random cohomology for quantum nerds - 2020-04-21 (page 1 of 2)

Balarka Sen

10:50

(You can see deleted messages by clicking on it and showing history; room owners can do that)

How's the situation in Italy right now? Stable yet?

Alessandro Codenotti

Yesterday for the first time the number of active cases decreased for the first time

(by 20, also the number of active cases decreasing is not necessarily good, it might just mean that a lot of people died, but in fact the number of new cases was also the lowest recorded in a month and a half)

We're still in a full lockdown until the 3rd of May and it's not clear what will happen afterward though, which restrictions (if any) will be lifted

Balarka Sen

Let's hope that's the end of the outbreak on your side

Alessandro Codenotti

It will probably take a while, we still have around 100k active cases and 2-3k new cases per day

And I'm only talking about tested and confirmed cases, the estimated numbers of total cases counting asymptomatic people are crazy

Balarka Sen

Ugh

We're at the beginning of the avalanche as well

Still not so bad but that can't be far

Balarka Sen

11:24

OK, given a topological space $X$, Spivak defines an end $\varepsilon$ of $X$ to be a function $\varepsilon$ on the compact subsets of $X$ such that $\varepsilon(K)$ is a connected component of $X \setminus K$ such that $K_1 \subset K_2$ implies $\varepsilon(K_2) \subset \varepsilon(K_1)$. This is the same thing as an element of $\varprojlim \pi_0(X \setminus K)$.

Call this $\mathcal{E}(X)$ and the end-compactification is as a set $X \cup \mathcal{E}(X)$. Define neighborhoods of an end $\varepsilon_0$ in here to be $N_K(\varepsilon_0) = \varepsilon_0(K) \cup \{\varepsilon : \varepsilon_0(K) = \varepsilon(K)\}$. This is the obvious topology

If $X$ is locally compact Hausdorff, connected and locally connected then the claim is $\overline{X} = X \cup \mathcal{E}(X)$ is compact Hausdorff

Mike Miller

We hear that NYC is expected to reopen by maybe September

School probably won't though since there's a lot of international students

Balarka Sen

NYC is getting hit hard right? There's like 3k cases every day

Balarka Sen

11:42

Clearly the right way to do this is by nets. Take any net $(x_\alpha)$ in $X$. If it entirely goes inside some compact set, then it must have a convergent subnet. So assume not; then for every compact set $K \subset X$ there is some $\alpha_K$ such that $x_{\alpha_K} \notin K$.

Choose any random end $\varepsilon$. This hypothesis tautologically implies $(x_\alpha)$ has $\varepsilon$ as an accumulation point.

Ah, but I want end-convergence as well. $(x_\alpha)$ need not belong to $X \subset \overline{X}$. Ends can converge to ends. Let's see how to phrase that.

I should be able to prove $\mathcal{E}(X)$ is compact first

That's probably where I need connected and locally connected.

Alessandro Codenotti

12:30

@MikeMiller That seems sightly optimistic, we're talking about September in the best case in Italy, but January seems likely

Mike Miller

I'm just repeating things NY State people have said

I expect what they mean is that in Sept restrictions will be loosened

Alessandro Codenotti

I know, I'm just saying that they have said optimistic things in my opinion

Mike Miller

Fair nuff, just making clear I am not trying to be an epidemiologist myself

Balarka Sen

@Alessandro I got the problem

I can write it down if you want

Alessandro Codenotti

Wait I'm still in that class

Balarka Sen

12:43

Oh ok

Alessandro Codenotti

Yoneda's embedding coming up

Balarka Sen

Oof

Alessandro Codenotti

Indeed. We did a review of (co)limits. Me is confused

Balarka Sen

Actually maybe I'll write it down in my school google classroom just to troll my instructor

I'll copy-paste it here later

Alessandro Codenotti

lol

Mike Miller

12:46

Why do you need the yoneda embedding.

Alessandro Codenotti

Because we want to talk about presheaves

Balarka Sen

lol

nuts

Alessandro Codenotti

I'd better not mention what the course is about because I'll be flooded by memes

Balarka Sen

Higher geometry

I remember you mentioned a course like that

Alessandro Codenotti

Did I? I don't know what higher geometry is

Balarka Sen

12:53

Oh is this infinity categories

Alessandro Codenotti

There is an higher category theory course in Bonn but I'm not doing that

It's a models of HoTT course lol

Balarka Sen

damn

Alessandro Codenotti

And apparently the easiest one is Voevodsky's model in simplicial sets

Mike Miller

13:11

Logic is respectable I suppose

Alessandro Codenotti

Apparently the category of graphs is a presheaf category. I don't know what to do with this information

Mike Miller

13:49

I guess that just means that a graph is two sets E, V equipped with two maps E -> V

Stupid

Alessandro Codenotti

Yeah exactly

Alessandro Codenotti

15:39

That was really just a slightly less trivial example than the category of presheaves over a point is just Set

Also we saw a nice way to think about limits that made it really obvious to me why the can be pulled out of the right side of hom-sets

Mike Miller

15:53

I still don't really get why people care

Just "I want closure under (co)limits really badly"?

Alessandro Codenotti

care about what exactly?

Balarka Sen

Convenient Categories

Mike Miller

Why bother talking about presheaf categories

Alessandro Codenotti

Because we want to talk about simplicial sets which are presheaves categories, I'm not sure what do we gain by doing the general case rather than just talk about simplicial sets though

Balarka Sen

Time to read my rant here

:3

I think I promptly forgot most of what I learnt about simplicial sets lol

Alessandro Codenotti

16:00

LoL

I'll learn them from the lecturer, I have no hurry to find out how they work

I went back to read about residually finite groups and suddenly there's projective limits again, there's no escape

Balarka Sen

Ripriprip

Alessandro Codenotti

Oh ok but there's an honest explicit description of what a projective limit of groups looks like as a subgroup of the product

Phew

A more serious student would now verify that it does have the correct universal property

Balarka Sen

Inverse limits are pretty much nice objects

Unless you have a fucked category to begin with

Alessandro Codenotti

Fair enough, but I only care about groups right now so everything is beautiful

Balarka Sen

Once a professor in TIFR was taking an innocent walk but suddenly decided to corner me and gave me the following puzzle "Are inverse limits of an inverse system of nonempty objects with epimorphisms between them always nonempty?"

Alessandro Codenotti

16:06

@AlessandroCodenotti this actually looks easy after thinking about it for a strictly positive amount of time

@BalarkaSen I guess not if your category is messed up and it's very easy to be an epi?

Balarka Sen

There are examples in Sets

:3

Alessandro Codenotti

Oh damn

I can construct examples in sets if choice fails lol

Balarka Sen

Lmao

Alessandro Codenotti

Choice is equivalent to the cartesian product of nonempty sets being nonempty

Balarka Sen

Unfortunately I think the example is consistent with choice

Alessandro Codenotti

16:10

Also that's a limit rather than an inverse limit

I know I was joking, let me think about it

Balarka Sen

My immediate instinct was to take the indexing set to be huge. I wanted to set up a vector field on a manifold whose flow blows up in finite backward time, and take inverse limit of the inverse system given by the flow

Indexing set is $\Bbb R$ ofc

But that can't work, because the morphisms are isomorphisms

He told me the complicated example afterwards

The indexing set is $P(\Bbb R)$, if that helps

Nah it's smaller than that.

Alessandro Codenotti

Oh ok I think I see it. The index set is $\omega_1$, and for each $\alpha\in\omega_1$, $X_\alpha$ is the set of order preserving injections $\alpha\to\Bbb Q$. Then an element of the inverse limit should be an injection $\omega_1\to\Bbb Q$

Balarka Sen

Yeah that's it

You can index over finite subsets of $\Bbb R$ and let $X_\alpha$ the set of injections to $\Bbb Z$ as well

Alessandro Codenotti

Yeah, once you think about sets of functions rather than sets it's easy to construct examples where the codomain is small and you're gluing together too many functions

I just wanted to do some honest group theory and now you tricked me into categorical nonsense!

I'm out

Balarka Sen

just set theory nonsense

Alessandro Codenotti

16:19

To be fair I had seen a similar construction, just not phrased in those terms

Balarka Sen

yeah i thought you might be able to say it immediately

Alessandro Codenotti

Maybe you remember, I mentioned this on discord, you can build in ZFC a tree of height $\omega_1$ with countable levels such that there is no uncountable branch through the tree

Balarka Sen

Ah this is exactly that kind of phenomenon

Of course

Very clear now

Alessandro Codenotti

And this is done by building a tree made of increasing functions from countable ordinals into Q ordered by extension

Balarka Sen

Yeah, the set of ends of the tree is like the inverse limit of the finite levels

Alessandro Codenotti

16:21

And you have no uncountable branch by the same reason, gluing together all the functions in the branch would inject $\omega_1$ into $\Bbb Q$

@BalarkaSen Yeah, for trees with finitely many successors for every edge (finite degree? I always forgot what that's called) that's also an exhaustion by compact sets

Balarka Sen

Yeah.

Alessandro Codenotti

And we're back to Cayley graphs

Gromov has entered the chat

Balarka Sen

Lmao the one man who puts all conversations in the same bucket

Everything we were thinking about today is suddenly the same thing

Aronszajn trees <-> Inverse limits <-> Space of ends

Alessandro Codenotti

16:46

I really like this intuition about inverse limits as branches through trees. But that's probably just because I spent a lot of time working with trees in set theory

There's also another kind of combinatorial trees considered in set theory which are called Kurepa trees, a $\kappa$-Kurepa tree has height $\kappa$, levels of size smaller than $\kappa$, and at least $\kappa^+$ branches. So it's like the opposite of an Aronszajn tree

Balarka Sen

$\kappa^+$ is the successor ordinal?

Weird notation

Alessandro Codenotti

successor cardinal

Balarka Sen

Ahok

ofc

Alessandro Codenotti

So like the first case would be a tree of height $\omega_1$ with countable levels and at least $\omega_2$ cofinal branches

Balarka Sen

Got it

Alessandro Codenotti

16:56

Turns out that "there is no Kurepa tree" is equiconsistent with "there is an inaccessible cardinal"

(Kurepa tree means $\omega_1$-Kurepa tree if the cardinal is not specified)

Balarka Sen

What's an inaccessible cardinal again

Mike Miller

Not a power set or a (small) union I thought

Maybe not a subset of a powerset or a small union

Alessandro Codenotti

It's a regular cardinal $\kappa$ for which $\lambda<\kappa\implies 2^\lambda<\kappa$

Regular essentially means that it is not the union of less than $\kappa$ cardinals all smaller than $\kappa$

Balarka Sen

Ah OK

Mike Miller

That's what I said right

Alessandro Codenotti

17:02

It's the kind of cardinals you want to get Grothendieck universes if you're an algebraic geometer

@MikeMiller yeah that's right

Mike Miller

I assume $\omega$ doesn't count

Alessandro Codenotti

Right, I should have said an uncountable cardinal such that blablabla

Mike Miller

Yeah yeah I was just nitpicking

I think we all agree countable sets exist

Alessandro Codenotti

Wildberger intensifies

Balarka Sen

Lmao

loch

17:06

what are Grothendieck universes ? (i've never bothered to learn it)

Balarka Sen

There's a Wildberger article on how the Alexander horned sphere does not exist

Alessandro Codenotti

They're sets $U$ closed under a bunch of operations, like pairs, powersets, unions indexed over elements of $U$ and maybe some more

The idea is that as long as you work with objects in $U$ you can do all constructions you care about without leaving $U$

Set theorists would say to pick $U=V_\kappa$ for a big enough inaccessible $\kappa$, but algebraic geometers like to formulate this is that other (but mostly equivalent) way

Balarka Sen

@Alessandro Can you give me a filter-free proof that every net admits a universal subnet

Alessandro Codenotti

I don't know what an universal subnet is

loch

hmm
ok
but i dont know why one needs such a notion

lol

Alessandro Codenotti

17:10

I never learned properly how nets work, I now how they are supposed to behave enough to handwave nets arguments lol

Balarka Sen

A universal net $(x_\alpha)$ in $X$ is something such that if $A \subset X$ is any subset, $(x_\alpha)$ is either eventually in $A$ or in $X \setminus A$

Alessandro Codenotti

@loch Apparently Grothendieck used them to avoid proper classes, instead of proper classes you just have sets which happen to live in a bigger universe than the current one

Balarka Sen

It's probably equivalent to Choice

Alessandro Codenotti

I'm not sure why that's better than dealing with proper classes

This is somewhat similar to the stratified universes approach of type theory

@BalarkaSen It smells of choice since ultrafilters $F$ on $X$ are characterized by either $A\in F$ or $X\setminus A\in F$ for all $A\subseteq X$

Balarka Sen

Ah ok

I don't know anything about filters

I learnt something once but promptly forgot

Alessandro Codenotti

17:14

But "every filter is contained in an ultrafilter" is known as the ultrafilter lemma and is weaker than choice

So I'd guess that you're statement is also equivalent to the ultrafilter lemma

@BalarkaSen They're definitely the correct way to prove Thyconoff's theorem

Balarka Sen

Is $\{x_\beta : \alpha \leq \beta\}$ a filter

Probably

So maybe I just put it inside your ultrafilter and funk with it

Alessandro Codenotti

I know nothing about nets sorry

Balarka Sen

Give me an accessible intro to filters so I can figure out if my thing follows from ultrafilter lemma or not

Alessandro Codenotti

Uhm I'm not sure

I think a topology book that does the filter approach would be more interesting for you than learning filters from a set theory book

Balarka Sen

Actually I'll just figure it out lol. A filter is a subset of $P(X)$ which is closed under intersection, union, and if $A$ is in the filter, $A \subseteq B$, $B$ is in the filter?

Alessandro Codenotti

17:23

no unions

Balarka Sen

Ya I'll just learn it by overhearing conversations it's ok

Alessandro Codenotti

And finite intersections

Balarka Sen

Ah ok

Cool

Alessandro Codenotti

Also usually people assume that the empty set is not an element of filters

(because otherwise you just have the whole of $\mathcal P(X)$, and a bunch of facts about filters fail for this trivial case so you always need to specify proper filter everywhere and it's easier to just exclude the annoying case)

Balarka Sen

Got it

If $\mathcal{A}$ has FIP then $\mathcal{A}$ is contained in an ultrafilter - that's the ultrafilter lemma. Should be easy to prove using AoC, lets see

Alessandro Codenotti

17:26

Well the point is that if A has the fip then A can be extended to a filter (no choice so far) and filters can be extended to ultrafilters

Balarka Sen

Oh fip is just so that it generates a filter

because of your emptyset axiom

Ah and filters extend to ultrafilters because simple Zorn's lemma

Alessandro Codenotti

Yep

Balarka Sen

Cool

Let $(x_\alpha)_I$ be a net on $X$. Then $\{x_\beta : \alpha \leq \beta\}$ is a filter, so by ultrafilter lemma contained in an ultrafilter $F$. The subnet indexed by $(A, \alpha)$, $A \in F$ and $\alpha \in I$ such that $x_\alpha \in A$ will be my obvious attempt at the universal subnet

Of course that works

Lol

Set theory is trivial

Everything is a tautology

Alessandro Codenotti

lol

Balarka Sen

It's exactly like algebraic geometry

All proofs are tautologies

Alessandro Codenotti

17:35

@BalarkaSen That's a big offense

Alessandro Codenotti

18:27

Trying to understand why groups are the inverse limit of their finitely generated subgroups @Balarka

This is obvious if I think in terms of gluing groups together, but I don't see it if I think about the inverse limit as a subgroup of the direct product

Balarka Sen

@Alessandro Huh? Did you mean direct limit?

Alessandro Codenotti

yes

I meant projective limit

I think projective and direct limit should be banned and only limit and colimit should be used, those names are too confusing

Balarka Sen

Projective is inverse and injective is direct for me

Direct limit of groups is not a subgroup of the product

Alessandro Codenotti

I swapped them lol

That happens every time

Balarka Sen

But I am totally confused about what you said. $G$ is indeed $\varinjlim_{H \; f.g} H$; that's not a subgroup of the product

It's a quotient garbage

Alessandro Codenotti

18:33

Ok wait let me make some order in my confusion

Oh ok I see it now

hmm maybe

Balarka Sen

What is the definition of direct limit of groups for you

Alessandro Codenotti

Disjoint union modulo uglyness

Balarka Sen

Yeah

Alessandro Codenotti

Either that or the universal property

Balarka Sen

Rule 1 is that universal property is always a property and never a definition. It gives no insight into the object

Alessandro Codenotti

18:40

Let's think in terms of the universal property. Is picking a group hom $G\to H$ the same as picking group homs $G_i\to H$ for all f.g. subgroups (with the obvious compatibility conditions)?

@BalarkaSen Right, but if you have a candidate object it's the easiest thing to think about

@AlessandroCodenotti Ok this is clear, I would have to check naturality in $H$ as well but I'm willing to believe that

Ok I'm convinced

Balarka Sen

Just note that if $H$ is a f.g subgroup of $G$ the inclusions $H \to G$ assemble to a map $\varinjlim_{H f.g.} \to G$. Tell me why this is an isomorphism

It's weird to think in terms of universal property. A direct limit is like a union, that's all you should focus on in a nice category (direct limit of groups is bad because is not really a subquotient but it's still OK to do this)

Alessandro Codenotti

I don't know I find it easier

@BalarkaSen Can't I just check that it is bijective from the "cocone diagram"

Balarka Sen

Are you trolling me

Alessandro Codenotti

The diagram showing the universal property of $\lim H_i$ I mean

Balarka Sen

It's a direct elementwise argument

Alessandro Codenotti

18:51

lol not sure what you mean

Mike Miller

Your arguments are the same

Just very different language imo

Balarka is saying that it's surjective because every element of G lies in an fg guy, and it's injective because any relation between finitely many elements lies in a finitely generated subgroup

Alessandro Codenotti

Yeah ok we were saying the same thing in a different way

Balarka Sen

It's injective because the inclusion maps $H \to G$ are injective, and the binding maps in the directed system are injective

Mike Miller

You're saying the same thing (any map from the colimit is determined by its values on fg subgroups, and if you can define a map on all the subgroups in a compatible way then it's globally defined)

@BalarkaSen I'm not even thinking of those as homomorphisms just subsets

Balarka Sen

Ok fine

Alessandro Codenotti

18:55

Like if I have $x\in G$ then the inclusion map $\langle x\rangle\to G$ + the universal property diagram show that $x$ is hit by $\lim H_i\to G$, so we have surjectivity

Injectivity is similar

Balarka Sen

That is an absurd way of thinking

Alessandro Codenotti

:C

Balarka Sen

Honest to god if I didn't know you and you told me the answer to my question was a cocone diagram I would have exploded on the spot

Fucking hell

(I'm laughing by the way, in case this sounds serious)

Alessandro Codenotti

So now stupid question. Say I have a group $G$, I write it as the limit over its f.g. subgroups and passing to Cayley graphs I get a limit in Top, what goes wrong if I call that limit the Cayley graph of $G$ and try to do geometry with it?

Balarka Sen

Seems like it could be very bad. What's the natural generating set?

Can you show it to me with $\Bbb Q$?

Alessandro Codenotti

19:00

Well I guess that's really just a messy way to get the Cayley graph of $G$ wrt the union of the generating sets picked for all the f.g. subgroups

Balarka Sen

I actually thought of a similar question but with inverse limits once

Mike Miller

@BalarkaSen It's one of these irritating things where sometimes the universal property takes the element-wise argument and makes it clear what's really going on, but that'snot always true

So here you have a universal property argument that really is just a rephrasing of the elementwise argument I think

@BalarkaSen I remember this

p-adic solenoids

gal(qbar,q)

Balarka Sen

lmao yeah

Now I know what I was heading towards was a perfectoid space but Peter Scholze beat me to the Fields medal

Mike Miller

Poor boy

OK I'm going to be an asshole

Balarka Sen

Actually Sullivan does geometry on solenoids in his "Localization, Periodicity, Galois Symmetry"

At the very end

It's too technical for me though

I should read it someday

Mike Miller

19:05

@BalarkaSen What's the correct statement of this that's generally valid

Balarka Sen

Yeah I don't know actually. It's true for fields as well, any field extension is direct limit of finite subextensions

I was thinking of something like that

Mike Miller

Vector spaces and sets too, obviously

Alessandro Codenotti

In the HoTT course the direct limit of some functor from an index category $X:I\to C$ was defined to be an object $\lim X(i)$ such that $\mathrm{Hom}_C(c,\lim X(i))\simeq \mathrm{Cone}(c,X(i))$ so that's why I wanted to think in terms of maps from the direct limit and maps from the factors / the universal property

I think it holds for modules too?

Balarka Sen

yah for sure

Mike Miller

I'm sure it's fine in any algebraic setting

Balarka Sen

19:07

It's a good question

Maybe Mike should think about it

Alessandro Codenotti

What does finitely generated mean outside of an algebraic setting?

Balarka Sen

and tell us the answer

Mike Miller

Compact object, probably

Balarka Sen

Lmao ^

ncatlab.org/nlab/show/…

Alessandro Codenotti

I was thinking something like if you have a "free functor" $F:Set\to C$ then an object $c\in C$ could be reasonably called finitely generated if there is an epimorphism $F(n)\to c$ for some $n$ but I don't know if this makes any sense

Mike Miller

19:10

Depends on the choice of F

Alessandro Codenotti

I want $F$ to be left adjoint to the forgetful functor $C\to Set$ for concrete $C$

That's what an honest "free object" should satisfy hopefully

Balarka Sen

Seems bad for Top

Mike Miller

The forgetful functor is a choice

Fine, "concrete C"

ncatlab.org/nlab/show/locally+finitely+presentable+category

It's probably this

Balarka Sen

Also apparently compact object in Top is not a compact space

Alessandro Codenotti

@BalarkaSen For top you only get finite spaces to be finitely generated with this approach

Balarka Sen

19:12

Lol

yeah that's what I meant

Alessandro Codenotti

I have no intuition what a f.g. topological space should or should not be though

Balarka Sen

You probably want something like every nice topological space is a direct limit of compact spaces as an answer to Mike's q

Mike Miller

Compact object (mathematics)

In mathematics, compact objects, also referred to as finitely presented objects, or objects of finite presentation, are objects in a category satisfying a certain finiteness condition. == Definition == An object X in a category C which admits all filtered colimits (also known as direct limits) is called compact if the functor Hom C ⁡ ( X , ⋅ ) : C → S e t s , Y ↦...

Balarka Sen

Which is probably true for compactly generated spaces

Mike Miller

Yeah exactly

Balarka Sen

19:13

So it seems like there should be better answer

Mike Miller

My cat stole a chicken tender from me

Balarka Sen

Rekt

Mike Miller

Just grabbed it with his mouth and walked away

Honestly he deserves it

Alessandro Codenotti

LOL

But yeah compact objects are finite discrete spaces, so also not the correct notion

Balarka Sen

This is what happens to category theorists

Mike Miller

19:17

Yeah a category theorist tells me these are called locally finitely presentable categories

That's enough for me

Balarka Sen

is this Kevin

Alessandro Codenotti

en.wikipedia.org/wiki/Finitely_generated_object by the way this is a thing, even though is doesn't look like a useful thing

Mike Miller

yes

See item 3 onm the examples page

Looks like any "category of algebraic objects" has this property

Great

That's everything I'm ever going to need to know about this before I die

Balarka Sen

Nice

Alessandro Codenotti

lol

Leaky Nun

19:31

@MikeMiller aww

Alessandro Codenotti

19:41

What's your cat called? @Mike

Mike Miller

Henri

Alessandro Codenotti

I think I might have asked before but I forgot in case

@MikeMiller nice, never heard of a cat named Henri

Balarka Sen

Henri Ca(r)t(an)

Mike Miller

Poincate

3

Alessandro Codenotti

I asked a category theory person to explain to me what locally finitely presentable means. I think I made a mistake

@MikeMiller that's genius

Balarka Sen

19:44

I forgot Poincare's first name was Henri

I should have figured it out

I should get a cat named Misha

Alessandro Codenotti

lol

I know a guy with a cat named Klein

Balarka Sen

Feline Klein

Alessandro Codenotti

Crazy guy, he's doing a PhD in real algebraic geometry

Balarka Sen

thats a nice subject

i should learn something about it at some point

Mike Miller

I didn't know you could do a PhD in that

Alessandro Codenotti

19:52

In Trento there's a professor who's research is mostly in that and that's his supervisor

Mike Miller

I downloaded a book by Jost about harmonic maps but the people who uploaded the PDF have replaced his intro with 10 pages of a book about K-theory of groups

?????

Alessandro Codenotti

LoL

Balarka Sen

NSA knows you Mike

they know what you secretly like to read

Alessandro Codenotti

I found a book on libgen once that had the chapters in the right order, but the pages within every chapter were backward

Mike Miller

lmao

Unfortunately this book is typewriter font and I don't know if I can do analysis in typewriter font

Alessandro Codenotti

19:54

I guess they scanned it one chapter at a time and messed up the order or something

Balarka Sen

"Yeah whatever I'll upload this shit anyway"

Alessandro Codenotti

@MikeMiller analysis is painful in every font

Mike Miller

I'll read it anyway, this looks like the best textbook source

Alessandro Codenotti

The correct choice is to think about a different topic with no analysis

Mike Miller

No

That explains your cocones

greetings, kronenfeld

Balarka Sen

20:01

mystical greetings

i forgot that used to be a thing

user714630

is that how I spelled it?

shoot

Mike Miller

i think so

user714630

yeah

Mike Miller

how's life, given the circumstances

user714630

not overly terrible, virtually attended a conference over the weekend which was kinda cool

main challenge is with teaching

Mike Miller

20:06

teaching sucks now yeah

doing my best

the online seminars feel a little weird but theyre not bad

user714630

one person did a chalk talk with the webcam on their board - a valiant attempt

Mike Miller

really hard to do a good talk without a tablet

slides could work i guess

Balarka Sen

beamer presentations can be fine

sniped but yeah

user714630

my students have an exam next week, fingers crossed that it isn't a technical failure as they have to upload their solutions

(whole spring quarter is virtual here)

Mike Miller

same here, we anticipate summer will be too. fall is unclear

i am going to do a take-home

aka, longer time limits

Leaky Nun

21:14

I think this won't be over until next year summer

Mike Miller

"this" is underdetermined

Transcript for

♦ $Mathematics$ random cohomology for quantum nerds

Transcript for

♦ random cohomology for quantum nerds

♦ $Mathematics$ random cohomology for quantum nerds