Mathematics - 2025-01-29

Thorgott

10:43

desperately trying to scan this 1.5k page source to find the part where it clarifies the relation between grouplike E_infty-algebras and commutative algebras

ILikeMathematics

14:37

How do you pronounce $K[X]$ in German?

Ben Steffan

@Thorgott ...are you talking about HA?

@ILikeMathematics "K X"?

ILikeMathematics

@BenSteffan Not something like "'K von X"?

Ben Steffan

No, definitely not that.

ILikeMathematics

@BenSteffan Alright

Ben Steffan

Generally I would avoid pronouncing that

Thorgott

14:55

@BenSteffan yeah

Ben Steffan

:/

Thorgott

it's not exactly easy to parse

@BenSteffan if I want to talk about the object, I just call it the "Polynomring über K", but if I read it as part of a more complicated epxression, then "K X"

Ben Steffan

@Thorgott yeah, that sounds right

@Thorgott I'm not quite sure what you're looking for here, but an $E_\infty$-algebra is a commutative algebra by definition

these are the same thing

Thorgott

in the non-archimedean algebra lecture this semester, we have $K[X]$, $K[[X]]$ and $K\langle X\rangle$

Ben Steffan

group-likeness is relevant only for things like May's recognition theorem: A grouplike $E_\infty$-algebra is an infinite loop space

(in $\mathcal{S}$)

Thorgott

15:00

well, I'm about to have a Zoom call with my advisor. if my doubts aren't cleared up after that, I'll return to this discussion :)

Ben Steffan

alright, good luck :)

Thorgott

15:22

nvm had to postpone a bit cause he's still in an important meeting

so when I say commutative algebra I mean an object in $\mathrm{CAlg}(\mathcal{S})$ in Lurie's notation

which should definitionally be given in terms of $N(\mathrm{Fin}_{\ast})$ and not $\mathbb{E}_{\infty}$

Ben Steffan

@Thorgott these are the same thing!

$N\mathrm{Fin}_* = \mathrm{Comm}^\otimes \simeq \mathbb{E}_\infty$

Thorgott

ah, so that's the part I was missing lol

Ben Steffan

yeah, now the question makes sense :)

Thorgott

ok, but it's only the grouplike commutative algebras in $\mathcal{S}$ that correspond to (connective) spectra, right?

this should be the recognition theorem in some form

Ben Steffan

A grouplike commutative algebra in $\mathcal{S}$ is (up to equivalence obv.) an infinite loop space, and therefore the same thing as $\Omega^\infty S$ of some spectrum $S$

that's the recognition theorem

in that sense the answer is yes

Thorgott

15:30

hmm, so I need to figure out where to get the grouplike hypothesis from for what I wanna do

Ben Steffan

it should also be true that $\Omega^\infty$ is an equivalence onto its essential image when restricted to connective spectra, but I don't see the proof immediately

Thorgott

hmm, perhaps the punchline is that $(\infty,\infty)$-categories are always deloopable

@BenSteffan should it be? deloopings aren't unique

Ben Steffan

deloopings aren't unique, but intuitively $\Omega^\infty S$ should already contain all the homotopical information about $S$ when $S$ is connective

Thorgott

but don't non-equivalent deloopings give non-equivalent spectra mapping to the same space under $\Omega^{\infty}$?

Ben Steffan

do they? all that counts is what the map does on homotopy groups, and these are already fully determined by $\Omega^\infty S$

okay, no, you're right: this is wrong

What $\Sigma^\infty \Omega^\infty S$ is when $S$ is connected was determined by Kuhn, and it's not $S$

Thorgott

15:47

I mean, equivalent spectra have to be in particular levelwise equivalent spaces, that's my unsophisticated thinking

Ben Steffan

yeah, it was silly of me to suggest this

Xander Henderson

@Thorgott I initially read that as "deplorable", and was ready to agree.

Thorgott

17:09

@Ben just in from the meeting: negative-dimensional manifolds -> L-theory

Ben Steffan

.......ok

sure, why not :))

Thorgott

if I understand 10% of what that means, I will put it in the "future research possibilities" section of my thesis

here's the (incredibly rough) summary:
there is a geometric realization functor from (infty,infty)-cats to spaces (left adjoint to the inclusion)
this induces a realization functor from symmetric monoidal (infty,infty)-cats to E_infty-spaces
this induces a realization functor from dualizable symmetric monoidal (infty,infty)-acts to grouplike E_infty-spaces i.e. connective spectra
if you take the (infty,infty)-cat of bordisms, this should yield the appropriate Thom spectrum

(explaining all this would be the maximal extent of my thesis)

now, the further generalization

you can take (infty,infty)-spectra, which are like spectra if you replace "space" with "(infty,infty)-cat" (and the loop space functor with endomorphisms of the basepoint)

this can alternatively be thought of as a higher category which has k-morphisms for all integer k

there's still a realization process for these (with some adjectives), which yields spectra, now not necessarily connected

if you take the algebraic cobordism category, where you consider Poincaré duality complexes instead of manifolds (and the algebraic definition of those totally makes sense for negative dimensions!), you get something which realizes to the L-theory spectrum

and apparently you can also describe the relation of L-theory and Grothendieck-Witt theory in this picture :)

(K-theory spectra can also be constructed in this manner)

(well, except this is all broadly conjectural, of course)

leslie townes

17:35

thorgott is it broadly conjectural in a 'moon shot' kind of sense, or 'this is pretty much what ought to be true but the details are going to be a slog' kind of sense

Ben Steffan

oh brave new world that has such mathematics in it

Thorgott

emphatically the latter

Ben Steffan

@leslietownes well people are having a lot of conferences and workshops on this kind of stuff right now so hopefully the latter :)

Thorgott

like, nobody would not believe this, I think, but the details may be hard to provide

leslie townes

haha thanks, this is nowhere near my field and im not up on current developments, so i have to bug you guys to fill me in on the big picture

seems like more people are working on this now than were when i left math (~ 2010)

Ben Steffan

17:43

I'm not sure how much of the framework you need to even conjecture this existed back in 2010

and what existed was probably too scattered for anyone except a few "experts' experts" to have an in-depth understanding of

it's a field where people sometimes refer to papers from 2015 and later as "classical"

leslie townes

well if i didn't already feel old, that certainly does it :D

Thorgott

well, I'm not the big picture guy either, I just regurgitate what the big picture guy told me

@BenSteffan HTT is ancient!

Ben Steffan

they call it the "bible" for a reason :)

Thorgott

I remember when Milnor was the bible, how times have changed

Ben Steffan

@leslietownes you are old and at peace, while we are young and still have an unending, unquenched thirst for suffering :)

Thorgott

17:48

@BenSteffan the framework can be part of the conjecture :)

GTMW was before 2010, which I think should serve as evidence for some of this (except I haven't really understood yet what it says)

leslie townes

what are these acronyms? secret codes? :)

Ben Steffan

shibboleths of the field

leslie townes

oh, i think i know what HTT is

Ben Steffan

but GTWM I don't know either

Thorgott

Galatius-Tillman-Madsen-Weiss

Ben Steffan

17:50

@Thorgott sure, but at some point you're just going too far out on a limb

leslie townes

i had personal acquaintance with lurie and others in that orbit, so i may have gotten the impression that way more people were doing that stuff back then than were actually doing it

it seemed like every other person was doing it

Thorgott

I don't think we ever told Grothendieck he was going too far

well, until he started talking about the fourteen mutants, I suppose

@leslietownes well, I was just a kid back then, but certainly a sizable amount of people is doing it nowadays

my thesis so far has references:
3 from the 80s
0 from the 90s
9 from the 00s
6 from the 10s
12 from the 20s

it's crazy how lopsided this is lol

(to be fair I'll eventually probably include a bunch more classic topology references once I write the historical context chapter, which should balance it out)

leslie townes

even that strikes me as pretty reasonable, given that cutting edge research is usually incremental and mostly in dialogue with the most recent other improvements

it would kind of be interesting to see how that distribution compares across subfields and times though

here's my histogram
40 x
50 xx
70 xxxx
80 xxx
90 xxxxx
00 xxxxxxxxxxxxxxxx

Thorgott

yeah, I suppose it is, but it's surprising to me (well, perhaps to be expected given that this is my first venture into anything resembling research, tho hardly cutting-edge)

yesterday, I've finally managed to prove a conjecture that my advisor gave me to me last year, which turned out to follow from the combination of results from half a dozen recent papers and a few easy observations, incremental indeed

leslie townes

the threshold for citation seems to be generally lower in other fields (i.e. people are both expected to cite more, and do cite more) but the shelf life of stuff in other fields is often shorter, so its not unusual to see a life sciences paper that cites like 50 papers and works in progress from the last 2-3 years and nothing else

Thorgott

18:04

mathematics has the benefit of results not losing their validity as time progresses (except for the inaccurate ones, of course), at worst they lose relevance

my citation threshold is also not that high, at one point I cite a paper to say "I took the naming convention from here"

leslie townes

if i'm being honest i probably didn't "really" need to cite the paper from the 40s in my thesis :) it was just the first and simplest example of the thing i was using, in the case i was using it

Thorgott

it's cool to go far back

perhaps I'll find an excuse to cite JHC whitehead

leslie townes

if you can do it without sending the reader off on some kind of historical expedition to understand why what you're citing actually says what you're using it for, it's kind of fun

Thorgott

yeah, unfortunately (well, not really) I'm doing pretty modern stuff

leslie townes

19:04

0

Q: A special diffeomorphism

Let $A$ be an open subset of $(M,g)$ closed riemannian manifold. Given $\varepsilon>0$, is it true that there exists a diffeomorphism $\phi:M\to M$ such that $$ \phi^{-1}(A) = Vol(M)-\varepsilon \qquad? $$

differential-geometry

likely to be closed in its present form, but an interesting question. feels like the answer is yes, or at least would be yes if vol(phi^{-1}(A)) were only required to lie somewhere in an interval (vol(m) - epsilon, vol(m)). but no idea what the proof would look like

or the context for why anyone would wonder about this

feels true subject to some massaging in view of arctic char's comment, i guess

Ben Steffan

well, if you replace $=$ with $\geq$...

but that question is really poorly proofread

Thorgott

so what they want to ask is whether we can deform $A$ to account for arbitrarily close to all of the volume of $M$?

Ben Steffan

that seems like the likely interpretation

Thorgott

@leslietownes that would imply the strict equality version

by an appropriate continuity argument

leslie townes

i mean surely there are enough diffeos to deform the volume of a proper open subset into an interval around the original volume

or maybe i don't have a good mental picture of what an open subset of an arbitrary riemannian manifold looks like

Ben Steffan

19:16

@Thorgott how so?

Thorgott

smh you're supposed to nod along and say "obviously"

Ben Steffan

well if you interpret "diffeo" to actually mean "isotopy" then :)

but the statement is still false for $A = M$

Thorgott

yeah, I suppose

I was actually imagining isotopies that 'bulge out' $A$ along balls

Ben Steffan

that would be strictly stronger than what the question seems to ask

Thorgott

but this idea is only sound if $A$ has reasonable boundary

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