Mathematics - 2012-07-17 (page 1 of 3)

The course will be on algebraic geometry of low dimensional algebraic
varieties – curves and surfaces. We will roughly follow Hartshorne’s
“Algebraic Geometry” Ch. IV,V. As needed we will briefly recall the
general algebraic-geometrical concepts covered in the first 3 chapters of
this book. If you had no previous experience with algebraic geometry,
the course still may be interesting to you, provided some background in
algebra, complex analysis and algebraic topology.

BenjaLim

god

@HenryT.Horton I realised one thing

you want to know what it is?

Henry T. Horton

Sure

BenjaLim

You see in my university

we have this degree whereby supposedly only "smart" people should enrol

once you're in there

you can skip lots of prereqs and do a lot of advanced courses

ok fine you may be able to do a course and get a high mark

but some people

but that does not mean the impression of maths that you have has changed

and often a lot of people don't have the maturity

for example

I think maturity has to do a lot with being able to draw analogies between different parts of maths

for example just a few days ago I was reading about the product and box topologies

first thing that came to my mind was direct product and direct sum

and how the two have different universal properties

and with direct product the problem often comes in that you only have a map into and not out of it

For example one of the problems is that the direct product does commute with tensoring @HenryT.Horton

So I guess drawing from that

I can see how come the box topology is preferred over the product topology

@BrandonCarter hey :D

Henry T. Horton

What? someone prefers the box topology!?

BenjaLim

sorry the other way round :D

Henry T. Horton

1:17 AM

Ok good

BenjaLim

see what I mean?

@HenryT.Horton For example when I started with the quotient topology

it was easy to grasp quickly because in algebra you deal so much with quotiening

but it's a little different though because in algebra you want to make sure that the thing you quotient out by

has to be such that multiplication is well defined

e.g. subgroup,ideal,subspace, submodule....

Henry T. Horton

I'll be back...

BenjaLim

@HenryT.Horton GOD DAMMIT ...............

Henry T. Horton

CHILL OUT MAN

BenjaLim

I'M MAD.

Henry T. Horton

1:26 AM

I have to get pizza

Do you want to deny me pizza?

BenjaLim

@HenryT.Horton Americans like pizza huh?

Henry T. Horton

Sure

BenjaLim

here we like kebab

or:

The food that almost all university students survive on (strangely enough):

Indomie Mi goreng

Indomie Mi goreng is an instant noodles product line made under the Indomie brand by the Indofood company, the world's largest instant noodle manufacturer, located in Indonesia. This product has entered the market since 1983 and currently available in many parts of the world includes America and Europe, African Region, Australasia and various regions in Asia. A Pack of cooked Indomie Mi Goreng comes as a plain fried noodle, but it can be served with various additional ingredients to enhance the taste. Popular additional ingredients include a sunny side egg or omelette, corned beef or...

I swear to god once there was a bunch of surfies all of them look so hungry

then they all start eating mi goreng

I know a friend at schoolies he survived for one week on mi goreng @HenryT.Horton

Alex Becker

@BenjaLim Ah, a familiar food indeed. Here we call it Ramen, and I had a bowl for lunch today.

BenjaLim

@AlexBecker Ramen is japanese

Alex Becker

1:33 AM

@BenjaLim Yes, but I was under impression this was the same type of thing. We call any brand of instant noodles "Ramen" in my experience.

BenjaLim

Instant noodles is perhaps a better word

@AlexBecker We have a common kitchen downstairs

I cook everyday but never have I just had instant noodles for lunch/dinner

usually I will cook some veggies

Alex Becker

@BenjaLim Usually I cook something a little better as well. But today we were running low on groceries, so I made due. My roommate is actually out shopping right now.

BenjaLim

@AlexBecker You share rooms at uni?

how much is your rent in USD?

it's about 1-1 now between USD and AUD I believe

I pay AUD 175 a week

unlimited internet

common kitchens, showers

electricity/water/gas for cooking included

I realise "gas" to us for americans is fuel for cars

Rajesh D

Hey @Ben : Are you an Aussie?

BenjaLim

@RajeshD how come?

Alex Becker

1:38 AM

@BenjaLim I have a single during the year, but I'm living in an apartment over the summer with a couple roommates. My summer rent runs 400/mo including utilities (no internet, but if I position my laptop just right I get uni wifi), but the rooms during the year are godawful expensive, so most people move out second or third year.

Rajesh D

@BenjaLim AUD = Australian Dollar?

BenjaLim

@RajeshD yes

@AlexBecker 400 a month is not bad

Rajesh D

then I guess you are

BenjaLim

I pay 175 a week

Alex Becker

@BenjaLim But my apartment looks like crap, so 400/mo is not that great a deal.

BenjaLim

1:39 AM

@AlexBecker picture?

google maps is good

@RajeshD Are you in india?

Rajesh D

yep

Alex Becker

@BenjaLim You can't see the holes in google earth, unfortunately. But let me find me.

BenjaLim

I like briyani

@RajeshD The original one cooked in a copper pot

Rajesh D

Dum ki Biryani!

BenjaLim

where you put dough round the sides and seal the lid and then put hot rocks on top

Alex Becker

1:40 AM

@BenjaLim maps.google.com/…

Rajesh D

they actually cook it in pot, Dum means pot.

BenjaLim

@RajeshD Like this one here: youtube.com/watch?v=pdGIqvXaCdc

Rajesh D

Anyway iys spicy and hot, do you like it spicy?

BenjaLim

@RajeshD yes

@AlexBecker no street view :(

Rajesh D

@Ben : you are an Indian aren't you

BenjaLim

1:42 AM

@RajeshD no. But I have many indian friends

Rajesh D

quite rare to see somone from west like spicy food

Alex Becker

@BenjaLim I assure you, you aren't missing much.

BenjaLim

@RajeshD You'd be surprised. I once went to a dim sim restaurant the whole restaurant was full of white people

@RajeshD And btw Thai is very popular here in australia

and I assure you thai food is not spicy but burning hot

@AlexBecker what do you usually cook?

@RajeshD pad thai, tomyum is so popular now.....

Alex Becker

@BenjaLim I'm a big fan of stir fries. But my roommates prefer sausages, so I do both of those pretty often.

BenjaLim

@RajeshD There are many asians now in australia

@AlexBecker in your past life i'm sure you were asian :D

Rajesh D

1:45 AM

I know

Peter Tamaroff

@BenjaLim Could you define "isolated point" in terms of beighborhoods?

BenjaLim

@PeterTamaroff I prefer the one in terms of closures

Alex Becker

@BenjaLim Me and another roommate are currently competing to make the best green beans. I think my green bean & bacon recipe will be hard to beat though.

Chuck Fernández

What is a PDF in mathematics?

Peter Tamaroff

@BenjaLim I need a definition in terms of nbhds.

Alex Becker

1:46 AM

@BenjaLim Quite possibly. My interest in math is another mark in favor of that theory. :)

BenjaLim

@AlexBecker why?

Rajesh D

@ChuckFernández prolly, prolly distribution function = PDF.

BenjaLim

@PeterTamaroff A point $x$ is said to be an isolated point of a set $E$ if there exists $\epsilon>0$ such that $B_\epsilon(x) \cap E = \{x\}$

Alex Becker

@BenjaLim At least the math department here has a lot of asians, and this is true of most other depts I've seen.

BenjaLim

@AlexBecker true..... but I think my interest in maths has nothing to do with being asian.....

@RajeshD I'm confused people like to divide indian food into north and south

but hydrebad

what is that north or south....

Peter Tamaroff

1:47 AM

@BenjaLim Sweet.

Alex Becker

@BenjaLim That definition doesn't look quite right. Perhaps you mean the intersection is $\{x\}$?

Mark Dominus

What sort of deck has 14 each of red, green, yellow, and black?

BenjaLim

@AlexBecker I usually take the one where we say $x \in X$ is isolated if $\{x\}$ is open

@AlexBecker actually chorizo is good but it's quite expensive

Mark Dominus

It seems that the answer is a Rook deck.

Thanks, Google.

Peter Tamaroff

@BenjaLim CHORIZO

That is a Spanish word.

BenjaLim

1:51 AM

yes

I like it a lot

@PeterTamaroff gazpacho

@PeterTamaroff paella

@PeterTamaroff My mexican friend once organised a pinata party

Rajesh D

@BenjaLim yes they are a bit different in general. but Biryani is invented by the Moghuls in North and by Nizam's in Hyderabad. The famous 'Hyderabadi Biryani' is that by Nizams and is quite popular all.

Peter Tamaroff

i0.kym-cdn.com/photos/images/newsfeed/000/200/420/…

BenjaLim

@PeterTamaroff and the next day we had real mexican en.wikipedia.org/wiki/Machaca

@PeterTamaroff where did you get that, why are you showing it to me?

Peter Tamaroff

@BenjaLim Just a funny meme.

BenjaLim

@RajeshD no offense, but sometimes my pakistan friends tell me that their briyani is superior to that found in india....

Mark Dominus

1:53 AM

I associate biryani, and Mogul food in general, with North India.

I do not know exactly where Hyderabad is.

BenjaLim

@RajeshD do you have dosas/rasam/apam/sambar?

Rajesh D

@BenjaLim Biryani is basically the speciality of the Nizam's especially Hyderabadi Biryani, well may be its their opinion. I have never tasted theirs.

Mark Dominus

Wow, Hyderabad is totally not in the north part.

Rajesh D

@BenjaLim Thats south Indian. What is cooked at my home quite regularly

Hyderabad is south

BenjaLim

@RajeshD oh wow so the food in hydrebad is south indian?

@RajeshD I like drumstick

Rajesh D

1:55 AM

yes

BenjaLim

the first time I had it I did not know you're only supposed to suck out the inside

Rajesh D

lol

BenjaLim

@RajeshD what was your breakfast, idli/chapati with coconut chutney? And red one or white one?

@RajeshD what do you speak?

Rajesh D

not yet prepared, but there are other usual things like upma, pongal, chapathi etc

Telugu

BenjaLim

ah ok

Rajesh D

1:57 AM

coconut chutney, thats the very usual thing.

Peter Tamaroff

@BenjaLim Could we just say "a point that is not a limit point is an isolated point"?

BenjaLim

@RajeshD any other?

tamil/malayalam

Rajesh D

@PeterTamaroff same there too

BenjaLim

@PeterTamaroff What if you negate the definition of limit point?

Rajesh D

they are also south indian

BenjaLim

1:57 AM

@RajeshD you like tollywood?

Rajesh D

but the tatse differs

ofcourse, but not a big fan

Peter Tamaroff

@BenjaLim Well yes, that would do.

BenjaLim

@RajeshD have you seen sivaji the boss?

@PeterTamaroff well actually

the definition of isolated point requires the point to be in the set

Mark Dominus

Would it be insane to refer to the point (0,1) as {0}×{1}?

BenjaLim

whereas for a limit point we don't have that

Rajesh D

1:59 AM

yeah I saw the promos only, missed it

BenjaLim

@RajeshD youtube.com/watch?v=-VJdkXG30XQ

Peter Tamaroff

@BenjaLim Potato, patata.

BenjaLim

@PeterTamaroff hmmm there's some small detail involved

Peter Tamaroff

@BenjaLim Yes, yes.

BenjaLim

@PeterTamaroff but usually it does not come in :D

who left chat

CLarue

2:12 AM

@MarkDominus following usual conventions, $(0,1)$ is an element of $\{0\}\times\{1\}$.

rather than the entire set

Mark Dominus

True.

I guess it would be somewhat insane.

Peter Tamaroff

@MarkDominus Yo Markus

Mark Dominus

Bon soir, Piotr.

Peter Tamaroff

@MarkDominus I'm reading about open and closed sets now. I saw you posted a topology question on main.

Mark Dominus

It's only sort of a topology question.

It's kind of an oddity.

Peter Tamaroff

2:19 AM

@MarkDominus I see.

Henry T. Horton

What's a topology?

Mark Dominus

Sometimes it's hard to choose tags.

@HenryT.Horton It's a structure on a set that says, in a very abstract way, what it means for points to be "close" to each other.

Chuck Fernández

its an object (X,T) X being a set and T being a set of subsets of X containing three properties:X,{} elements of T

for any O1 and O2 in T, O1intersectO2 in T

Mark Dominus

@HenryT.Horton It's an abstraction of the properties of open balls in $R^n$.

Peter Tamaroff

@MarkDominus =D

Mark Dominus

2:23 AM

Why =D?

Chuck Fernández

for any O1,O2,O2...On in T O1unionO2unionO3....Union On element of T

Peter Tamaroff

@MarkDominus It is a nice characterization.

Which I find enlightening

Mark Dominus

@ChuckFernández No. You need to allow arbitrary unions, not just finite unions.

This is very important.

Chuck Fernández

ok

Peter Tamaroff

@ChuckFernández The intersections are finite, but the unions are arbitrary.

Chuck Fernández

2:24 AM

ts kind of hard to type here

Im reading topology without tears

Robert Smith

Sorry to interrupt... I have a quick question regarding the definition of rate entropy.. Does any want to help me?

Henry T. Horton

Is that a shampoo?

Peter Tamaroff

That is, given an indexing set $I$, if $O_{\alpha}$ whenever $\alpha \in I$ is in the topology $\mathcal T$, then $$\bigcup_{\alpha \in I} O_\alpha$$ is also in $\mathcal T$.

Mark Dominus

@HenryT.Horton Only if your head can be considered to be a hairy ball.

Peter Tamaroff

@HenryT.Horton A book. But I don't liked the intro much.

It just starts with topologies. No explanation, no discussion.

Quite dull, IMO

Mark Dominus

2:27 AM

Too many topology discussions take this approach, in my opinion.

Peter Tamaroff

@MarkDominus Which approach?

Chuck Fernández

I started reading Bert Mendelson but it was too confusing

Peter Tamaroff

@ChuckFernández I'm using that one, and I'm loving it.

Mark Dominus

"Here are three axioms. Now we follow with 17 definitions and then prove a bunch pf theorems without any reference to why the intersections are finite and the unions are not."

Henry T. Horton

As you probably know, one of the best approaches is to start with the balls

Peter Tamaroff

2:29 AM

@MarkDominus Right. I don't like that.

@HenryT.Horton LOL though I guess it is not intended as a joke

@ChuckFernández Though it is a little tough at first.

Mark Dominus

I was fortunate that my first exposure to topology was from Rudin, which starts with metric spaces.

Peter Tamaroff

It took me some time to really start studying from it.

It usually grabbed it and read from it, and wimped out.

Chuck Fernández

But topology without tears covers more topics

Peter Tamaroff

You'll know when you're ready.

@ChuckFernández But Mendelson gives the elements or basics, which I find are important to know first. You need to build strong pilars.

Chuck Fernández

Mendelson focuses a lot on the euclidean topology

How far are you in mendelsons book?

Peter Tamaroff

2:36 AM

@ChuckFernández Page 55. Section 6 of CH2

Still have a looooooong way to go.

Mark Dominus

@PeterTamaroff V.I. Arnol'd has a superb essay criticizing this style of mathematical instruction. His example is group theory. He complains that it is almost always presented in terms of the three or four abstract axioms, but this approach is much less clear than defining a group as a complete collection of permutation mappings.

Chuck Fernández

thats more or less where i stopped

i figure ill finish it after i read twt

Mark Dominus

The abstract approach is not even more general, since Cayley's theorem says that every group can be understood as a set of bijections.

Peter Tamaroff

@ChuckFernández Why did you stop?

Chuck Fernández

it took me too much time to read a page

Peter Tamaroff

2:38 AM

@MarkDominus I see. It is good it is being debunked.

@ChuckFernández "It takes time ♫♪"

Mark Dominus

Here it is: "On Teaching Mathematics", by V.I. Arnol'd.

Chuck Fernández

mathematics is a part of physics? i have never felt more insulted

Mark Dominus

"The determinant of a matrix is an (oriented) volume of the parallelepiped whose edges are its columns. If the students are told this secret (which is carefully hidden in the purified algebraic education), then the whole theory of determinants becomes a clear chapter of the theory of poly-linear forms. If determinants are defined otherwise, then any sensible person will forever hate all the determinants, Jacobians and the implicit function theorem."

Peter Tamaroff

@MarkDominus Why would he even start with "Mathematics is a part of physics" if he wants any mathematician to read that?

Mark Dominus

He didn't start with "Mathematics is a part of physics". He started with "V.I. Arnold". That might be enough to get many mathematicians to read it.

Henry T. Horton

2:43 AM

@MarkDominus I was tutoring a linear algebra student the other day and told her about the volume interpretation of determinants

I'm cool

robjohn

@HenryT.Horton Do your determinants go to 11?

Rajesh D

Hi @robjohn : I was looking for you

robjohn

@RajeshD Found me you have...

Rajesh D

I have question related to this here

robjohn

@RajeshD okay... what's the question?

Rajesh D

2:51 AM

There you have dealt with one singularity. But I want to know what happens when there are two singularities, say $f$ jumping at $x=d_1$ and $f'$ jumping at $x=d_2$. How is the behaviour of $f_h$ (Hilbert transform) at $x = d_1$ and that of $f_h^{(1)}$ at $x = d_2$?

@robjohn

the later is the derivative of the hilbert transform of $f$

robjohn

@RajeshD The Hilbert transform smooths anything not local, so if there are two discontinuities, there will be a similar blow up at each discontinuity

user19161

@MarkDominus I dislike his differential equations books.

Mark Dominus

@JasperLoy What did you dislike about them?

Rajesh D

you mean $f_h$ blows up at $x = d_1$, that is fine, But how does the derivative of it's hilbert transform $f_h^{(1)}$ blow up at $x = d_2$? I am not able to prove it. @robjohn

Mark Dominus

The only Arnol'd book I have read is Mathematical Foundations of Classical Mechanics, which I thought was superb.

user19161

2:58 AM

@MarkDominus To me, the best approach motivates topological spaces by introducing metric spaces briefly before proceeding to the theorems in a more general setting. This saves time yet does not forgo intuition.

Mark Dominus

@JasperLoy I agree.

robjohn

@RajeshD what is causing the problem?

Rajesh D

@robjohn In this case the two discontinuties are not at same level, one is a jump of $f$ and the other is a jump of $f'$. I don't know how to deal with the jump of $f'$

user19161

@MarkDominus I only glanced through them though. It seems that the proofs are a little handwaving. I am thinking more about his ODE text.

user19161

Hello @hen! I hope the guys did not bully you today!

Rajesh D

3:01 AM

Because $f_h^{(1)}+g_h$ is not smooth for some $g$ as $f_h$ already has a blowup at $x = d_1$. @robjohn

robjohn

@RajeshD The You can break up the function into the sum of two $C^\infty$ functions except that one has the one discontinuity and the other has the other.

Henry T. Horton

My eyes

Rajesh D

@robjohn be back after a thought

user19161

@MarkDominus Hmm, that would be {(0,1)} instead wouldn't it? Unless you are doing some kind of correspondence with an alternative definition.

user19161

@BenjaLim In my present life, I'm not sure I'm human.

Peter Tamaroff

3:15 AM

Check this one guys

Yes, I was certainly a little bored.

Henry T. Horton

Hello dudebros

Peter Tamaroff

@HenryT.Horton When did you leave?

Henry T. Horton

I don't know

I'm always watching

Peter Tamaroff

@HenryT.Horton Creeeeeeeeeeeeeepy.

John Calsbeek

Terminology question: Is there a name for a vector x that you get by permuting and possibly negating the components of X, such that X ⊥ Y?

Henry T. Horton

3:27 AM

Trigonometry questions have been quite popular the past couple days

John Calsbeek

er, *name for a vector Y

Peter Tamaroff

@HenryT.Horton More like lots of fellas started trig courses.

Henry T. Horton

@JohnCalsbeek I've never heard of such a thing by any special name

Peter Tamaroff

►

anon

@JohnCalsbeek One could say $y\in S_n x\cap x^\perp$ (the intersection of $y$'s orbit under the permutation action and $x$'s orthogonal complement). Dunno about a name though.

Peter Tamaroff

3:32 AM

@anon I am disappoint.

anon

Actually, one would need to upgrade $S_n$ with reflections..

$C_2^n \rtimes S_n$?

John Calsbeek

I'm just curious if there's a Googleable term, I don't necessarily need it in concise notation.

I suspect there isn't one, but it seems like the sort of thing someone somewhere would've put a name to.

Peter Tamaroff

@JohnCalsbeek You can call them Calsbeekian vectors while we're at it.

John Calsbeek

@PeterTamaroff That just trips right off the tongue, doesn't it?

Peter Tamaroff

@JohnCalsbeek Well, it is very similar to Wronskian!

John Calsbeek

3:45 AM

@PeterTamaroff Introducing a thing gives you the right to stick your silly last name on it. I think that's international law. You have to be reasonable if you're merely the first person to name something :(

Henry T. Horton

Stigler's law of eponymy

Stigler's law of eponymy is a process proposed by University of Chicago statistics professor Stephen Stigler in his 1980 publication "Stigler’s law of eponymy". In its simplest and strongest form it says: "No scientific discovery is named after its original discoverer." Stigler named the sociologist Robert K. Merton as the discoverer of "Stigler's law", consciously making "Stigler's law" exemplify Stigler's law. Derivation Historical acclaim for discoveries is often allotted to persons of notoriety who bring attention to an idea that is not yet widely known, whether or not that person w...

John Calsbeek

I didn't say anyone followed international law, now, did I?

Also, isn't Gauss kind of a strong counterexample to that? =)

Peter Tamaroff

I'm off now.

Bye byes.

anon

4:04 AM

@PeterTamaroff srsly?

Rajesh D

$\frac{1}{\pi}$ ..... $\dfrac{1}{\pi}$

@robjohn If this is possible then it works. But can we always breakup a function in such a way?

robjohn

4:28 AM

@RajeshD The Hilbert Transform is linear, so use a smooth cutoff function to separate the singularities

$H(\varphi f)+H((1-\varphi)f)=Hf$

@RajeshD each of $\varphi f$ and $(1-\varphi)f$ have one singularity

Rajesh D

4:57 AM

@robjohn What is the notation $H$ and $\phi$, could you tell me in words so that I can google them

Oh, $H$ is Hilbert transform I guess, (not Heaveside). What is $\varphi f$?

@anon : Do you have any idea what $\varphi f$ means?

user19161

5:18 AM

@HenryT.Horton So am I.

Rajesh D

5:47 AM

@robjohn I now understand. Beautiful and elegant!. I should prolly kill myself for not knowing and not figuring out myself about the existence of these beautiful creatures called "smooth cutoff functions". Really its a handicap to not know of the existence of $\varphi$'s.

Jonas Teuwen

7:29 AM

Moooooo.

robjohn

@RajeshD $H$ is the Hilbert Transform, $\varphi$ is a cutoff function that is $1$ where one singularity is and $0$ where the other is.

Rajesh D

@robjohn Thanks @robjohn . I got it.

robjohn

@RajeshD I replied before I saw the second comment. I'm glad that you got it :-)

@RajeshD Sorry to take so long. I had to go afk.

Rajesh D

No problem. Now I got everything about this problem cleared of. But I want to extend this to circular hilbert transform of periodic functions. I'll try it in next few days.

robjohn

@RajeshD It should involve the same ideas.

Rajesh D

7:37 AM

ok

Transcript for

$Mathematics$ Mathematics