth, do you remember the construction of the universal covering space?

fading vale Mar 8, 2021, 7:33 AM

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vaguely

clear storm Mar 8, 2021, 7:33 AM

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i wasnt precise withj my phrasing

sleek thicket Mar 8, 2021, 7:33 AM

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well unique lifts and a trivial monodromy and all tell us what it has to be

fading vale Mar 8, 2021, 7:33 AM

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i just want ot make sure im clear on the thing sei is talking abt

sleek thicket Mar 8, 2021, 7:34 AM

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any universal cover has to be identifiable with the space of homotopy classes of paths in X

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the fiber over p being all paths starting at p

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so you can construct it as such

clear storm Mar 8, 2021, 7:34 AM

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given xand y on a circle you have a path between them. you have equivalence relation via homotopy.

sleek thicket Mar 8, 2021, 7:34 AM

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I forget the topology, I think it might be ugly

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iirc the way I saw it was by declaring the "take first endpoint" map to be a local homeomorphism

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on sets where the base is simply connected

clear storm Mar 8, 2021, 7:35 AM

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the topology is quite weiord

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you defined it via genetrating set

sleek thicket Mar 8, 2021, 7:35 AM

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sei, could you share the reference you're using?

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It sounds a little nonstandard

clear storm Mar 8, 2021, 7:35 AM

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wait i have a link

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from stack exchange

sleek thicket Mar 8, 2021, 7:36 AM

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like, thinking of the set of arrows in the fundamental groupoid as being the universal cover or whatever

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even though it is true

clear storm Mar 8, 2021, 7:36 AM

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https://math.stackexchange.com/questions/4052739/topology-of-fundamental-groupoid

Mathematics Stack Exchange

Topology of fundamental groupoid.

Given fundamental groupoid $\Pi_1(S^1)$ of the circle, how can one define a topology on it? The information on nlab did little help other than the fact that it can be done since $S^1$ is path-conne...

fading vale Mar 8, 2021, 7:36 AM

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ohhh

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ive seen this i think

clear storm Mar 8, 2021, 7:36 AM

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now i understand how topology is defined

sleek thicket Mar 8, 2021, 7:36 AM

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did you check the reference in hatcher?

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but yeah this is one way to define the topology

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you take the base of sets from being semilocally simply connected

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or whatever

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and declare those to be evenly covered

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it's been a while since I looked at this construction

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although this only works for nice spaces

clear storm Mar 8, 2021, 7:37 AM

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i tried looking for it in hatcher but didnt manage

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maybe somewhere in covering spaces chapter

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yes

sleek thicket Mar 8, 2021, 7:37 AM

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yeah, that's where I meant

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I like the presentation of the universal cover in Introduction to Topological Manifolds by Lee

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it's very explicit and done in great detail

fading vale Mar 8, 2021, 7:38 AM

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sleek thicket like, thinking of the set of arrows in the fundamental groupoid as being the uni...

this is basically (partly) how dieck does it by associating the cover to the Hom functor lmao

sleek thicket Mar 8, 2021, 7:38 AM

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ah gotcha

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~~sham stop simping for lee for one minute challenge~~

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oh actually

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i think topology and groupoids may have done this

fading vale Mar 8, 2021, 7:39 AM

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the way i saw this presented was very umm

sleek thicket Mar 8, 2021, 7:39 AM

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thinking

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no

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topology and groupoids takes a cover of the fundamental groupoid

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it develops a theory of covering groupoids

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and then constructs a space from such a thing

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yes

clear storm Mar 8, 2021, 7:40 AM

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hmm i think given an explicit representative of an arbitrary path(morphism) in the groupoid i feel like there is natural way to come up with an isomorphism

sleek thicket Mar 8, 2021, 7:41 AM

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an isomorphism between what

clear storm Mar 8, 2021, 7:41 AM

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homeomprohism

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lol

sleek thicket Mar 8, 2021, 7:41 AM

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between what?

fading vale Mar 8, 2021, 7:41 AM

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tom dieck basically does this by getting an equivalence of the categories COV_B and TRA_B = [Pi(B), SET] where for a functor Phi: Pi(B) -> SET you get the cover thats basically uhh, take the disjoint unions of all the sets in the image of Phi and send Phi(c) to c for c in B lol

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and then the universal cover is associated with Hom(b, -)

clear storm Mar 8, 2021, 7:41 AM

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betyween fundamental groupoid of circle and S cross R

sleek thicket Mar 8, 2021, 7:41 AM

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ah

clear storm Mar 8, 2021, 7:41 AM

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i.e infinite cilynder

sleek thicket Mar 8, 2021, 7:42 AM

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i do not think you can produce an isomorphism from a single path

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wait

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hang on

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I am confused

fading vale Mar 8, 2021, 7:42 AM

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i saw this done from van kampen

sleek thicket Mar 8, 2021, 7:42 AM

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so you have a topology on the set of arrows

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and you have a topology on the set of objects

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how are you combing these?

fading vale Mar 8, 2021, 7:42 AM

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with a topological groupoid G defined where the object space is S^1 and the morphism space is S^1 x R

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and van kampen established the isomorphism

clear storm Mar 8, 2021, 7:43 AM

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well the topology of the underlying set stau the same

fading vale Mar 8, 2021, 7:43 AM

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onto Pi(S^1)

clear storm Mar 8, 2021, 7:43 AM

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right

sleek thicket Mar 8, 2021, 7:43 AM

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oh I see, I'm being silly

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the universal cover isn't the space of all paths

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you fix a starting point

fading vale Mar 8, 2021, 7:43 AM

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yea

sleek thicket Mar 8, 2021, 7:43 AM

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ahhh okay

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im unconfused now

clear storm Mar 8, 2021, 7:43 AM

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because the set of objects in the groupoid is just the underlying s@ace

fading vale Mar 8, 2021, 7:43 AM

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its associated with the Hom functor from the grpoid

clear storm Mar 8, 2021, 7:43 AM

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space

fading vale Mar 8, 2021, 7:43 AM

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it sends the paths from b to c up to htpy to c

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where b is the basepoint

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(this sounds right, right sei lmao)

sleek thicket Mar 8, 2021, 7:44 AM

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so when we say Pi_1(S^1) is homeomorphic to S^1 x R we're talking about the set of arrows

fading vale Mar 8, 2021, 7:44 AM

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i presume so

sleek thicket Mar 8, 2021, 7:44 AM

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what's "it" moth?

fading vale Mar 8, 2021, 7:44 AM

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the Hom set stuff

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and the universal cover

sleek thicket Mar 8, 2021, 7:44 AM

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i mean that sounds like the projection out of the universal cover

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yeah

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but in this case we have more stuff

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we have arrows with different bases

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so the idea should be like

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you have a path class p : x -> y

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in Arr(Pi_1(S^1))

fading vale Mar 8, 2021, 7:45 AM

#

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this is basically in tom dieck

sleek thicket Mar 8, 2021, 7:45 AM

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you can send this to (x, (lift of p to universal cover)(1))

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this doesn't feel right

clear storm Mar 8, 2021, 7:45 AM

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sleek thicket so when we say Pi_1(S^1) is homeomorphic to S^1 x R we're talking about the set ...

ahhhh i see

fading vale Mar 8, 2021, 7:45 AM

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if u want to read it sei

sleek thicket Mar 8, 2021, 7:45 AM

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fading vale

this looks right, yeah

fading vale Mar 8, 2021, 7:46 AM

#

#

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2.6.1 is van kampen btw lol

clear storm Mar 8, 2021, 7:46 AM

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whats the reference??

fading vale Mar 8, 2021, 7:46 AM

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tammo tom dieck

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algebraic topology

clear storm Mar 8, 2021, 7:46 AM

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this looks like what i needed

fading vale Mar 8, 2021, 7:46 AM

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he does most of the basic pi_1 stuff from the perspective of groupoids

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it might help

sleek thicket Mar 8, 2021, 7:46 AM

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this is an interesting question

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I should read dieck

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i was gonna read uhhh

fading vale Mar 8, 2021, 7:47 AM

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its a good book

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inb4 lurie

sleek thicket Mar 8, 2021, 7:47 AM

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algebraic topology from the homotopical viewpoint

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feel like im supposed to learn alg top at some point

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learn it good, actually

fading vale Mar 8, 2021, 7:47 AM

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impossible

sleek thicket Mar 8, 2021, 7:47 AM

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Retweet this https://twitter.com/grassmannian/status/1368815776069603333?s=19

Brendan ☘️ (@grassmannian)

trying to be straight passing so when I call someone a fag it sounds like a slur and not a term of endearment

fading vale Mar 8, 2021, 7:48 AM

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i cant im leaving twitter

sleek thicket Mar 8, 2021, 7:48 AM

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I feel like your followers would like it

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oh no

cedar pebble Mar 8, 2021, 7:48 AM

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https://math.mit.edu/~hood/spectral_sequences/tmf/

sleek thicket Mar 8, 2021, 7:48 AM

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Good for you

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Twitter is bad

fading vale Mar 8, 2021, 7:48 AM

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ive been off it for 2 days

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the amount ive gotten done

sleek thicket Mar 8, 2021, 7:48 AM

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Hey I just went to a spectral sequences talk

fading vale Mar 8, 2021, 7:48 AM

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has gone way up

clear storm Mar 8, 2021, 7:48 AM

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anyway

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thanks guys

fading vale Mar 8, 2021, 7:48 AM

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np

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https://www.maths.ed.ac.uk/~v1ranick/papers/diecktop.pdf

clear storm Mar 8, 2021, 7:48 AM

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i didnt actually expect anyone to help

fading vale Mar 8, 2021, 7:48 AM

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heres a (legal) link to dieck

sleek thicket Mar 8, 2021, 7:48 AM

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oh no ng

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this is

cedar pebble Mar 8, 2021, 7:48 AM

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when you enable all the display options for the differentials

Screen_Shot_2021-03-08_at_2.46.18_AM.png

sleek thicket Mar 8, 2021, 7:48 AM

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very very bad

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jesus fucking christ

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it slowed down my laptop

fading vale Mar 8, 2021, 7:48 AM

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nG why are you bringing black magic into the chat

cedar pebble Mar 8, 2021, 7:48 AM

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https://math.mit.edu/~hood/spectral_sequences/

sleek thicket Mar 8, 2021, 7:49 AM

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i think this was max's talk https://math.mit.edu/~hood/spectral_sequences/latex/SSS-KZ3.pdf

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iirc

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no

fading vale Mar 8, 2021, 7:49 AM

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gamer moment

sleek thicket Mar 8, 2021, 7:49 AM

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lol

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anyways yeah i went to a talk on spectral sequences tonight

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and learned about the GSS

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very cool

fading vale Mar 8, 2021, 7:50 AM

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do u ever think about how probably at some point all the ways we talk and think about this stuff will seem like

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weird and dumb

cedar pebble Mar 8, 2021, 7:50 AM

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oh my god it shows every step

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monkaGIGA

fading vale Mar 8, 2021, 7:50 AM

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when some better perspective is uncovered

sleek thicket Mar 8, 2021, 7:50 AM

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absolutely moth

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spectral sequences seem so artificial

fading vale Mar 8, 2021, 7:50 AM

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and itll be like "lol they struggled so much with htpy groups of spheres"

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"dummies"

sleek thicket Mar 8, 2021, 7:51 AM

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Atiyah Hirzebruch spectral sequence computing $\mathit{KO}^*(\mathbb{RP}^{14})$

gentle ospreyBOT Mar 8, 2021, 7:51 AM

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!!Shamrock!!**

sleek thicket Mar 8, 2021, 7:51 AM

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RP^14

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why

fading vale Mar 8, 2021, 7:51 AM

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"thats literally so easy just do the blah blah blah"

fading vale Mar 8, 2021, 7:52 AM

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sleek thicket RP^14

its out of kindness sham

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RP^15 would crash ur laptop

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smug

sleek thicket Mar 8, 2021, 7:52 AM

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lmao

cedar pebble Mar 8, 2021, 7:52 AM

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haha ng is such a moron all he does is calculate integrals imagine doing something so tedious

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also algebraic topologists:

Screen_Shot_2021-03-08_at_2.51.51_AM.png

sleek thicket Mar 8, 2021, 7:53 AM

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i have said this

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many times

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you can search it in chat logs

gritty widget Mar 8, 2021, 7:53 AM

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Oh, I thought you were a girl

sleek thicket Mar 8, 2021, 7:53 AM

#

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lol

cedar pebble Mar 8, 2021, 7:54 AM

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kek

sleek thicket Mar 8, 2021, 7:54 AM

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ahhhhhhh

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i had a thing due

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for homological algbera

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did not finish it

fading vale Mar 8, 2021, 7:54 AM

#

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holy shit

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weirdga

sleek thicket Mar 8, 2021, 7:54 AM

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i had nothing else on this weekend

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lmao moth owned

fading vale Mar 8, 2021, 7:55 AM

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tbf this is from like umm

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wait when did i join

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like 2.25 years ago?

sleek thicket Mar 8, 2021, 7:55 AM

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lmfao owned

fading vale Mar 8, 2021, 7:55 AM

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mfw a math discord changed my life

sleek thicket Mar 8, 2021, 7:55 AM

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literally 4x

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if i were a high schooler i would simply resist math brain worms

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(no i wouldn't)

fading vale Mar 8, 2021, 7:55 AM

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i was naive and 14

sleek thicket Mar 8, 2021, 7:55 AM

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(i didn't)

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wow really?

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youre like benjamin button

fading vale Mar 8, 2021, 7:56 AM

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yea when i joined

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i didnt start doing math seriously until like

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right b4 i turned 15 tho

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also on the topic of reminiscing this time last year i was freaking out about applying to sumac and now its due in 2 days and idek if i wanna go

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oh well

sleek thicket Mar 8, 2021, 7:59 AM

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tfw your sick burn goes unnoticed

fading vale Mar 8, 2021, 7:59 AM

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idk what benjamin button is

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im normal

sleek thicket Mar 8, 2021, 7:59 AM

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wtf

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how

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benjamin button aged backwards

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is the reference

fading vale Mar 8, 2021, 7:59 AM

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i hate u

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i will devastate ur entire lineage

sleek thicket Mar 8, 2021, 8:00 AM

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lol

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how do you not know who benjamin button is

fading vale Mar 8, 2021, 8:01 AM

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why would i know who benjamin button is!!

sleek thicket Mar 8, 2021, 8:02 AM

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it is a very common cultural touchstone!!!

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ive never even seen it

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or read it

fading vale Mar 8, 2021, 8:02 AM

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ur just a boomer

sleek thicket Mar 8, 2021, 8:02 AM

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no

fading vale Mar 8, 2021, 8:02 AM

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yes!

sleek thicket Mar 8, 2021, 8:02 AM

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i retain my youth

fading vale Mar 8, 2021, 8:02 AM

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☘️

sleek thicket Mar 8, 2021, 8:02 AM

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you will not shake me

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☘️

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🦣

fading vale Mar 8, 2021, 8:02 AM

#

sadMothman

sleek thicket Mar 8, 2021, 8:04 AM

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many such cases

fading vale Mar 8, 2021, 8:04 AM

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how do i motivate myself to do the problems for this summer camp when it covers intro AT and id literally just be paying like several thousand dollars to learn weaksauce version of intro AT online

sleek thicket Mar 8, 2021, 8:04 AM

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¯_(ツ)_/¯

fading vale Mar 8, 2021, 8:04 AM

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man.

tough imp Mar 8, 2021, 8:30 AM

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fading vale

Literally like 10 times my amount wtf

sleek thicket Mar 8, 2021, 9:26 AM

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👀

pastel linden Mar 8, 2021, 9:27 AM

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I was going to ask what the point of actually doing a weaksauce summer camp for money is

sleek thicket Mar 8, 2021, 9:27 AM

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well I mean like

pastel linden Mar 8, 2021, 9:27 AM

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is it to have physical clout of AT skills

sleek thicket Mar 8, 2021, 9:27 AM

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It's only weaksauce because moth is absurdly chad

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I am defending the honor of this program

gray dirge Mar 8, 2021, 9:28 AM

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lol. Weaksauce is better than no sauce? As always, the amount you have to pay is important.

sleek thicket Mar 8, 2021, 9:28 AM

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Also, community and mentors and stuff

pastel linden Mar 8, 2021, 9:28 AM

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it's genuinely insane that they pay undergrads to do math if you get into reus

gray dirge Mar 8, 2021, 9:28 AM

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I'm not defending the program but it does matter how much it costs. 1 cent? Maybe it's worth it.

sleek thicket Mar 8, 2021, 9:28 AM

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yeah haha

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Well

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I'm not going to get paid

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but mostly

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which is very good

pastel linden Mar 8, 2021, 9:29 AM

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if I get both my research grant and my reu that's 7800 for the summer of just learning math

sleek thicket Mar 8, 2021, 9:29 AM

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That would be so based

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What's the research grant ?

pastel linden Mar 8, 2021, 9:29 AM

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to write a harmonic analysis exposition with my professor

gray dirge Mar 8, 2021, 9:29 AM

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I'm confused are you in high school now?

pastel linden Mar 8, 2021, 9:29 AM

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It's not really research

sleek thicket Mar 8, 2021, 9:29 AM

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That would be really cool!

pastel linden Mar 8, 2021, 9:30 AM

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it would be insane to have that opportunity but there's one grant and it's up for grabs for all math majors

sleek thicket Mar 8, 2021, 9:30 AM

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Dang yeah not very likely

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I don't know of anything like that here

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There's some kind of outstanding junior arward I meant to apply to but I can't find any info

pastel linden Mar 8, 2021, 9:31 AM

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I do think it's kind of likely that I'll get into that reu, since it turns out my professor knows the director

sleek thicket Mar 8, 2021, 9:31 AM

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Doesn't seem to have been awarded last year

gray dirge Mar 8, 2021, 9:31 AM

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It's just called "normal" research. If you know enough and are hard-working enough or creative enough or both to contribute you might get someone to work with you on research.

sleek thicket Mar 8, 2021, 9:31 AM

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what

pastel linden Mar 8, 2021, 9:32 AM

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email the department and say to them "I am an outstanding student and I want money"

sleek thicket Mar 8, 2021, 9:32 AM

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lol

gray dirge Mar 8, 2021, 9:32 AM

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No, you have to prove you know what you know and try to get "hired." It's just like every other job.

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Oh, and bury bodies. Don't tell anyone you harmed your past advisor ...

pastel linden Mar 8, 2021, 9:33 AM

#

??????

gray dirge Mar 8, 2021, 9:33 AM

#

I'm kidding.

#

Seriously, you have to be skilled enough to do well if someone gives you a chance. (life lesson)

sleek thicket Mar 8, 2021, 9:34 AM

#

you seem to be saying a lot of stuff disconnected from the rest of the conversation and I don't understand what you're trying to say

sleek thicket Mar 8, 2021, 9:34 AM

#

gray dirge It's just called "normal" research. If you know enough and are hard-working eno...

like with this

sleek thicket Mar 8, 2021, 9:34 AM

#

gray dirge No, you have to prove you know what you know and try to get "hired." It's just ...

Or this

gray dirge Mar 8, 2021, 9:35 AM

#

You're asking about REUs and calling some shams. I'm saying if you're good enough you have the option of possibly doing real research.

sleek thicket Mar 8, 2021, 9:35 AM

#

I'm not calling any of them shams

gray dirge Mar 8, 2021, 9:35 AM

#

I am.

pastel linden Mar 8, 2021, 9:35 AM

#

I wasn't calling the reu a sham, I just meant that exposition with my professor isn't research, it's a reading/expository project

gray dirge Mar 8, 2021, 9:37 AM

#

Fair enough. I lost the thread of the conversation.

sleek thicket Mar 8, 2021, 9:38 AM

#

Moth ("many such cases") is a high school student who is trying to decide whether they should apply to a summer program (which costs money) when they already know a lot of algebraic topology

#

Bacono is an undergrad applying to both an REU and a research grant, the research grant is really for an expositional project with their professor. Both of these would pay them

#

I'm an undergrad and have been admitted to the university of chicago REU which is very prestigious but accepts lots of applicants and can't distribute funding evenly. I was whining that I wouldn't be paid for this reu, unlike most

#

I think that's the conversation in summary?

gray dirge Mar 8, 2021, 9:44 AM

#

I'm headed out :). My head hurts from helping others with calculus.

fading vale Mar 8, 2021, 3:35 PM

#

society if tom dieck clearly stated what the topology on this G-set is

#

pretty sure its the discrete topology

#

i hope.

marsh forge Mar 8, 2021, 4:12 PM

#

Either A) assume it has a given topology (i.e. its just some arb top space w action)

#

B) assume its discrete

#

C) it's a group that has a topology like SU(n)

#

i think context makes which of these it is obvious

fading vale Mar 8, 2021, 4:32 PM

#

marsh forge B) assume its discrete

i think its discrete but it genuinely isnt that clear from context

#

i mean i dont think this works otherwise tho

#

he just says "let F be a G-set, p: E -> B a cover and take p_F: E x_G F -> B"

#

but i dont think the fibers r discrete if F doesnt have the discrete topology so

marsh forge Mar 8, 2021, 4:35 PM

#

If they just say G-set and not like, topological G-set

#

i think discrete is the only reasonable assumption

gritty widget Mar 8, 2021, 5:21 PM

#

hey pals, can you have a neighborhood U of p, where V is uncountable?

median glade Mar 8, 2021, 5:22 PM

#

can you have an open set containing p that is uncountable?

#

is that what you mean?

gritty widget Mar 8, 2021, 5:24 PM

#

Correct me if im wrong, I thought U is p in X to a point in V, where V is in X,T. Is that right?

fading vale Mar 8, 2021, 5:25 PM

#

ummm

#

im not sure what you mean

#

a neighborhood of p is just a subset U with an open set V where p in V subset U

gritty widget Mar 8, 2021, 5:26 PM

#

Excuse the bad drawing.

#

is that not what U is?

fading vale Mar 8, 2021, 5:27 PM

#

im not really sure what this is

#

are U and V the sets?

#

or points?

gritty widget Mar 8, 2021, 5:27 PM

#

Circles are sets, red dots are points

fading vale Mar 8, 2021, 5:28 PM

#

I thought U is p in X to a point in V, where V is in X,T.
im not totally sure what you mean by this but a neighborhood doesnt involve two points, or an interval between two points (necessarily)

#

like a neighborhood of p is just a set U whose interior contains p

#

theres nothing wrong with your drawing im just not sure what you mean by "to a point in V"

gritty widget Mar 8, 2021, 5:29 PM

#

fading vale > I thought U is p in X to a point in V, where V is in X,T. im not totally sure ...

That in this drawing, can the set V be uncountable

fading vale Mar 8, 2021, 5:29 PM

#

a neighborhood can be uncountable yes

#

a neighborhood can have any cardinality depending on the underlying set and the topology placed on it

#

the interval [0, 1] is a neighborhood of 0.5 in the reals with the standard topology

gritty widget Mar 8, 2021, 5:30 PM

#

yeah, gotcha

#

Ok, thanks

median glade Mar 8, 2021, 5:34 PM

#

fading vale a neighborhood of p is just a subset U with an open set V where p in V subset U

reeeeee

#

Do you actually use that definition?

fading vale Mar 8, 2021, 5:35 PM

#

thats literally the defn

#

thonkEyes

median glade Mar 8, 2021, 5:35 PM

#

that seems real awkward compared to just defining it to be the open set V

fading vale Mar 8, 2021, 5:35 PM

#

wut

wanton marsh Mar 8, 2021, 5:35 PM

#

what

fading vale Mar 8, 2021, 5:35 PM

#

closed sets can be neighborhoods too

median glade Mar 8, 2021, 5:35 PM

#

I learned it as "A neighborhood of x is an open set containing x"

fading vale Mar 8, 2021, 5:35 PM

#

ummmm thats not correct lol

wanton marsh Mar 8, 2021, 5:35 PM

#

that would be an open neighbourhood

fading vale Mar 8, 2021, 5:35 PM

#

^

median glade Mar 8, 2021, 5:35 PM

#

It was in Munkres

fading vale Mar 8, 2021, 5:35 PM

#

dan

#

thats surprising

#

neighborhood means that x is in the interior of V

median glade Mar 8, 2021, 5:36 PM

#

how often do you use "closed neighborhood" though?

fading vale Mar 8, 2021, 5:36 PM

#

i mean fairly frequently

median glade Mar 8, 2021, 5:36 PM

#

really?

fading vale Mar 8, 2021, 5:37 PM

#

for one thing separation isnt equivalent

#

by open neighborhoods vs neighborhoods in general

median glade Mar 8, 2021, 5:37 PM

#

wdym?

#

for reference

fading vale Mar 8, 2021, 5:38 PM

#

saying that any two points can be separated by closed neighborhoods is a stronger condition than saying they can be separated by neighborhoods in general

median glade Mar 8, 2021, 5:38 PM

#

yeah

fading vale Mar 8, 2021, 5:38 PM

#

https://math.stackexchange.com/questions/24016/example-of-hausdorff-space-x-s-t-c-bx-does-not-separate-points stack exchange example

Mathematics Stack Exchange

Example of Hausdorff space $X$ s.t. $C_b(X)$ does not separate points?

We know the Stone-Weierstrass theorem for locally compact Hausdorff spaces (LCH) which states the following:
Theorem: Suppose $X$ is LCH. A subalgebra $\mathcal{A}$ of $C_0(X)$ is dense if and o...

median glade Mar 8, 2021, 5:38 PM

#

the way munkres phrased that is: There exist neighborhoods U, V, with the closure of U distinct from V, and vice versa

fading vale Mar 8, 2021, 5:38 PM

#

also if you want to do something like say

median glade Mar 8, 2021, 5:38 PM

#

which is equivalent

fading vale Mar 8, 2021, 5:38 PM

#

neighborhood bases

median glade Mar 8, 2021, 5:39 PM

#

also Lee backs my pov up

fading vale Mar 8, 2021, 5:40 PM

#

when is that book from?

median glade Mar 8, 2021, 5:40 PM

#

Introduction to Topological Manifolds (2011)

fading vale Mar 8, 2021, 5:41 PM

#

wacky

#

the terminology is pretty non standardized unfortunately

median glade Mar 8, 2021, 5:41 PM

#

I feel like it's wacky to not do it this way

fading vale Mar 8, 2021, 5:41 PM

#

not really

#

like it makes intuitive sense

#

the idea of a neighborhood is that we have wiggle room around a point

median glade Mar 8, 2021, 5:41 PM

#

it makes sense to me, in the sense that any statement about a space has to use the open set structure

#

so if I want to talk about wiggle room, it makes no sense to use a general set

#

the wiggle room is an open set

fading vale Mar 8, 2021, 5:42 PM

#

both defn use the open set structure

#

thats wack

#

every book ive used has done that

median glade Mar 8, 2021, 5:42 PM

#

dieck does it moth's way

fading vale Mar 8, 2021, 5:42 PM

#

dieck does

#

bredon also does

#

as does hatcher in his notes iirc

#

i think the idea intuitively is that the point is the interior of the set

median glade Mar 8, 2021, 5:43 PM

#

it's less of a problem if you're coming from the "neighborhood = open set" school

#

because then you read "open neighborhood" everywhere, and it's unambiguous

#

but I guess it's usually clear through how they use the neighborhood, that they mean its open

#

like idk, it seems weird to take a neighborhood, and then immediately discard it to work with an open set

#

why not just take the open set directly

#

😛

fading vale Mar 8, 2021, 5:44 PM

#

i think ive done exercises involving closed neighborhoods specifically

#

but its been too long to remember anything specific

median glade Mar 8, 2021, 5:45 PM

#

I mean, at the end of the day, it doesn't really matter

fading vale Mar 8, 2021, 5:45 PM

#

the terminology is non standardized cuz no one cares about point set egg_hank

median glade Mar 8, 2021, 5:45 PM

#

if you need to talk about closed neighborhoods, you can just mention open neighborhoods whose closure has a certain property right

fading vale Mar 8, 2021, 5:45 PM

#

yea

#

just take closure

#

if ur working w/ general neighborhoods just take the interior (or the arbitrary open set contained within)

median glade Mar 8, 2021, 5:46 PM

#

for example like, did you learn regular as "every open neighborhood of x contains a closed neighborhood?"

#

or well, did you see that version of regular

fading vale Mar 8, 2021, 5:47 PM

#

i learned regular as any closed set C and point p, p notin C, admit disjoint open neighborhoods

median glade Mar 8, 2021, 5:47 PM

#

yeah that's the definition

fading vale Mar 8, 2021, 5:48 PM

#

honestly couldnt even tell u if i saw urs

median glade Mar 8, 2021, 5:48 PM

#

but an equivalent, and often more useful version is: for any point p, and open U containing p, there exists an open V containing p, with cl(V) contained in U

fading vale Mar 8, 2021, 5:48 PM

#

ive seen that

median glade Mar 8, 2021, 5:48 PM

#

right yeah

fading vale Mar 8, 2021, 5:49 PM

#

(i think)

#

idk i guess it ultimately doesnt really matter because

median glade Mar 8, 2021, 5:49 PM

#

and this is equivalent to saying that any open neighborhood of a point contains a closed neighborhood

fading vale Mar 8, 2021, 5:49 PM

#

yea

#

i guess the defn ive seen seems more natural to me also because its a little more general

#

and exposure obviously

median glade Mar 8, 2021, 5:50 PM

#

nah it's the better definition

#

going from hausdorff -> regular -> normal makes sense this way

fading vale Mar 8, 2021, 5:50 PM

#

mhm

median glade Mar 8, 2021, 5:50 PM

#

you can separate larger and larger things

#

an equivalent formulation of hausdorff is that if x != y, then x has a closed neighborhood not containing y

#

8^)

wanton marsh Mar 8, 2021, 5:51 PM

#

closed neighborhood is not the same as closure of open neighborhood

fading vale Mar 8, 2021, 5:52 PM

#

this is true you run into problems with empty interiors

#

but i mean

#

if the interior is empty then it cant be a neighborhood of any point anyway

median glade Mar 8, 2021, 5:53 PM

#

doesn't closed nbrhood mean: closed set, containing an open set?

fading vale Mar 8, 2021, 5:53 PM

#

idk maybe there are other counterexamples i cant think of off the top of my head

median glade Mar 8, 2021, 5:53 PM

#

closed set containing is indeed not great

#

because you might have a boundary point

fading vale Mar 8, 2021, 5:54 PM

#

thats fair

median glade Mar 8, 2021, 5:54 PM

#

indeed

long coyote Mar 9, 2021, 1:04 AM

#

Let R_K be the K topology, what does R_K/K look like

#

the base of R_K is (a,b)U(a,b)\K

thin bramble Mar 9, 2021, 2:30 AM

#

Anyone know Geometric Algebra?

flint cove Mar 9, 2021, 2:51 AM

#

thin bramble Anyone know Geometric Algebra?

I briefly went over some lecture notes and was quite overwhelmed by the bazillion operations that have been introduced

thin bramble Mar 9, 2021, 2:53 AM

#

flint cove I briefly went over some lecture notes and was quite overwhelmed by the bazillio...

I have a couple of questions. can I DM you?

flint cove Mar 9, 2021, 2:53 AM

#

sure, but I don't recall much of it, so not sure if I can actually help you with anything

zinc venture Mar 9, 2021, 9:11 AM

#

Hoping this would be the right channel to ask/discuss this...

#

I am trying to find the gradient on a color image.
I understand the usual approach is to covert the image to grayscale, then take the gradient there. However, a clear problem is that e.g. a fully saturated color of some lightness would be the same color as gray, creating no gradient - obviously a problem.
If we were to take the gradients in each color channel separately and added those up though, then something like a pure green transitioning to a pure red would have two gradients of equal magnitude but opposing direction, which would cancel out, leaving us, again, with no gradient - obviously also a problem.

gritty widget Mar 9, 2021, 9:14 AM

#

This is probably not the right channel 🙂 but maybe you could do something like taking the l2 norm of the channel separated gradients

#

Is there a reason that grayscale doesn't work for you?

zinc venture Mar 9, 2021, 9:16 AM

#

Different colors of equal intensity would turn gray creating no edges where there clearly are some. Like clouds in the sky - light gray and light blue are the same in grayscale.

#

I really looked at all the channels, and I don't know... #calculus maybe?

gritty widget Mar 9, 2021, 9:20 AM

#

Tbh I feel like this isn't a math question

#

Maybe it could fit in #numerical-analysis

zinc venture Mar 9, 2021, 9:25 AM

#

No programming Discords really know what to make of it either. Too theoretical for their tastes. 😅
Digital Signal Processing, but that's too niche to have a separate community.
The norm you suggested - do I not lose the direction component with that, leaving only the magnitude, not allowing me to find which way the gradient goes?

gritty widget Mar 9, 2021, 9:28 AM

#

I guess so, but I'm not sure what better you can do if you want a scalar output

#

That said I'm not particularly familiar with image processing so I don't know if there are standard ways to handle this situation

zinc venture Mar 9, 2021, 9:30 AM

#

That's fair. Maybe there is no good solution. I'll keep thinking/searching.

split needle Mar 9, 2021, 11:44 AM

#

try here: https://computergraphics.stackexchange.com/ and you can also convert your image to a different space such as HSL/HSV to separate "color" from "intensity" and take the gradient only on the "color" part. It also feels like you're asking an XY question and are already proposing a solution without telling us what you actually want to achieve. @zinc venture

brave gate Mar 9, 2021, 2:58 PM

#

What is the meaning of $dx=x^2$ Is there any way for me to get rid of the dx?

gentle ospreyBOT Mar 9, 2021, 2:58 PM

#

R3P34T3R

gritty widget Mar 9, 2021, 3:03 PM

#

context?

brave gate Mar 9, 2021, 3:04 PM

#

i incorrectly solved a differential equation

#

and this is what i was left with

#

and i was wondering what this thing could be

gritty widget Mar 9, 2021, 3:04 PM

#

what's the differential equation

brave gate Mar 9, 2021, 3:04 PM

#

i dont want you to solve the equation

#

i just used separation of variables incorrectly

gritty widget Mar 9, 2021, 3:05 PM

#

#multivariable-calculus

brave gate Mar 9, 2021, 3:05 PM

#

still im curious

#

wouldnt this be considered a differential?

tepid depot Mar 9, 2021, 3:06 PM

#

@brave gate you could think of dx as a differential form, but then it wouldn't equal x^2. this is like if you got the equation 0=1, it just means you messed up

brave gate Mar 9, 2021, 3:06 PM

#

so when you have an equation made up of differential forms it cannot contain a 'normal' function?

tepid depot Mar 9, 2021, 3:08 PM

#

there's this way of creating a formal algebraic system which assigns meanings to stuff like dx, dy, df, etc.

#

that system is consistent with the manipulations you do in ODEs and whatnot

#

in that system, dx means something specific, and x^2 means something else specific, and those two things are different things

#

dx isn't really a function like x^2 is

#

so this equation doesn't really make sense

brave gate Mar 9, 2021, 3:10 PM

#

ahhhh

#

i was kind of hoping if it wasnt like this

#

but i understand

ivory dragon Mar 9, 2021, 3:10 PM

#

~~set d = x~~

tepid depot Mar 9, 2021, 3:11 PM

#

cool. if you want to learn more about this topic the thing you'd google is differential forms. you can learn the algebra rules easily but the formalism rigorously is pretty advanced

brave gate Mar 9, 2021, 3:13 PM

#

oh yeah

#

i knew of them before

#

and looked them up this time too

#

somehow i didnt realize what i was looking for was false

#

thank you so much tooru

urban yarrow Mar 9, 2021, 3:42 PM

#

cancel x out on both sides you get d=x

gritty widget Mar 9, 2021, 3:43 PM

#

make a bad post about it on twitter and you'll cancel the entire equation

gritty widget Mar 9, 2021, 3:59 PM

#

Suppose we have a exact sequence:
0->A->B->C->D->0
Is it true in general that AxB is isomorphic to CxD

gritty widget Mar 9, 2021, 4:17 PM

#

If anyone is interested the answer is no, take D=0 and pick an exact sequence that does not split

urban yarrow Mar 9, 2021, 6:06 PM

#

This sequence looks pretty boring in general, but i guess you can try $A \times C$ and $B\times D$ in some specific categories. I’m not sure what will come out though

gentle ospreyBOT Mar 9, 2021, 6:06 PM

#

Aki

obtuse meteor Mar 9, 2021, 6:07 PM

#

what's a good way to tell if a covering is regular?

#

so like

#

if I have a covering of a wedge of two circles

#

and it has 3 vertices

#

and how I've lifted the edges is like

#

edge a has a loop on the left vertex

#

edge b has a loop on the right vertex

#

and there are transposition looking things from the middle vertex to the other two

#

this feels non-regular but I can't figure out why

#

like I kinda get it intuitively because there's a symmetry break but

#

hrm

obtuse meteor Mar 9, 2021, 6:28 PM

#

@marsh forge can I placate your anger with topology?

#

ah wait I think I have a big brain way of doing this

#

so like if you look at the left and right vertex

#

you get an induced fundamental group from each

#

the left one has an a in it

#

the right one has a b in it

#

they can't be the same

#

because if they were it would be the whole group

#

which has index 1

#

not 3

marsh forge Mar 9, 2021, 6:30 PM

#

everyone always things im angry

#

im watching puppies on tiktok during that entire convo

obtuse meteor Mar 9, 2021, 6:30 PM

#

lol

#

puppies good

marsh forge Mar 9, 2021, 6:31 PM

#

i showed my gf a bunch of cute ones

obtuse meteor Mar 9, 2021, 6:31 PM

#

and so there can't be a deck transformation

#

that's sweet ^^

marsh forge Mar 9, 2021, 6:31 PM

#

oh god

obtuse meteor Mar 9, 2021, 6:31 PM

#

does this argument make sense?

marsh forge Mar 9, 2021, 6:31 PM

#

coho theres a fast heuristic of like

#

if you can visualize it

#

any starting point should have similar paths

#

so like, name the lifts properly

#

and see if doing the same path on two points

#

looks the same

#

i also think there might be a more formal criterion

obtuse meteor Mar 9, 2021, 6:32 PM

#

maybe

#

but what I said works :)

marsh forge Mar 9, 2021, 6:32 PM

#

yeah i mean if there is a symmetry break

#

you can formalize this at the group theoretic level easily

obtuse meteor Mar 9, 2021, 6:32 PM

#

not if you're bad at algebra like me max

marsh forge Mar 9, 2021, 6:33 PM

#

basically anything that is a loop at one of the verticies

#

has to be a loop at the others

#

and that breaks pi_1 stuff

obtuse meteor Mar 9, 2021, 6:33 PM

#

mhm

warm hedge Mar 9, 2021, 7:49 PM

#

question 😛 completeness of a space isnt a topological property, right ?

wanton marsh Mar 9, 2021, 7:50 PM

#

completeness of a space ?

warm hedge Mar 9, 2021, 7:50 PM

#

if a space is complete (i dont know the english word i guess 😛 )

wanton marsh Mar 9, 2021, 7:51 PM

#

don't you need a metric space for that ?

warm hedge Mar 9, 2021, 7:51 PM

#

yeah

wanton marsh Mar 9, 2021, 7:51 PM

#

so you want to find two homeomorphic metric spaces, only one of which is complete ?

urban yarrow Mar 9, 2021, 7:53 PM

#

R and an open interval

warm hedge Mar 9, 2021, 7:54 PM

#

oh yeah right

#

so it isnt topological property

#

but a space to be called polish needs to be complete and at the same time it is a topological property , how this is working ?

urban yarrow Mar 9, 2021, 7:57 PM

#

it doesn’t need to be complete itself

bitter yoke Mar 9, 2021, 7:57 PM

#

polish only requires that you're homeomorphic to complete right?

urban yarrow Mar 9, 2021, 7:57 PM

#

open interval is also polish

warm hedge Mar 9, 2021, 8:02 PM

#

the open interval can be complete with the discrete metric ?

hard wind Mar 9, 2021, 8:04 PM

#

#

Need help on part b). My guess is I need to find a particular retraction to a subspace that has fundamental group Z^N, but I am not exactly sure how to find that.

urban yarrow Mar 9, 2021, 8:08 PM

#

warm hedge the open interval can be complete with the discrete metric ?

yes discrete always implies complete

wanton marsh Mar 9, 2021, 8:13 PM

#

hard wind Need help on part b). My guess is I need to find a particular retraction to a su...

or you can start by finding countably many morphisms into Z

hard wind Mar 9, 2021, 8:15 PM

#

Okay... so we have countably many retractions r_n which are mapping to one particular circle. This gives us countably many morphisms from Hawaiian fundamental group into Z

#

I'm just not sure how to go about multiplying them all together

warm hedge Mar 9, 2021, 8:18 PM

#

I see

#

Polish spaces is something new for me and i am trying to understand them

urban yarrow Mar 9, 2021, 8:26 PM

#

it has a weird name

#

similar to tropical geometry

#

i think mathematicians are just not talented enough to invent names holothink

warm hedge Mar 9, 2021, 8:52 PM

#

ohh i see

long coyote Mar 9, 2021, 9:13 PM

#

why do we need surjective for a quotient map

marsh forge Mar 9, 2021, 9:14 PM

#

The most important reason is just motivational imo

#

A quotient is a very general notion in mathematics

#

and it almost universally means like

#

apply some equivalence relation to object A to get object B

#

so you want B to 'come from' A

#

if the map is not surjective this means something is in B that didn't come from A

ivory dragon Mar 9, 2021, 9:19 PM

#

yeah its not like, divinely inspired; its more that quotient maps are introduced almost exclusively to study quotient spaces

#

and having a nonsurjective map doesnt make sense if you want your codomain to be determined entirely by an equivalence relation (i.e. a quotienting) on your domain

long coyote Mar 9, 2021, 9:23 PM

#

thank you for detailed comments

digital peak Mar 9, 2021, 9:38 PM

#

I don't get it

#

if I have say a lie group with a chart around identity

#

then I could rewrite multiplication "in coordinates"

#

i.e I'd have a map Psi : R^n x R^n -> R^n

#

and we can show that Psi(x, y) = x + y + B(x, y) + o(x^2+y^2)

#

and supposedly associativity of Psi gives us associativity of B?

#

but I just don't see that happening?

#

like, sure, if we expand Psi(Psi(x, y), z) we will get B(B(x, y), z)

#

but that won't be the only term of third order

#

cancer computation:

#

#

the B B thing is what I want but the other cubic term won't go away sully

#

hmm this tensor is symmetric in i,j

#

it's equal to a similar tensor that's symmetric in j,k

#

means its symmetric in i,j,k

#

there exist multiple totally symmetric nondegenerate tensors in rank 3 though

long coyote Mar 9, 2021, 11:09 PM

#

how to show $\mathbb{R}^n/B\simeq\mathbb{R}^n$ where B is a closed unit ball

gentle ospreyBOT Mar 9, 2021, 11:09 PM

#

亜城木夢叶

sleek thicket Mar 9, 2021, 11:15 PM

#

the way I would think about this is that R^n is homeomorphic to the space (S^{n-1} times [0, infinity))/(S^{n-1} x {0})

#

does that make sense?

#

basically, the map sending (x, r) to rx is a quotient map from S^(n-1) × [0,infty) to R^n

#

then to prove your claim you can just work on the radius and ignore the coordinate on the sphere

#

Because collapsing B to a point has the effect of collapsing [0,1] to a point in [ 0,infty)

#

I guess I'll take the 🤔 as a no?

long coyote Mar 9, 2021, 11:26 PM

#

no, don't follow that

sleek thicket Mar 9, 2021, 11:28 PM

#

A simpler way to say what I'm saying is that everything is radially symmetric, so you just have a find a nice function s(r) from [0,infinity) to [0,infinity) sending [0,1] to 0 and bijective everywhere else, and then you can define a function F : R^n -> R^n by F(x) = r(|x|)/|x| * x which squishes the unit ball to a point

#

So I guess even further, to show something is homeomorphic to a quotient you need to exhibit a quotient map going the right things together, right?

#

So the way you should think to prove R^n/B ≈ R^n is by exhibiting a quotient map R^n -> R^n squashing the unit ball to a point and injective on everything else

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I'm saying that to find such a function you want to just think about the norm of a point and work with that, do everything in a radially symmetric way

long coyote Mar 9, 2021, 11:33 PM

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now, i understand what you mean, thanks

sleek thicket Mar 9, 2021, 11:36 PM

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Cool

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Does it make sense that R^n \ {0} is homeomorphic to S^(n-1) × (0, infinity)?

long coyote Mar 9, 2021, 11:46 PM

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yes

sleek thicket Mar 9, 2021, 11:55 PM

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Yeah, so this extends to the case where you add in the origin and look at all of R^n

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But then in S^(n-1) × [0, infty) you have to crush the sphere S^(n-1) × {0} to a point

digital peak Mar 10, 2021, 10:55 AM

#

turns out

#

it's not even true

#

counterexample is the most obvious coordinate representation of SL2(R)

#

((a, b), (c, d)) mapsto (a-1, b, c)

gritty widget Mar 10, 2021, 5:45 PM

#

So lets say I have some function f that is continuous on a subset D of R^n. Is this equivalent to it fulfilling the preimage conditions in D with subspace topology?

median glade Mar 10, 2021, 5:48 PM

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Usually this terminology means that f is defined only on D, and continuous on D (with the subspace topology)

#

so yeah

marsh forge Mar 10, 2021, 5:48 PM

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Thats not true for R^n

#

you have a notion of point-continuity

#

Contuinity on a subset is at least stronger. To see that it is at least stronger, the preimage of an open set of the image of D must be open in R^n and in particular this means it will be open in the subspace topology of D.

#

but it might be equivalent

#

let me think

#

On open subsets D it should coincide

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and Im forgetting the details of continuity on a closed subset

sleek thicket Mar 10, 2021, 5:52 PM

#

what about sequences

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Can we use those to check continuity without worrying about being relatively open?

marsh forge Mar 10, 2021, 5:53 PM

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I'm not sure because how do you control sequences not taking place in D

sleek thicket Mar 10, 2021, 5:53 PM

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ah yeah

gritty widget Mar 10, 2021, 5:53 PM

#

D can be closed as an example and then the preimages are no necessarily open in R^n

marsh forge Mar 10, 2021, 5:53 PM

#

is the indicator for [0,1] conitnuous in R

gritty widget Mar 10, 2021, 5:53 PM

#

but they will be an intersection of an open set and D afaik

marsh forge Mar 10, 2021, 5:53 PM

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or sorry continuous on [0,1]

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Yeah desc thats what we are worried about

gritty widget Mar 10, 2021, 5:54 PM

#

ye but they will be open in the subspace topology so it should be fine

marsh forge Mar 10, 2021, 5:55 PM

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Yes so one direction is simple

#

but I don't think if you have a function f : R^n\to R^n

#

and f is continuous restricted to the subspace topology

#

that necessarily makes it cont on D

#

as a subset

#

Actually yeah take the indiciator of [0,1]. This is continuous on [0,1] as a subspace but not as a subset

#

because the sequence approching 0 from the left (choose your favorite) won't be preserved

#

@sleek thicket ur idea worked in reverse i think hahaha

sleek thicket Mar 10, 2021, 5:57 PM

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Oh lol

marsh forge Mar 10, 2021, 5:58 PM

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this was a good exercise im gonna put in on a pset if i ever teach point set

gritty widget Mar 10, 2021, 5:58 PM

#

I'd consider it continuous in [0,1] as a subset tho

marsh forge Mar 10, 2021, 5:58 PM

#

It is by definition not continuous at 0 or 1

gritty widget Mar 10, 2021, 5:58 PM

#

wait isnt this indicator =1 at 0 and 1

marsh forge Mar 10, 2021, 5:58 PM

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yep

#

but like

#

take a sequence approaching 0 from the left

#

you get nothing but 0s

gritty widget Mar 10, 2021, 5:59 PM

#

Ye so the discontinuity is outside the subset

marsh forge Mar 10, 2021, 5:59 PM

#

yes

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so it is continuous as a subspace function

#

but not as a subset

#

like the function f : R^n \to R^n is not continuous on [0,1]

#

but the function it induces [0,1]\to R^n is genuinely continuous

gritty widget Mar 10, 2021, 6:00 PM

#

If its discontinuous on the boundary thats fine for what im doing tho

marsh forge Mar 10, 2021, 6:00 PM

#

well

#

i think this should work fine

gritty widget Mar 10, 2021, 6:00 PM

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as long as it isnt discontinuous approaching from inside so to say

marsh forge Mar 10, 2021, 6:00 PM

#

pass to the interior of your set

#

yes continuity on a subset is certainly stronger

#

and if you have continuity as a subset you can pass to the interior

#

or more importantly like

#

you should be able to say that if D is a subset of R^n

#

and f is continuous on the interior of D as a subset of R^n

#

then f is continuous as a function on D as a subspace topology

#

i think something like this should work

gritty widget Mar 10, 2021, 6:02 PM

#

there has to be something more because i could have the boundary completely messed up

marsh forge Mar 10, 2021, 6:02 PM

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yeah its too strong i think

#

maybe if you explain your context I can be more helpful

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probably something like connected with nonempty interior might suffice

gritty widget Mar 10, 2021, 6:03 PM

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Well its stability analysis on a connected subset of R^n

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I can make additional restrictions without much loss like borel

#

So I am interested in the preimages of a function V:D->R D\subseteq R^n

#

I am additionaly demanding V to have bounded sublevel sets

#

The sublevel sets of V are invariant sets of the dynamical system

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If D is R^n then these are compact for preimages of closed bdd intervals which is very useful

#

So basically I'd like to characterise the continuity of V on D so that I get the preimages, and I think the subspace topology of D does that?

marsh forge Mar 10, 2021, 6:12 PM

#

I would see if you can prove it directly without worrying about lemmas

gritty widget Mar 10, 2021, 6:14 PM

#

Feels like it should just jump out at me

#

I mean it has to mean something when we say we have a function from a subset

marsh forge Mar 10, 2021, 6:16 PM

#

V on D will certainly be continuous

#

if it's continuous on D as a subset of R^n

#

as I described above

gritty widget Mar 10, 2021, 6:16 PM

#

Well yes, but what if it doesnt exist outside of D

marsh forge Mar 10, 2021, 6:16 PM

#

the issue is that V can be continuous even without that (I think my example above about [0,1] fits your hypotheses, for example)

#

what do you mean

#

you can't talk about the continuity of a function on a space for which the function doesn't exist

gritty widget Mar 10, 2021, 6:18 PM

#

I only care if its continuous in D, and continuous on the boundary from within D, outside D the function doesnt matter

marsh forge Mar 10, 2021, 6:19 PM

#

I really am not understanding you then

gritty widget Mar 10, 2021, 6:19 PM

#

so like [0,1] indicator, i dont care what happens in R-[0,1] it is of no importance in the context

marsh forge Mar 10, 2021, 6:19 PM

#

what exactly are you trying to do

#

If you want to show V: D\to R is continuous

#

you just have to show it

#

you can use epsilon-delta stuff for this

gritty widget Mar 10, 2021, 6:19 PM

#

I have given V is C1 on a subset D

#

I wonder if this means it has open preimages of open sets in the subspace D in the subspace topology

marsh forge Mar 10, 2021, 6:20 PM

#

it does

gritty widget Mar 10, 2021, 6:20 PM

#

noice

marsh forge Mar 10, 2021, 6:20 PM

#

there are more open sets in the subspace topology

#

open subsets of D

gritty widget Mar 10, 2021, 6:20 PM

#

yes

marsh forge Mar 10, 2021, 6:20 PM

#

so its easier to be continuous

gritty widget Mar 10, 2021, 6:21 PM

#

It is continuous in the metric sense within the subset at least

marsh forge Mar 10, 2021, 6:21 PM

#

no its continuous in the subspace as well

#

just take the sequential characterization

#

if all sequences are preserved

#

certainly sequences in D are

gritty widget Mar 10, 2021, 6:22 PM

#

ye

#

so V:D->R, preimage of [0,1] as an example would be closed by continuity, and bounded by assumption on V

#

But not compact depending on what the boundry is like

marsh forge Mar 10, 2021, 6:24 PM

#

yeah you need V to be proper to get the latter

gritty widget Mar 10, 2021, 6:25 PM

#

It is proper if D is compact because it would be an intersection of compact sets

#

Or if V goes to infinity at all boundaries not contained in D i suppose (V is bounded below by 0)

#

Thanks for clearing this up, although I still need to read up on some things 😛

strong heron Mar 10, 2021, 7:16 PM

#

Any 1-form can be expressed as a finite sum $\displaystyle \sum_{i} f_i dg_i$ for some smooth functions $f_i$ and $g_i$. Here, by finite, I mean locally finite. In other words, I think we need to do some partitions of unity stuff here.

gentle ospreyBOT Mar 10, 2021, 7:17 PM

#

Yash

strong heron Mar 10, 2021, 7:17 PM

#

Help?

#

I can find suitable f_i I think. But I'm unable to see how to get the g_i...

#

Any help would be much appreciated...

gritty widget Mar 10, 2021, 7:28 PM

#

can't you just do this in each coordinate chart and then patch it together with a partition of unity

strong heron Mar 10, 2021, 7:28 PM

#

Yes, but how do you define the f_i and g_i globally?

gritty widget Mar 10, 2021, 7:30 PM

#

to be zero outside of the coordinate neighbourhoods possibly?

#

then multiply by the bump functions in the POU to get something smooth

#

i haven't checked the details but something like that should work

strong heron Mar 10, 2021, 7:31 PM

#

I agree, something like that has to work, but my question is basically that I'm unable to make it precise...

#

I can show you what I attempted

#

#

The f_i's above are defined globally. But what about the g_i's? I cannot figure that part out.

strong heron Mar 10, 2021, 8:01 PM

#

@gritty widget You were saying something?

south minnow Mar 10, 2021, 8:10 PM

#

Is there any way to find the possible number of topological orderings of a tree. (that is a tree like looking undirected graph

#

shit, I'm not a math major

#

lol, my bad

empty pond Mar 10, 2021, 8:59 PM

#

hey I have a question and MasterTernimus said this is a better place to ask it

#

$$
\sinh^{-1}(2\pi p) + 2\pi p \sqrt{4 \pi^2 p^2+1} = \frac{4\pi v t}{d}
$$

gentle ospreyBOT Mar 10, 2021, 9:00 PM

#

もの

empty pond Mar 10, 2021, 9:00 PM

#

I'm trying to isolate p

#

I can also show how I got to this equation if you think there might be an easier way to get around this

#

here's my original question from #help-1 : #help-1 message

gritty widget Mar 11, 2021, 12:22 AM

#

I want to see that if V is a symplectic vector space, and J is a compatible complex structure then J sends a lagrangian subspace W to another lagrangian subspace and their intersection is trivial

#

I've show that the image of W is indeed a lagrangian subspace. but I can't see why the intersection has to be trivial

gritty widget Mar 11, 2021, 12:38 AM

#

if $g$ is an inner product on $V$ such that the compatibility equation $g(u,v)=\omega(u,Jv)$ is satisfied, suppose that $w \in J(W)\cap W$. then $w=Jw'$ for some $w'\in W$. then $$g(w,w) = \omega(w,Jw) = \omega(w, J^2w')= -\omega(w,w') = 0,$$ where the last equality follows from the fact that $W$ is lagrangian. since $g$ is positive-definite, $w = 0$, which proves that $J(W) \cap W = 0$

#

omega and w a little hard to read lol

gentle ospreyBOT Mar 11, 2021, 12:41 AM

#

(T*Terra, dqⁱ ∧ dpᵢ)

gritty widget Mar 11, 2021, 12:45 AM

#

lagrangian splittings catThink

#

ah yes thank you

rotund thicket Mar 11, 2021, 12:56 AM

#

Would $S^1 \times D$ where $D$ is the disk of unit radius be a torus with its insides filled?

gentle ospreyBOT Mar 11, 2021, 12:56 AM

#

Trichloromethane

gritty widget Mar 11, 2021, 12:57 AM

#

gritty widget ah yes thank you

thanks for the practice, my symplectic class is gonna do this exact lemma next class opencry

rotund thicket Mar 11, 2021, 12:57 AM

#

That's the word I was looking for lol

#

What about T x I

#

where I = [0, 1]

#

Is that also a torus?

#

just beefier

#

Like it would have inner radius still 1 but the outer radius would be 3 instead of 2

#

Uhhhhh

#

Actually

#

idk

gritty widget Mar 11, 2021, 12:59 AM

#

weird shape

rotund thicket Mar 11, 2021, 1:00 AM

#

but its basically just a torus

#

but thiccer

gritty widget Mar 11, 2021, 1:00 AM

#

im trying to imagine like, a torus, but with an interval attached to every point

#

a [0, 1]-bundle over T opencry

rotund thicket Mar 11, 2021, 1:01 AM

#

I mean because you choose a point on the surface of the torus and just go out in the perpendicular

#

welpm, i gotta draw it

gritty widget Mar 11, 2021, 1:02 AM

#

bro share some of your 4d paper

rotund thicket Mar 11, 2021, 1:03 AM

#

ill just make flipart

sleek thicket Mar 11, 2021, 1:04 AM

#

can't you just extrude in like they're saying

#

like

#

take a

#

t u b u l a r n e i b o r h o o d

#

by compactness you can choose a constant radius

#

That looks like what they're describing yeah?

#

and it's diffeomorphic to T × I

rotund thicket Mar 11, 2021, 1:05 AM

#

that's good enough for me

#

lol

#

so S^1 x D is the same thing as T x I basically

sleek thicket Mar 11, 2021, 1:06 AM

#

Yeah I think you're right :)

#

Just trying to convince tterra

#

Hmm, I don't think these are the same

gritty widget Mar 11, 2021, 1:06 AM

#

i still don't have a response from fields

rotund thicket Mar 11, 2021, 1:07 AM

#

I mean that's how my prof said to think about S^1 x I

#

you just choose a point on the circle and go some distance

sleek thicket Mar 11, 2021, 1:08 AM

#

I think of it as a cylinder

rotund thicket Mar 11, 2021, 1:08 AM

#

I mean the thinking is the same

#

I just go out some distance

#

whether in the same plane

#

or the other perpendicular one

sleek thicket Mar 11, 2021, 1:09 AM

#

Hmm I think I disagree with what I was saying

#

earlier

#

Oh no

#

I see

#

So the thick torus S^1×D is filled in

#

in the middle

#

T×I still has a hole

#

In the tube

rotund thicket Mar 11, 2021, 1:10 AM

#

Does it?

sleek thicket Mar 11, 2021, 1:10 AM

#

Yup

#

You haven't filled anything in

#

Just extruded out the sides a little

rotund thicket Mar 11, 2021, 1:10 AM

#

what about the inner radius going outward

sleek thicket Mar 11, 2021, 1:10 AM

#

Think about it like this: take a torus and taken outward pointing a normal vector to it

#

Attach a stick in the direction of the normal vector at each point

rotund thicket Mar 11, 2021, 1:11 AM

#

Ohhhhhhhhhh

sleek thicket Mar 11, 2021, 1:11 AM

#

That's T^2 × I

rotund thicket Mar 11, 2021, 1:11 AM

#

I see, i was thinking that I could, in your analogy, take the negative of the normal when im on the inner radius

sleek thicket Mar 11, 2021, 1:12 AM

#

Well you can take the inward normal

#

But you can't fill everything in

#

You have to make sure the little sticks don't intersect

rotund thicket Mar 11, 2021, 1:12 AM

#

So like half of the torus is filled?

sleek thicket Mar 11, 2021, 1:12 AM

#

That's not how I'm thinking about it

#

It's like if you took a solid torus

#

And then took a smaller solid torus contained in it

#

And cut that out

rotund thicket Mar 11, 2021, 1:13 AM

#

i see

sleek thicket Mar 11, 2021, 1:13 AM

#

like, T×I = (S^1 × I) × S^1

#

S^1 × I can be thought of as an annulus in the plane

#

yeah?

#

So take the surface of revolution of the annulus

rotund thicket Mar 11, 2021, 1:13 AM

#

Yeah breaking it up into its components makes it easier to think about

#

thanks!

#

So you can just commute the products?

sleek thicket Mar 11, 2021, 1:16 AM

#

Yup

#

Up to homeomorphism

#

Geometrically this is like a reflection of R^3

#

Being applied to the shape

#

Swapping the y and z coordinates

rotund thicket Mar 11, 2021, 1:17 AM

#

that's cool

#

Wait so that means that S^2 x I is homeomorphic (?) to T x I?

#

Im getting that S^2 x I is a 3d annulus

sleek thicket Mar 11, 2021, 1:27 AM

#

no

rotund thicket Mar 11, 2021, 1:28 AM

#

Oh crap i was adding the exponents :p

sleek thicket Mar 11, 2021, 1:28 AM

#

What specifically do you mean by 3d annulus?

rotund thicket Mar 11, 2021, 1:28 AM

#

i mean a cyllinder

sleek thicket Mar 11, 2021, 1:28 AM

#

I think S^2 × I and T×I are both sort of this

rotund thicket Mar 11, 2021, 1:28 AM

#

with a hollow center

sleek thicket Mar 11, 2021, 1:28 AM

#

I think that would just be S^2

#

a hollow cylinder I mean

rotund thicket Mar 11, 2021, 1:29 AM

#

Oh wait I mixed myself up, S^2 x I is a sphere with a hollowed out center

sleek thicket Mar 11, 2021, 1:29 AM

#

careful with your words

#

A sphere is a shell

#

Like S^2

#

also "ball with a hollow center" describes a sphere imo

rotund thicket Mar 11, 2021, 1:30 AM

#

yeah

#

that's what i mean

sleek thicket Mar 11, 2021, 1:30 AM

#

Here you have some bits hollow and some not

rotund thicket Mar 11, 2021, 1:30 AM

#

ok, ball=filled, sphere=shell

sleek thicket Mar 11, 2021, 1:30 AM

#

anyways yeah that's what S^2×I looks like

#

Whereas T×I is like the surface of revolution of an annulus

rotund thicket Mar 11, 2021, 1:31 AM

#

but its still filled around the inner torus?

#

here's my drawings

#

please dont judge

sleek thicket Mar 11, 2021, 1:32 AM

#

No judgment here haha

#

Let's start at the bottom right

rotund thicket Mar 11, 2021, 1:33 AM

#

ok

sleek thicket Mar 11, 2021, 1:33 AM

#

I agree with this drawing

#

Also S^2 × I

#

Take a closed ball and scoop out an open ball inside

#

Of eg half the radius

rotund thicket Mar 11, 2021, 1:34 AM

#

I'm imagining some mortal combat stuff rn lol

sleek thicket Mar 11, 2021, 1:34 AM

#

lmao

#

I'm having trouble telling apart your drawings of S^1×D and T×I

rotund thicket Mar 11, 2021, 1:34 AM

#

Yeah so S^1×D is the solid torus

sleek thicket Mar 11, 2021, 1:34 AM

#

Right

rotund thicket Mar 11, 2021, 1:35 AM

#

and then T×I is like S^2×I in that we scooped out the child of the parent object

sleek thicket Mar 11, 2021, 1:35 AM

#

Yup, exactly

rotund thicket Mar 11, 2021, 1:35 AM

#

ok dope

sleek thicket Mar 11, 2021, 1:35 AM

#

wait, I'm totally wrong about T×I being like an annulus but you rotate it

#

That will give you S^2×I

#

Sorry for the confusion!

rotund thicket Mar 11, 2021, 1:36 AM

#

So they're flipped?

sleek thicket Mar 11, 2021, 1:36 AM

#

Your drawings/mental models are right

#

I just said something wrong earlier

rotund thicket Mar 11, 2021, 1:37 AM

#

yeah yeah like the difference between it being filled and a surface

#

makes sense now

#

so basically multiplying by I causes hollowness

sleek thicket Mar 11, 2021, 1:49 AM

#

Hmm, sort of

#

The objects you started with were "hollow", right?

#

I think what's going on is sort of the difference between a cylinder and a cone

#

Like, the cylinder is homeomorphic to an annulus whereas the cone is homeomorphic to a disk

#

Hollow vs filled

#

There's lots general definitions of a cylinder and a cone on a topological space (the above are the case where your base is the circle)

#

The cylinder on X is X × I, the cone is more complicated (you need to have seen quotient spaces to define it)

#

In the case of a sphere the cylinder is the hollow thing S^2 × I whereas the cone is the filled in ball

#

This isn't exactly right for the torus examples though, so maybe I'm wrong

quartz edge Mar 11, 2021, 6:00 AM

#

if the eigenvalues of the second fundamental form of a surface at a point are the corresponding principal curvatures, what then are the eigenvalues of the first fundamental form? are they the principal 'speeds' of travel wrt. the parameterization?

quartz edge Mar 11, 2021, 6:26 AM

#

it would seem so from prodding at it

#

yes

gentle ospreyBOT Mar 11, 2021, 6:27 AM

#

gristle

quartz edge Mar 11, 2021, 6:28 AM

#

my linear algebra is weak as i'm just now studying it, i merely know how to compute these things so the geometric intuition can be lost on me at times

#

for instance i have trouble visualizing what it means to multiply a tangent vector by I_P

#

what sort of transformation does that entail

#

like what does the new vector correspond to

#

i can compute lengths, areas, prove certain things analytically, but i guess understanding what this thing is as an object is confusing to me

#

@gritty widget what do you mean when you say 'induces the inner product'

#

this terminology is something i feel i ought to be able to use but the words don't really make any sense to me

#

what is meant by induce

#

and why is the inner product referred to as inner

gentle ospreyBOT Mar 11, 2021, 6:47 AM

#

~S^1
Compile Error! Click the errors reaction for more information.
(You may edit your message to recompile.)

shut moat Mar 11, 2021, 6:48 AM

#

IMO it might be easier to get an intuition for what $\mathbf{v}^T I_P$ is. It's the function that takes $\mathbf{w} \mapsto \langle \mathbf{v}, \mathbf{w}\rangle$

gentle ospreyBOT Mar 11, 2021, 6:48 AM

#

~S^1

shut moat Mar 11, 2021, 6:50 AM

#

quartz edge <@456226577798135808> what do you mean when you say 'induces the inner product'

Basically, if you have any bilinear function $B$, you can express it as $B(\mathbf{v}, \mathbf{w}) = \mathbf{v}^T [B] \mathbf{w}$ for some matrix $[B]$. Slimvesus was just making this distinction between the matrix that gives the inner product and the inner product itself

gentle ospreyBOT Mar 11, 2021, 6:50 AM

#

~S^1

shut moat Mar 11, 2021, 6:51 AM

#

and dunno why it's called inner lul

quartz edge Mar 11, 2021, 6:51 AM

#

ye

#

that much seems reasonable to me

#

also

#

for

gentle ospreyBOT Mar 11, 2021, 6:52 AM

#

gristle

quartz edge Mar 11, 2021, 6:52 AM

#

we need E=G to be able to say this is the same as

#

yeah?

shut moat Mar 11, 2021, 6:53 AM

#

that second expression doesn't make sense

gentle ospreyBOT Mar 11, 2021, 6:53 AM

#

gristle

shut moat Mar 11, 2021, 6:53 AM

#

still doesn't really make sense

sleek thicket Mar 11, 2021, 6:53 AM

#

Wym approx?

shut moat Mar 11, 2021, 6:53 AM

#

I think you want to exchange v and w?

sleek thicket Mar 11, 2021, 6:53 AM

#

That makes sense to me?

quartz edge Mar 11, 2021, 6:54 AM

#

the dimensions work out

shut moat Mar 11, 2021, 6:54 AM

#

oh wait

#

I read this as I_P(v.w)

#

smh smh

quartz edge Mar 11, 2021, 6:54 AM

#

but yea it relies on symmetry of I_P right?

sleek thicket Mar 11, 2021, 6:54 AM

#

They're asking whether you need E=G for v^T I_p w = (I_p v)^T w

#

Yes gristle

quartz edge Mar 11, 2021, 6:54 AM

#

i meant err

#

WELL

#

you dont need E=G

sleek thicket Mar 11, 2021, 6:54 AM

#

The inner product of a and b can be written as a^T b

quartz edge Mar 11, 2021, 6:55 AM

#

it always works actually

sleek thicket Mar 11, 2021, 6:55 AM

#

quartz edge you dont need E=G

Nope

#

but you need symmetry

quartz edge Mar 11, 2021, 6:55 AM

#

cuz F=F

#

ye

shut moat Mar 11, 2021, 6:55 AM

#

yeah

quartz edge Mar 11, 2021, 6:55 AM

#

nice

#

i like that

#

well

#

so lemme get this straight

sleek thicket Mar 11, 2021, 6:56 AM

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hmmm

quartz edge Mar 11, 2021, 6:56 AM

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I_P v is a function

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it takes another vector w and spits out <v, w>

sleek thicket Mar 11, 2021, 6:56 AM

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This is very sticky

quartz edge Mar 11, 2021, 6:56 AM

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this much is clear

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if sticky

sleek thicket Mar 11, 2021, 6:56 AM

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I would say I_p v is a vector

quartz edge Mar 11, 2021, 6:56 AM

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yea

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same

sleek thicket Mar 11, 2021, 6:56 AM

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You're multiplying a matrix by a vector