Mathematics

I am loving it

I am gonna start working through it in the summer.

the algebra is slightly intense but there is a good appendix

once this semester ends

ya'll are too smart for me

i'm still studying galois stuff :(

12:29 AM

I finally get why its called algebraic geometry now

@ZachHauk galois stuff is cool

you're studying AG?

that's my topic of interest

@AliCaglayan Yeah I want to have perfect understanding for commutative algebra. Also, I am slowly moving in allufi chapter 0.

I am in page 131 now.

But I can't do too much because I have to study for my subjects this semester + TA work

@Ali are youundergrad?

@ZachHauk yes

but my goal is to finish allufi chapter 0 during the summer. I am very excited to go to spectral sequence stuff and derived categories.

12:30 AM

waht yaer?

1

sorry

i can't write

I want to maybe work in mirror symmetry + quadratic forms @AliCaglayan

@Adeek I met someone working in mirror symmetry in the past

same day I met Reid

@AliCaglayan I know one prof working in mirror symmetry and my other prof working in quadratic forms.

12:31 AM

a lot went on that day and I remember so little

I think I will ask the other guy to also supervise me as well. This mirror symmetry stuff seems very interesting.

So I am looking at Nullstellensatz today

The interplay between geometry and algebra is amazing

very cool

affine algebraic sets V <--> coordinate rings k[V]

cool

12:32 AM

points of V <--> maximal ideals of k[V]

affine algebraic subsets in V <--> radical ideals of k[V]

subvarieties in V <--> prime ideals in k[V]

oh

very cool.

@AliCaglayan you might be interested in my blog btw

and morphisms f : V -> W correspond to k-algebra homomorphisms k[W] -> k[V]

@Adeek throw us a link

@AliCaglayan spectral-categories.blogspot.ca

it been 2 month since I have updated it, but I will update it this week.

Maybe thursday or friday.

@Adeek I'll give it a read this week

I have come up with some geometric images for thinking about the correspondances

cool.

12:35 AM

you can think of affine algebraic sets as some union of circles

think of a venn diagram of 3 circles

have you heard of this derived categories @AliCaglayan ?

this corresponds to some coordinate ring

It seems very interesting

@Adeek I have heard of them however I know very little about them

affine algebraic subsets you can think of as 2 of the 3 circles

oh I see.

12:36 AM

subvarieties you can think of as 1 of the circles

oh

btw I was reviewing some stuff in commutative algebra. Maybe in your knowledge of geometry so far you can answer it to me.

go on

So we used zorn lemma to prove $Rad(R) = \cap P$ where P is taken to be prime ideal.

Is there a geometric way to see this ?

I mean inclusion $Rad(R) \subset \cap P$ is obvious.

but is there geometric way to see the inclusion $\cap P \subset Rad(R)$ ?

here Rad is the radical of R

that is the ideal whose elements are nilpotent elements.

hmm I will think about it

could you guys maybe talk about something i understand

/s

This looks like commutative algebra, with Radicals and Nilradicals and prime ideals and what not

12:44 AM

well we are trying to draw a picture about it

How about we talk about something everybody understands, like why can you never divide by 0 in the set of real numbers :P

0 doesnt have an inverse

ya goober

that applies to all fields <3

Why doesn't it have an inverse?

because it can't multiply with anything to give 1

@skullpetrol cuz it's a field...

12:50 AM

Exactly. @AliCaglayan

who was that "exactly" to?

:[

yeah @AliCaglayan

yeah @ZachHauk this is commutative algebra with Radical being the set whose elements are nilpotent elements

someone talk to me about maths I've already studied

I just did :P

@Adeek I am not really sure how to think of the nilradical

Geometrically that is

@Adeek actually the nilradical is going to be $\operatorname{rad} I/I$

that is the nilradical of k[x]/I

So for some algebraic set V

the ideal of V is I(V)

the nilpotent radical is rad I(V) / I(V)

1:00 AM

very cool @AliCaglayan

@Adeek try asking on the main I don't think I can come up with a better picture