Lurie mentions at the start of his thesis that the intersection of a curve with itself should be a categorical ring (but that one should immediately generalise to some theory of infinity categorical rings because he is interested in the derived intersection in Serre's intersection formula). Is the algebraic geometry of just categorical rings written down anywhere?

@EricPeterson wish I knew :-(

I just copied the link, but perhaps it is relevant that I didn't type anything before typing the link?

DREW! your name is on the wall!

and so is @EricPeterson's

you're on deck, for our seminar.

and eric is on deck to be on deck

Cool cool

It's going to be fun. I'm giving a practise talk tomorrow just to make sure I'm on point for you guys

hah

noice

I wish I could get rid of the 'work in progress' from the abstract though

hah.

i wish i could add work in progress to... something

exciting times

lol

What are you talking about Eric?

craig+me stuff

Oh cool. I should ask Craig about that one day. I like to talk about my work though

i'll send you the talk notes, just don't spill the beans to beardsley

:)

there, via gmail

Man, you are so organised

i was supposed to give a talk a month ago at stanford so i figured i'd use the opportunity to get ahead and practice this one

which is why i'm somewhat ahead of the game & over-thorough

Well without spoiling the fun for Jon, you're at least saying some words that I'm also saying. So the two talks kind of go nicely near each other. (Assuming people go to both)

yeah, i think it's good timing too

if anything it might be picard overload

there are worse things than that though

Nah this is cool. I like this

hey guys

so

beans.

soooo.......

are you for them, or against them?

i'm containing them.

a good number of them.

beans \subseteq jon beardsley

i'm giving up on math anyway, so nobody needs to worry about spilling any beans

oh, those beans

it's weird that that statement leaves open the possibility that beans = jon beardsley

i think you need to talk to a type theorist

i'm in a slump. hopefully when i go to lousiana and talk with andrew for possibly 27 straight hours, i'll get excited about math again, or at least, have something to write about.

yeah, it can be really invigorating to talk to people

plus when i get back, eric peterson will be somewhere in baltimore. so i can try to keep whatever salch buzz i've got going alive

man, you sound like a junkie

that's pretty much how i treat everything in my life.

i'm not exactly a 'middle-of-the-road' kind of guy

you guys seem quirky haha; hi, I'm new to here! Aaron told me to join in on the fun

oh man hi Dmitry!

are you a student of mathematics?

Yes - I am a first year student at Duke. But despite my department's interests, I really want to become a (higher) category/homotopy theorist! I reached out to Aaron (whom I knew from undergrad) upon seeing him post a facebook status on infinity categories, and he recommended I hop on board here!

totally man!

welcome.

thank you! what's your story?

we kinda bug out in here sometimes, but other times we actually talk about real math.

yeah I'm chill with either haha, I also probably know far less math than you guys, but hopefully I'll pick much up through osmosis in reading what you all chat about / be able to ask questions here

i'm a 4th year (jesus.... 4 years) graduate student at hopkins.i guess i do algebraic topology. i'm also (like most of us in here) pretty into algebraic geometry.

i guess homotopy theory though is sort of expanding enough at this point to encompass just about anything you want it to.

Yeah I've been a Grothendieck fanboy ever since catching ear of his sweeping vision (the rising sea metaphor)/perspective (relative point of view and such), so I'd really like to see where his brilliance manifested. I'm mostly interested in the more abstract "structural" aspects of Alg Geo than say more classical problems - but maybe that's just me being weird

It seems as though mathematical fields aren't organized "horizontally" in the sense that one is higher up on the "noun to adjective" spectrum, so to speak - e.g. I hear of "homotopical" treatments

homotopy theory seems to be very "high up" this ladder

so I guess it can shine light on a lot of things

well, I'm not sure about that. it seems to provide some interesting perspectives, but it's unclear still precisely whether or not it can actually solve problems

haha! Do you think solving problems is "the point" ?

i mean, sure sure, the Kervaire Invariant Problem, and like, 1 or 2 others i guess...

hm.

well, i guess i used to not think that.

and i'm still in love with the big picture.

yeah right now I don't think that, I almost view all of this stuff as more of intellectual hedonism - a mind high for, as you just said, the big picture of math

but sometimes i get way too deep into it, and i'm like, wait, what the hell am I doing

as in, you can go pretty far without actually doing anything.

yeah that makes sense - I'm sure it has the hazard of becoming existentially troubling

I feel like I'm the type of person who would do that - just explore really "high level" ideas for its own sake

I guess I'll get to play around with ideas, until I have to actually do something with them in my thesis

Yeah. I guess that's been my experience more or less.

I'm in the "now I have to do something" phase.

the point of this pursuit seems to be though to have fun exploring things that interest you - so of course to sustain that you have to contribute to the community's conversation on progress in solving problems, but as long as you can do that I see no issue with exploring your heart's desire - but perhaps i'm "young and dumb"

what got you into this direction of math interests?

well, that's definitely part of it. but i think, for me, it's also about wanting to feel like i've accomplished something, even more so than contributing, lol.

hmmmmm good question.

haha i was just telling a friend about tmf today, and another guy (a math postdoc!) who happened to be listening in was like: "well okay, but what's the point?"

i think i started out being interested in category theory and algebraic geometry. when i got to hopkins, my now advisor, Jack Morava, started telling me about connections to algebraic topology and homotopytheory, and i got pretty into it.

i get that non-mathematicians are going to be asking that question, but it was kind of annoying that i was supposed to tell him "oh, well there are deep connections to number theory, e.g. the riemann zeta function controls K(1)-local homotopy groups"

yeah, i mean, i guess i don't mean that so much. i mean, that's important to me in the sense of what it does, the fact that it pops up and is connected to other stuff.

the mere existence of tmf is amazing, and deserves to be written up in great detail, and explained and understood.

what's wrong though with just telling amazing conceptual stories via math?

erg. i never thought i'd be on this end of this conversation.

my point is just that he's been conditioned not to appreciate something unless it's connected to something he's heard about, e.g. the riemann hypothesis

yeah, i don't agree with that at all.

i.e. his position, is what i don't agree with.

right, yeah i'd expect not

it's just, i don't know, i guess i'm feeling pretty disheartened right now anyway, lol.

i'm sorry to hear that

yeah. haha. no worries. as i said, hopefully things will pick back up soon.

it seems like it'd be pretty intense to have jack as an advisor

(not that i've ever met him)

What's the lore around this guy?

i think part of my being disheartened is the fact that there's a particular abstract structure i'm looking at, and it seems like there are a LOT of things in algebraic topology that are examples of it, but i just can't figure out why that's relevant, or rather, what to do with that.

/ what's the abstract structure?

so, i'm in this position where i'm like, well, abstract thing 1 is abstract thing 2. but i can't see why anyone else should care, lol.

but maybe, for writing a thesis, that shouldn't matter.

so, okay, Jack Morava is an algebraic topologist who one might say connected algebraic topology to algebraic geometry.

jack morava has always been a few decades ahead of things; most of the foundations of chromatic homotopy theory (especially its usage of algebraic geometry) is due to him i think

oh right, dmitry if you read later in that slideshow you'll see "Morava K-theory" and "Morava E-theory" -- same morava

i've had one mathematician tell me that Jack Morava is the greatest mathematician of our time (perhaps a little hyperbole), and another tell me that Jack typically gives talks on things that are only understood about 10 years later.

haha wow!! that's crazy

I wish I knew who people were before I went into grad school

but alas lol

yes, i didn't know anything about the big wide world of homotopy theory either

(chromatic or otherwise)

me neither.

i didn't know anything about math

i didn't even know where to apply, or anything. lol.

I feel like analogous to how when you take "early college math" it's all geared towards engineering stuff - when you learn a lot of "late undergrad math" it's still geared away from more "kooky" things like homotopy theory. I feel like the community wants to push people into things like PDE's

yeah it's true.

that reaallly irks me!

we could teach people group theory in 8th grade, but we don't.

i got to teach group theory to this awesome 6th grader one time

he had some sort of learning disability, but was really incredible at math

probably just some mild form of autism

that's really cool. it's unfortunate we railroad people into analysis essentially.

anyways, i explained it in terms of colors

that's really really neat.

(like, different elements of the group are different colors, and they mix in certain ways)

then i started to tell him about quotient groups, and he just casually pointed out that this wouldn't work unless the subgroup was normal

haha. awesome.

not in those words, obviously. but he saw it immediately

woaaahhh!

btw: what's the "morita" relationship determined by a faithfully flat map of rings A-->B? i.e. if the map is faithfully flat, then the category of "descent data" over B is equivalent to the category of A-modules, but what's the relationship of the category of B modules to the category of descent data? it seems unlikely that A and B are morita equivalet.

i forget, this should be in COCTALOS. but i'd imagine that the category of descent data forgets down to B-modules

i reaaaally doubt that all you need for a morita equivalence is a flat map

no i agree.

yeah, i mean, i think the precise way to say it is that a B-module with descent datum is the same as a B-module which is also a $B\otimes_A B$-comodule

so, that definitely shouldn't be all B-modules.

and yeah, i guess there shouldn't be any kind of obvious or general relationship between that category and the category of B-modules.

hm i was under the impression that there was a restatement of james periodicity for complex projective spaces, but i can't find such a statement now, and looking back at all these breadcrumbs i've left for myself i've carefully avoided quoting such a statement

lol

hm. i guess i've heard of the james splitting, but not james periodicity

real james periodicity is the assertion that the J-groups of RP^k are finite, so in particular that the tautological bundle is torsion, and then Thom(RP^k; n*L) = Susp^n RP^k for some n

something like this is also true for BSigma_p, i thought maybe it was also true for profinite CP^n but now i'm not so sure

ah. yeah. i dunno.

so it seems that truly nobody has written down a basis for the homology (mod 2) of MSO inside of the homology of MO in terms of the standard generators for the latter.

wtf.

i actually emailed david pengelley about this.

there are bases for H_*(MO) which make writing down H_*(MSO) therein rather easy, but it's not even clear how such bases are related to the standard bases.

WTF!

correct

it seems like, if one wanted to solve this problem, one would probably want to work out how pengelley's basis is related to the usual basis of sw-classes

since he gives it in terms of specific manifolds.

but i'm not convinced that's tractable at all.

lordy.

this breadcrumb says that there is a james periodicity theorem for each j in BZ/p^j, but that their associated n don't limit to something nice as j goes to infinity. that's that then, nvm

how do i get better at leaving myself notes that make sense

lol. this is a breadcrumb you left for yourself?

hahah.

the problem is, you can't anticipate the precise piece of information you'll want in the future.

i have the same issue.

i write tons, but i never seem to coherently write down the fact that will be the most useful to me in the future.

@EricPeterson do u know, is there any interesting theory for, i dunno, something like "real oriented spectra" i.e. with a ring map from MSO or something?

(notably not BO, which i suspect isn't very interesting right?)

no, i don't really know anything. maybe with some work i could describe what ring maps MSO --> E look like for E complex oriented and finite height, but i wouldn't expect such a description to be interesting to anyone

yeah. hm. thanks.

i miss the bear

haha really?

don't go back!

pick something new.

yeah. this chat is a lot less goofy without it

like an octopus

lol

new q: mathoverflow.net/questions/146240/…

fyi, the formal schemes associated to Loops SU(n) and to BU(n) are wildly different: the latter is sort of like words of length n in the free commutative monoid on BU(1), the former is like the whole free monoid on CP^(n-1)

if that helps explain the difference in filtration type

hmmmm yeah that's good to know.

i don't think you need preaching to, but i can't advocate these formal scheme descriptions strongly enough. they're ultra-compact and let you toss around spaces pretty fearlessly

yeah, i know. i definitely need to read more of strickland's papers

i can't keep all these little doodads straight.

but see... the "formal group" associated to BU is the same one associated to \Omega SU right?

yes

in the colimit on BU(n), you are removing the restriction on how long the words in your 'free monoid' can be; in the colimit on Loops SU(n), you are removing the restriction that the letters in your free monoid come from CP^(n-1) rather than CP^infty

without restrictions, you get the same things on both sides

does the letter perspective jibe with the notion of divisors?

:) both can be expressed in terms of divisors

$BU(n)_E = \operatorname{Div}_n^+ BU(1)_E$, whereas $\Omega SU(n)_E = \operatorname{Div}_0 \mathbb{C}\mathrm{P}^{n-1}_E$

aahhhh okay sure. god. hrmmmmmmm. so weird. lol

itym gorgeous

hahah, took me a second to decode itym

i've been saying fwiw and otoh in my emails with jack. i dunno if he knows what that means or not.

oh he wrote back this morning, saying he was getting settled from some traveling but would write something soon

awesome!

yeah. he went to dinner tonight after the seminar with us, it was fun.

also this tantalizing bit cl.ly/image/2m0g0z1a0525

hmmmmm

hmmmmmmmm

:)

the only thing i know from hodge theory is the hodge decomposition

and even then, just of HH, not even of like, whatever the hell hodge theory is supposed to decompose, like, manifolds, or schemes or wahtever.

:) dolbeaut cohomology

i took a complex geometry class my first year here, i think that's one of the smartest decisions i've made in graduate school

should i someday have students of my own, i'll probably push that decision onto them too

@JonBeardsley i had one thought about the X(n) a long time ago that is maybe useful: CP^infty is supposed to be the end-all space for complex vector bundles because there is a splitting principle. however, for a space X of complex dimension <= n, any map X --> CP^infty is necessarily homotopy to one through the (2n)-skeleton, CP^n

maybe there is something useful to thinking that an X(n)-orientation means you have complex orientations for vector bundles on suitably 'small' unstable complexes

& maybe not

@EricPeterson I'm taking one now. I agree that it was an awesome decision.

@JonBeardsley oh, and then a couple other things: first, BU(n) is not an H-space, so MU(n) is not naturally a ring spectrum in the same way MU and X(n) are, which helps to indicate why no one uses it. also, like tyler said a while ago, if you orient rank n bundles then using a determinant trick you get a chern class for rank 1 bundles, and so a whole complex orientation

i should probably delete those comments. argh.

those comments were not reflective of anyone in this room, btw.

@EricPeterson that's a good point. I realized last night that i had forgotten all about MU(n)

if you're interested in posting these comments as an answer, i'd certainly accept it.

anyone know somewhere i can look up the classical fact that given BG-->BH I get a fibration H/G-->BG-->BH?

like, I sort of get it in terms of, I dunno, thinking about looping back something like BH/BG or something. but I want to know the exact mechanics of what's going on here.

haha, somehow this seems likely to be in May's "classifying spaces and fibrations" book!

hrm anyone got a digital copy?

@JonBeardsley math.uchicago.edu/~may/BOOKSMaster.html

thanks!

Peter puts all the books he's legally allowed to on his web site.

And possibly some he's not really legally allowed to.

Sweet.

Augh, but why for he make so hard for to read?

In May's notation, the standard classifying space BG would be B(\ast,G,\ast) right?

oh yeah, he says that on page 31

Aha, my question is proposition 8.8 in may's book

Jon, are you talking to yourself? :)

yes!

i basically just really helped myself out.

thanks self.

sure thing jon.

I've now found two really useful equivalences in that book of May's in the past year. Which of course makes sense, considering that I'm dicking around so much with the bar construction on loop groups and lie groups and so forth.

Probably a lot of what I do is written down in one of May's books somewhere

Haha.

so the mod 2 Dyer-Lashof algebra can be thought of as the mod 2 homology of QS^0

and I guess what we usually think of as the algebra structure comes from interpreting the homology elements as operations and then composing them

but there should also be a commutative algebra structure coming from the group structure on QS^0 itself

how do we write that in terms of the usual algebra structure?