Geometry & Topology: Dank Ass Ganja

I have a chat room open: Atiyah Macdonald. Whenever I get stuck on something in my notes, I just write the question on there in bold and leave it. Eventually someone comes around and answers it or gives me some good hints, it's been pretty good for me.

I'm taking a course in algebraic number theory

Exterior

ooh I've heard that's a very interesting field

Robert Cardona

and starting on hartshorne's algebraic geometry under a student of hartshorne

Exterior

! lucky !

Robert Cardona

i really don't enjoy it; i'm finding it too computational (ANT).

hopefully it gets better :)

Exterior

personally I shy away from number theory

I like conceptual stuff

I don't like calculating

Robert Cardona

3:33 PM

same here, if only because I actually have to work with actual numbers

same here!

Exterior

looks like we have similar tastes :D

I really enjoy category theory, although I only know the basics

Robert Cardona

I'm very interested in algebriac geometry/homological algebra because I'm really interested in algebraic k-theory

which requires strong foundations in the above two fields

theres a great cat book you should check out: abstract and concrete categories

Exterior

I don't know anything about $K$-theory, but homological algebra is really cool! Hope to learn algebraic geometry some day

I often look in it

Robert Cardona

I love it; has tons of examples/theory.

Exterior

Usually I look at CWM, Borceux, and the $n$lab

Sometimes Kashiwara's Categories and Sheaves, but it's hard

Robert Cardona

3:35 PM

it's my favorite cat book so far. another one which was recommended by john baez at uc riverside is : conceptual mathematics: an introduction to categories, which ends up leaving you with a nice bit of topos theory

Exterior

I have a physical copy of it, but it's super basic

Robert Cardona

yeah, baez is big on the $n$-lab, $n$-cat cafe (or whatever it's called)

Exterior

Baez is generally speaking awesome

Robert Cardona

It's funny you should bring that book up; I added it to my wish list a few weeks ago :P

Exterior

hehe

alot of higher category people seem really snobby

then again, everything we mere mortals do must seem so dull :p

Robert Cardona

3:37 PM

yeah. I had a buddy of mine email back and forth with him when he was applying to grad schools. the general advice was: people who want to go into cat theory for the sake of it should go to europe, there is not cat theory in us.

:P

there's a funny von neumann anectdote similar to that.

Exterior

oh? I thought I knew them all :p do share

Robert Cardona

one if his friends saw him talking to his son, he was able to bring himself down to the toddlers level so well, he then thought a bit and realized, von neumann was probably doing the same thing when talking to people in general: bringing himself down do a much lower level.

Exterior

hahaha surely there is truth to that

Grothendieck must have been like that

Robert Cardona

I probably found it on one of the huge math quote community wikis on mathoverflow or math.stackexchange.

grothendieck was doing like 14 hours of math for like 10-15 years straight.

Exterior

Incredible

Robert Cardona

3:40 PM

have you heard the grothendieck prime anecdone?

anecdote

Exterior

haha yeaqh

yeah*

57 was it?

Robert Cardona

yeah

Exterior

dropbox.com/sh/d9si12xs1dfebrr/AABRhaQpbQphhSFxUUcJojvva?dl=0

Robert Cardona

"you mean an actual prime?"

you have a book by Kleimann

Exterior

I gathered a bunch of short biographies on him

Robert Cardona

3:41 PM

he and Altman have a really nice book on introductory commutative algebra

Exterior

I was thinking about learning algebra from Aluffi

his book has a nice feel to it

Robert Cardona

I've got a copy of it

yeah

it's very categorical from the beginning

the only problem I've found with it is that it doesn't go into field theory and galois theory that deep.

Exterior

(I'm an electrical engineering undergrad, so I'm waaaaaaaaaaaaaaaaay behind where I'd like to be)

mm, I'll remember that

Robert Cardona

I was a computer eningeering undergrad, so tell me about it...

my favorite algebra book thus far is: rotmans modern algebra

Exterior

did you move to math or finish computers?

Rotman writes about everything lol

Robert Cardona

3:43 PM

I double majored; so I got both at the same time. I'm currently working as a software dev. getting my masters in pure math.

yeah: topo, homological algebra, algebra, galois theory :P

Exterior

Must be an interesting guy. Although he doesn't write like a topologist

(nice! :p)

Robert Cardona

I applied to one phd program in germany, this semester, so hopefully I can devote myself fully to math.

Exterior

good luck!

Robert Cardona

he doesn't, but his topo book is a great precursor to his homological algebra book

thanks!

I already had skype interview, so now it's just waiting.

Exterior

are you content with the interview?

Robert Cardona

3:45 PM

thanks for the links, downloaded them.

yeah, I thought it went well.

Exterior

If you give me your email I'll share some.. more interesting things :p

Robert Cardona

if you go to my profile (math.stackexchange)

it links to my github account

my email should be there

k g2g back to work!

Exterior

ah, just a sec

awesome meeting you by the way!

Robert Cardona

same.

it's funny: I just went through your questions list and realized I've starred a few :P

Exterior

satisfaction

okie, check your mail

Alright, I'll let you continue your work. Switching to lurking mode

Exterior

5:42 PM

Basic topology question: Suppose I have the cylinder $X\times [0,1]$ and I identify the bases. What do I get?

Exterior

5:53 PM

@iwriteonbananas let's switch to here, perhaps?

iwriteonbananas

@Exterior start with this

i dunno how to explain this lol

just wrap it aruond until u get this

Exterior

ok, I understand

my problem is this:

start with $X$ and define $e_0 :x\mapsto (x,0)$ and $e_1 :x\mapsto (x,1)$. I want the pushout of $X\times [0,1]\leftarrow X \rightarrow X\times [0,1]$ where the maps are $e_i$.

This should be the disjoint union of the cylinders, with bases identified

oh..

iwriteonbananas

ok i dont even know what a pushout is and judging from your posts on MSE, you know far more than me

Exterior

oh, okay.

lol.

iwriteonbananas

:D

sry

Exterior

6:00 PM

lol no actually you've resolved my issue! let me explain:

iwriteonbananas

ok :D

Exterior

I thought I wanted to glue the bases of a cylinder. What I really wanted was to take two identical cylinders, and glue them along a common base

which yields a cylinder

iwriteonbananas

ok i see

Exterior

thanks :D

Exterior

6:47 PM

@RobertCardona homotopy theory is so elegant :D

Exterior

6:57 PM

@AlexanderGruber, I enjoyed reading your answer here math.stackexchange.com/questions/493915/…

Balarka Sen

7:50 PM

interesting room

@Exterior hehe, the computational stuff in number theory is actually kind of tiresome. but they can be fun sometimes.

Exterior

@BalarkaSen I believe you, I wouldn't know myself :p

I'm at a stage where I get all hyped up about elegant formalism

Balarka Sen

analytic number theorists like computing. they'd write up huge formulas and huge sums and claim that dis is a mellin transform of a modularform of dat and so on :P

Exterior

that sounds like my engineering courses (bleh :()

But your profile says you're also studying algebraic topology now

Balarka Sen

most of the stuff in analytic number theory can be interpreted arithmetic geometrically though

Exterior

now that actually sounds fascinating, but I don't know anything about it either

Balarka Sen

7:55 PM

for example, i've heard people say modular forms can be interpreted as sections of vector bundles on modular surfaces. i'm dying to know more about that.

Exterior

man, there's so much to discover

Balarka Sen

@Exterior yeah. i'm studying homology right now.

Exterior

which homology? singular?

Balarka Sen

yeah

Exterior

cool

I'm trying to refine my understanding of it

Balarka Sen

7:56 PM

the ideas can be a bit subtle. i'm trying to "see" what's going on behind all the algebraic mess.

Exterior

did you see simplicial homology?

Balarka Sen

yeah, that one is easy to see

Exterior

well singular homology is useless for computations, but it's useful because the groups contain all the singular simplices from the get-go, so you don't need to refine and subdivide as you go along

Balarka Sen

but not good for work. that simplicial homology is homotopy invariant would take a thousand page to prove without singular homology

Exterior

yep

Balarka Sen

7:58 PM

@Exterior no, no, singular homology is actually great for computation

Exterior

only via the theorems

you wouldn't actually calculate groups element-wise

Balarka Sen

yeah, $C_\bullet(X)$ are huge in the singular chain complex.

Exterior

I should have said direct computation i guess

Balarka Sen

there're some "tricks" involved

Exterior

I guess the long exact sequence is what makes everything possible

computation-wise

Balarka Sen

7:59 PM

yeah. the long exact sequence is a long version of the \pi_1 short exact sequence

you can use that to interpret Mayer-Vietoris as a long version of the Van Kampen theorem

it's actually weird, how homology and fundamental groups are analogous

Exterior

tell me about it..

that's where category theory is useful though

Balarka Sen

i think the correct formal way to prove the analogy would be to exploit the chain homotopy idea

yeah

Exterior

what I'm trying to do today is to learn basic abstract homotopy theory and then incoroprate it conceptually to the homotopy-invariance of singular homology story

incorporate*

Balarka Sen

sounds fun.

do you understand chain homotopy, really?

Exterior

it is!

Balarka Sen

8:02 PM

i find it confusing

Exterior

I spent alot of time trying to understand it :D let's put what I think I know to the test

Balarka Sen

ok, cool. tell me about your understanding. i'm all ears.

Exterior

okay so we start of with homotopic $f,g$ and we want to show the induced homomorphisms on homology are the same. So we need some kind of analog of homotopy on the level of chain complexes

(just a sec, I'm compiling my notes so I can use them more easily)

Balarka Sen

right

Exterior

Okay, so how does homotopy $f\simeq g$ work in $\mathsf{Top}$? It starts by raising dimension - turning a space $X$ into a "cylinder" $X\times [0,1]$. Second, it provides a way to get back $f$ and $g$ by restricting to the bases of the cylinder

Balarka Sen

8:07 PM

right, ok

Exterior

in light of these, it is appealing to "raise dimension" by taking a graded homomorphism of degree +1

Balarka Sen

mhm

Exterior

intuitively, each morphism $P_n:S_n(X)\rightarrow S_{n+1}(Y)$ takes a singular $n$-simplex and "thickens it" into a singular prism $P(\sigma):\Delta ^n \times I\rightarrow Y$

Balarka Sen

go on

Exterior

we want the restriction to the top base to be the image $g_# (\sigma)$ and the bottom restriction to be $f_# (\sigma)$. We also want a continuous deformation along the height of this prism (cone, really). These things are provided by the homotopy down at the topological level

so we want to take a walk (in the direction "up") on images of the compositions $F\circ (\sigma \times 1_I)$ where $F$ is the homotopy $F:f\simeq g$

Now I need to find a good picture. Do you have a copy of Rotman's algebraic topology to take a look at?

Balarka Sen

8:14 PM

uh, no, actually. I didn't like Rotman.

Exterior

let's see where it is in hatcher

there should be a drawing of a prism there somewhere

Balarka Sen

what picture do you have in mind?

oh prism. sure, i can imagine that.

Exterior

ok, so let's call this prism $P$. It is comprised of the top and bottom bases, and the side faces (it's better to think of those faces as just the "side body" at this level, I think)

Balarka Sen

mmhmm

Exterior

the "side body" is really just the prism restricted to the boundary of the image of $\sigma$

i.e to the vertices of the bases - you build up only the skeleton of the prism

make sense?

Balarka Sen

8:16 PM

okay

Exterior

cool, we're just one step away from (almost) getting the chain homotopy relation:

let's try to describe the boundary of this prism

Balarka Sen

no, no, i see how the chain homotopy relation follows from this.

Exterior

okay, direct me

Balarka Sen

you haven't quite explained why the chain homotopy is really a "homotopy"

Exterior

ah! cool

Balarka Sen

8:18 PM

there must be a natural interpretation

Exterior

there is

are you familiar with the tensor product of chain complxes?

complexes*

Balarka Sen

nope. tell me about it.

Exterior

okay. Basically, the tensor product is a functor that gives $\mathsf{Ch}_\bullet$ a (closed) monoidal structure. Look up the definition on the nlab real quick

ncatlab.org/nlab/show/tensor+product+of+chain+complexes

So we want to translate the weird thing called chain homotopy to a homotopy in $\mathsf{Ch}_\bullet$, whatever that is

Balarka Sen

ok, i get it.

Exterior

purely informally, it should be an arrow like $X\otimes I\rightarrow Y$ for chain complexes $X,Y,I$, but it's unclear what $I$ should be. It should somehow play the role of an interval

Balarka Sen

8:21 PM

well just take $I$ to be the chain complex of the interval $[0, 1]$.

that's the most natural thing to do

Exterior

yep, simple

now one has to actually take a look at how to define homotopy abstractly

to do this quickly and easily, I recommend Kamps and Porter's book

a few hours were more than enough for me, and I'm slow

Now comes the punchline

Balarka Sen

ok, interesting. i'll have a look later tomorrow.

Exterior

every homotopy in the sense we just described determines and is determined by a chain homotopy

so really, these things are the same

Balarka Sen

ah!

that's really super duper interesting

Exterior

it took me weeks of looking around to actually get this all in place

2

Q: Homotopy and chain homotopy determine each other

In continuation of this previous question, I'm having problems with the following proposition from Kamps and Porter's Abstract Homotopy and Simple Homotopy Theory: Proposition (3.7). [page 210] If $H:X\otimes I \rightarrow Y$ is a homotopy between $f$ and $g$, then defining $h(z)=H(0,...

Balarka Sen

8:24 PM

actually i was thinking of how to take products of chain complexes work but the best idea i had was to use a double complex

and double complex makes it quite unclear how we should assign the homotopy

... or does it?

Exterior

it's possible that higher nonsense can be the punchline, but double complexes did not pop out for me

for me the tensor-hom adjunction does the trick

it's not too hard to guess how to define the internal hom in $\mathsf{Ch}_\bullet$ object-wise

although the differential does involve much suffering

Balarka Sen

yeah, but tensor product of chain complex is still a bit unnatural to me.

Exterior

yes, but! now that you have internal hom, the adjunction dictates the tensor product

it's all about the closed-monoidal structure suddenly

math.stackexchange.com/questions/1122480/… there's a brilliant example of how to work out tensor from internal-hom backwards

I haven't been able to work it out in $\mathsf{Ch}_\bullet$, so I'd love some help on that, but conceptually, it's good enough for me for now: internal-hom dictates tensor

Balarka Sen

i've always loathed abstract nonsense, so i guess this might take a bit time

Exterior

I'm pretty sure more motivation for the tensor product lies in the theory of CW complexes, which is geometric in nature

Balarka Sen

8:28 PM

but anyway, it'd dead of a night in here so i'd better look at all that tomorrow

Exterior

so maybe you should look there instead

oh, okay

:D

Balarka Sen

thanks for all the ideas btw @Exterior!

Exterior

Any advice for me?

Balarka Sen

what kind of advice do you want?

:)

Exterior

for learning mathematics of course! Number theory for instance

Balarka Sen

8:31 PM

ah, but you'd find better advices from me on how to make sausages. anyhow, the bit of number theory i like is algebraic number theory, and modern algebraic NT consists of all sorts of arithmetic geometric stuff, so if you really want to learn number theory, keep sticking to the algebraic side and try understand things geometrically.

you like galois theory?

Exterior

don't know any.. I'm an electrical engineering undergrad, I don't know anything

Balarka Sen

switch to math if you want :P i know many people who did that.

anyway, you might do studying a bit about galois theory

Exterior

I want to, but first, gotta understand algebraic topology better

Anyway, I won't hold you up :P
Come by, I'm often around here.

Balarka Sen

sure.

Exterior

Nice meeting you

Balarka Sen

8:34 PM

to you too.

you haven't seen any galois theory, @Exterior?

Exterior

none at all

Balarka Sen

okay... but you know about covering spaces right?

Exterior

a tiny little bit

Balarka Sen

good, good, so let me try to translate galois theory to covering spaces. that'd make sense, i guess.

Exterior

ooh sounds interesting

Balarka Sen

8:36 PM

ok, so, you take a path connected, locally path connected and semilocally blah base space $X$

Exterior

mhm

Balarka Sen

$E$ and $E'$ be two covers of $X$.

a homeomorphism $f : E \to E'$ such that the diagram consisting of $E \to X$, $E' \to X$ and $f : E \to E'$ commutes is called a deck transformation, which i think you're familiar with.

Exterior

I vaguely recall reading stuff

but keep going, I'm gonna go over this later tonight

Balarka Sen

yeah, and now the group of all deck transformations of $E$, a cover of $X$, is called deck transformation group of the covering map $p : E \to X$ which I'll denote by $Aut(E, X)$

Exterior

ok

Balarka Sen

8:39 PM

so, visually, this is the group of all transformations of some cover of $X$ that leaves $X$ alone

Exterior

mhm

Balarka Sen

fact : $Aut(\widetilde{X}, X) \cong \pi_1(X)$ where $\widetilde{X}$ is the universal cover of $X$.

Exterior

I think that makes sense to me

Balarka Sen

yeah, you can prove this by lifting stuff

Exterior

I mean visually somehow

yeah okay

Balarka Sen

8:42 PM

so now, by the deck transformation groups, you have a correspondence between covers of $X$ and groups.

{ covers of X } --> {subgroups of pi_1(X)} by $E \to Aut(E, X)$ where $E$ is some cover of $X$.

Exterior

mhm

Balarka Sen

in fact, you can prove using a little effort that this correspondence is bijective

this correspondence also has nice properties. if $E' \to E \to X$ are a bunch of regular covers (they're some "nice" covering spaces), then there is a short exact sequence $1 \to Aut(E, X) \to Aut(E', X) \to Aut(E', E) \to 1$

Exterior

groups extensions - begin! :p

Balarka Sen

ok, so i take you know fields?

Exterior

no :'(

ahh I gotta get the hell out of electrical engineerin

Balarka Sen

8:47 PM

ah, that's bad. you should read up about fields, they are very important

iwriteonbananas

@BalarkaSen what is a regular cover? u mean a normal cover?

Balarka Sen

yeah bananas

iwriteonbananas

ok

Exterior

Do you have any book in mind for a short introduction?

Balarka Sen

@Exterior the geometric version of galois theory is what i have told you above.

Exterior

8:48 PM

something that gives intuition, not just formalism

I mean for fields

wait.. that's pretty amazing

Balarka Sen

the algebraic and standard version of galois theory comes up from fields

iwriteonbananas

also the universal covering space covers any other covering space

Balarka Sen

and until Grothendieck, nobody knew why there're so much similarity between two apparently different theories

Exterior

enter A Long March Through Galois Theory

Balarka Sen

it's quite amazing. i think i can skip the techincalities and tell a bit more about it and it's relation to number theory :

Exterior

8:50 PM

by all means

Balarka Sen

$\overline{\Bbb Q}$ be the set of all algebraics.

iwriteonbananas

keep talking about covering spaces

Balarka Sen

Define $Aut(\overline{\Bbb Q}/\Bbb Q)$ to be the group of all automorphisms of algebraics fixing teh rationals pointwise.

Exterior

what's an algebraic? Not an algebraic set I take it

Balarka Sen

i mean set of all algebraic numbers

Exterior

8:52 PM

okie

fixed pointwise i.e fixedpoints?

Balarka Sen

yeah

Exterior

okie

Balarka Sen

so now this group G = Aut(\bar Q/Q) is huge. nobody knows a "good" representation of this group

Exterior

what's the cardinality, for starters?

Balarka Sen

and it is known that study of images of G onto matrix groups GL_n(F_p) reveal a lot of number theoretic information.

@Exterior cardinality of the continuum, if i'm not mistaken

Exterior

8:54 PM

okay, really unpleasant

Balarka Sen

yeah.

but now, if you study fields, you'll see G is a lot similar to Aut(\tilde X, X), the deck transformation group

what Grothendieck did was to treat G as if it's \pi_1(X) for some X and that revealed a lot of information about how G looks like

Exterior

That makes sense for Grothendieck.. what was $X$?

Balarka Sen

this group G is also known as the absolute Galois group over Q. modern algebraic number theorists are mad about this group.

Exterior

in other words is there actually a way to assign spaces?

Balarka Sen

@Exterior i dunno much about it but as far as i know Grothendieck found a way to generalize fundamental groups to algebraic varieties, i.e., totally discrete mental stuff

Exterior

8:58 PM

typical Grothendieck.. what a monster

so what's absolute about this group?

Balarka Sen

and X was Spec Z, the set of all primes with zariski topology

Exterior

that's some deep stuff..

Balarka Sen

@Exterior it's just that every galois group over Q is a subgroup of G

just like universal covers

Exterior

this sounds bizarre

like all these words don't belong in the same sentence

Balarka Sen

it is. the analogy is downright maddening if you think about it

Exterior

9:01 PM

Sounds like something you could easily dedicate your life to

Balarka Sen

haha

well, i guess that's enough rough ideas for one day

Exterior

you blew me away at the end there

Balarka Sen

my best advice to you is to give up electrical engineering

... and now i got to sleep

byes

Exterior

yeah.. I want to.. Probably will.

bye!

banana whatcha up to?

iwriteonbananas

im eating food, was enjoying balarkas rant

u?

Exterior

9:04 PM

room topic changed to Algebraic Topology & Homological Algebra: Discussion in informal spirit [algebraic-topology] [category-theory] [homological-algebra] [homotopy-theory]

trying to tidy up a bit of my notes, gonna go eat in a bit

have you seen Birdman?

(no spoilers please)

iwriteonbananas

yeah everybody dies in the end

Exterior

helpful as always

iwriteonbananas

just kidding, havent seen it

have u seen whiplash?

Exterior

no