Mathematics - 2012-08-07 (page 1 of 6)

Hard to say why. He looks for insight. That seems like an obvious thing to do but I don't know, he puts it into practice about as well as anyone I've seen.

@PeterTamaroff It depends on what you mean.

In a sense, almost everyone uses topology.

Peter Tamaroff

1:49 AM

@DylanMoreland I mean, does topology (or any of its branches) play an important role in modern number theory?

Dylan Moreland

I don't think there are serious uses for general topology.

Peter Tamaroff

@DylanMoreland OK. But say, algebraic topology?

Dylan Moreland

There's still a lot of wiggle room there.

At a high enough level algebraic topology seems indistinguishable from hardcore algebraic geometry to me.

Look at what Lurie is doing, for example.

Matt has been doing a lot with the cohomology of arithmetic groups. That's something to look into. That's a wild subject.

The book by Ken Brown is a good one.

Peter Tamaroff

@DylanMoreland Who's Lurie?

Dylan Moreland

But I would say that the purely algebraic machinery you'd find in Hatcher is usually not enough to tell you what you want.

You need some other input. Often that has to come from Lie algebra methods or representation theory or blah blah.

But computing homology of certain manifolds with rational coefficients? That sounds like algebraic topology to me.

Jacob Lurie.

Peter Tamaroff

1:55 AM

@DylanMoreland I'm just a soon to be freshman. I was "complaining" today on how I'm starting univ. kinda late.

Dylan Moreland

It's also interesting to note that topologists have uses for modular forms. See this, for example.

Peter Tamaroff

@DylanMoreland Dylan, sorry to disappoint you, but I just know very few elementary number theory.

Dylan Moreland

But I don't see how any of that interacts.

I'm not disappointed. I don't know how else to answer these questions, though.

Peter Tamaroff

@DylanMoreland Hehehe that is a good point.

Dylan Moreland

Back in a bit, maybe. Good luck with topology!

Peter Tamaroff

1:59 AM

@BenjaLim You there. I have a question.

mixedmath

2:28 AM

quiet room now

leo

YEP

Peter Tamaroff

@mixedmath Hey

@leo Hola!

mixedmath

oh hiya

leo

@PeterTamaroff hola!

Peter Tamaroff

@mixedmath I have a question about Munkres' book.

@leo How's it going?

mixedmath

2:32 AM

ok, I seem to be answering lots of algebraic topology questions today

but some regular topology should be okay too

leo

@PeterTamaroff tomorrow I start new period

new semester

mixedmath

where are you from, that it's called new period?

Peter Tamaroff

@mixedmath Oh, but it is about the layout.

mixedmath

oh, alright

Peter Tamaroff

@mixedmath Hehhehee, he's from Costa Rica, if I recall correctly.

mixedmath

2:33 AM

should I go and grab my copy?

Peter Tamaroff

@mixedmath Nah.

leo

yes I am :-)

Peter Tamaroff

Just, why is the Set Theory chapter so "heavy"?

mixedmath

good memory

leo

it is periodo o ciclo in spanish

mixedmath

2:34 AM

I don't know what you mean. How is it 'heavy' ?

Cycle sounds better.

but semestre

no es un semestre?

is used as well

@mixedmath yes

mixedmath

ah, okok. pura vida ;p

leo

2:35 AM

@mixedmath Pura vida! :-)

Peter Tamaroff

@mixedmath Well, I mean it seems to be very detailed.

leo

why some books says: "Point Set Topology" and no just "Topology"?

mixedmath

I don't seem to recall that it was any more detailed than anything else. Would you prefer it to not be very detailed?

Peter Tamaroff

@leo Point Set means General. I would think they would deal with say, Algebraic Topology, but rather the elements of Topology.

leo

oh

I see

Peter Tamaroff

2:38 AM

@mixedmath Dunno. It's just I've been doing quite a little topology and I never used stuff in there.

mixedmath

oh, that's really funny. So, it turns out that I never really learned much topology outside of what one picks up in analysis classes before I learned algebraic topology and differential geometry

my reference for stuff I didnt know was Munkres, and I've found that I've used more or less all of it at some time or another

or at least, all of teh early chapters

Peter Tamaroff

@mixedmath Hehehe. Well, I happened to get hooked on topology when I started with Rudin. Do you happen to have a copy?

I wanna show you something.

mixedmath

hmm, let me see if I have that here.

leo

lol

Peter Tamaroff

@leo I happen to read that daily. Some of them are really fun, some are borderline crap.

mixedmath

2:42 AM

it must be at my office - I know I have it, but it's not here (at my home)

Peter Tamaroff

@mixedmath OK, nvm, I'll show you.

Wanna learn some topology in 10 seconds? Here we go!

mixedmath

that's lovely. gogogo, topologize!

Peter Tamaroff

It's like: BAM! There you go boy!

mixedmath

It's "perfect" if it's closed and all points are limit points?

who knew?

Nathan Bouscal

I love definition lists like that. Just, here's everything, now try to develop an understanding of it.

Peter Tamaroff

2:46 AM

@mixedmath Are you a professor?

Nathan Bouscal

Much better than having to hunt through the text to find what it is that you need to learn.

mixedmath

no, I'm a second-year graduate student at Brown University

Peter Tamaroff

@mixedmath Oh, that's why you have an office.

You're in your early 20s I take it?

mixedmath

yes indeed

Peter Tamaroff

@mixedmath And what are you working on?

mixedmath

2:49 AM

I work under Dr. Jeff Hoffstein on analytic number theory. He does multiple dirichlet series (which is to regular dirichlet series what multivariable calc is to calc) and automorphic forms. I haven't focused yet, but I'm looking into prehomogoneous vector spaces and the work of Bhargava

(if that means anything to you - I can give some references, etc., if you want more)

Peter Tamaroff

@mixedmath Hehehe not really. I barely remember reading what Dirichlet series were (in terms of Dirichlet characters, which are IIRC some kind of cyclic groups) but I never really studied anything of it.

I am interested in Analytic NT, though.

anon

close enough

mixedmath

nice

Peter Tamaroff

But now I want to get into Real Analysis and Algebra (mainly Linear and Abstract, which I really know very little about)

And Topology.

I am blindedly obsessed with topology. Fuck me, but well...

mixedmath

that's exciting. Stay with what you love

Nathan Bouscal

2:53 AM

Yeah, that reminds me a bit of me. I desperately need to study calc-related things to prepare for the Math GRE, but I'm just studying algebra and topology instead because they're more fun.

Peter Tamaroff

@NathanBouscal GRE?

mixedmath

oh, that's a different, much sadder story. You need to know that calc for the GRE

leo

analysis is fun

Nathan Bouscal

@PeterTamaroff Exam to get into grad school

mixedmath

the GRE is a test taken in the US to get into grad schools.

Nathan Bouscal

2:54 AM

@mixedmath I know =/

Peter Tamaroff

@mixedmath But should Calculus be left as "given" like, how did you graduated if you don't know calc....???

Nathan Bouscal

I didn't graduate with a math degree

Peter Tamaroff

@NathanBouscal Oh, my bad,

Nathan Bouscal

I do know calc, I'm just very rusty on a lot of the applications of it

I'm a really atypical case in general though

If a grad school ever takes me it will be really amusing :)

Peter Tamaroff

@mixedmath I'm a little sad that I have to start working this week. It might sound stupid or desconsiderate, since work is something many would love to have. I'm not greedy or anything, I would only like to have lots of money to have more time to study (or give off to charity and "good deeds": like schools and unis)

Nathan Bouscal

2:57 AM

@PeterTamaroff Most of us feel that way

mixedmath

if it makes you feel better, Brown's attitude towards the math GRE is really funny. It gives a lower cutoff point (say, 60-65 percentile or something): if you have less, than you don

Nathan Bouscal

The only thing I want money for is so that I don't have to waste time working for the rent

mixedmath

t get in, but that's about that

Nathan Bouscal

Well, my problem is I need a good GRE score to make up for the fact that I got a mediocre GPA in an unrelated major at a mediocre school :P

mixedmath

@PeterTamaroff what will you be doing?

Peter Tamaroff

2:58 AM

@mixedmath I'm a tennis teacher. =)

mixedmath

oh, well, you're right again

Nathan Bouscal

I know I can get a very good score, I'm an excellent test taker, I just need to study the stupid calculus

Peter Tamaroff

@NathanBouscal HEY!

Calculus is NOT stupid!

Nathan Bouscal

Haha

It just isn't exciting for me. At all.

anon

it's replay value is too low

Nathan Bouscal

3:01 AM

Yeah, I feel like it doesn't help me understand other things in the way that learning algebra or topology does.

Peter Tamaroff

@anon What does that mean?

Nathan Bouscal

Every time I learn something new in algebra, I am able to understand so much more other stuff that I read.

Peter Tamaroff

@NathanBouscal Are you using any particular book?

Nathan Bouscal

For which?

I'm studying out of about 10 books right now :P

anon

@PeterTamaroff http://en.wikipedia.org/wiki/Replay_value

mixedmath

3:02 AM

one might argue that 10 books is too many ;p

Nathan Bouscal

Yes, and one would likely be correct :)

I am just impatient

Peter Tamaroff

@MarianoSuárez-Alvarez Hello. Quite the day in the city today, huh?

Nathan Bouscal

I want to learn everything all at once, but my brain doesn't quite work that well.

Peter Tamaroff

@NathanBouscal Wut?

Nathan Bouscal

@PeterTamaroff What what?

Peter Tamaroff

3:03 AM

@NathanBouscal I mean a book for Algebra.

Nathan Bouscal

Ah. For Algebra I'm currently studying out of Dummit & Foote and also out of Artin

I also have Hungerford sitting on my shelf but haven't read any of it yet.

And I have a linear algebra pdf of Hoffman and Kunze

Peter Tamaroff

Does anyone think it is kind of weird to write $O=\bigcup_{x\in O}B_{x}$?

anon

no

Peter Tamaroff

@anon Just checking. Seems OK to me.

Bleh.

anon

no, a construction of O wouldn't presuppose what elements are in O.

Peter Tamaroff

3:08 AM

@anon Yes. I derpped.

anon

I would call it an observation

Peter Tamaroff

@leo

This one is good

smbc-comics.com/?db=comics&id=1618#comic

leo

@PeterTamaroff yes indeed

leo

3:27 AM

@PeterTamaroff what about this one

Peter Tamaroff

@leo HAHAHHAHAHAHAHA

leo

smbc-comics.com/?db=comics&id=656#comic

Mariano Suárez-Alvarez

@PeterTamaroff Yeah :-/

Peter Tamaroff

@MarianoSuárez-Alvarez I heard there was a train wreck once again, which interrupted the subway news broadcast. =/ Pretty chaotic.

Mariano Suárez-Alvarez

yeah

it is getting pretty absurd

Peter Tamaroff

3:35 AM

@MarianoSuárez-Alvarez I'm baffled by what some politicians are saying. It seems just unreal. I'm not a very political person so to speak, but it is just too much.

Have you seen this?

"El 25 de mayo de 2003, asumió como presidente de la Nación, el Dr. Néstor
Carlos Kirchner. Uno de sus objetivos fue evitar que continúe la desmovilización
política hasta ese momento causada por la represión, frustración y la indiferencia
popular.
Rápidamente la línea general de acción que desplegó el presidente Kirchner fue
clara y concreta.
**Sincerarse en el lenguaje y las conductas, y que cada dirigente asuma, sin hipocresía, su verdadera expresión política. Superar con nuevas ideas las divisiones internas de las fuerzas mayoritarias que estaban estancadas. Dar cauce y referencia nacion…

Mariano Suárez-Alvarez

wehre did you get that? :-)

Peter Tamaroff

@MarianoSuárez-Alvarez Its in the page of the Ministry of ${\rm int}(\text Argentina)$

Mariano Suárez-Alvarez

very cute

Peter Tamaroff

@MarianoSuárez-Alvarez What do you mean by that?

Mariano Suárez-Alvarez

«cute» as in «scary»...

Peter Tamaroff

3:45 AM

@MarianoSuárez-Alvarez Strange use of words, but I agree.

I'm mostly scared about the delusion they have. Sometimes I don't know if they are purposedly lying or if they just are psychotic.

Principio de la transposición. Cargar sobre el adversario los propios errores o defectos, respondiendo el ataque con el ataque. Si no puedes negar las malas noticias, inventa otras que las distraigan. ...directo del manual de Goebbels

Its kind of cliché doing that comparison, but what the hell.

Mariano Suárez-Alvarez

thanks for the link

very interesting

Peter Tamaroff

@MarianoSuárez-Alvarez It is very interesting, yes. And it is up to date.

But, as someone I know said, "it won't work on the elderly, but rather on certain young people"

"Queridos compatriotas, la Argentina cambió. Sólo nos faltan las Islas Malvinas
para sentir que nuestra patria está completa"

Mariano Suárez-Alvarez

4:07 AM

Sisi, esa frase es extraordinaria.

Peter Tamaroff

@MarianoSuárez-Alvarez Pregunta: que aplicacion tiene la topologia $A$ y la topologia $\ell$ de $\Bbb R$. La primera es la topologia que surge de la base $(a,b)$ mas todos los conjuntos de la forma $(a,b)-\{1,1/2,1/3,\cdots\}$ y la $\ell$ es la "lower limit topology" con base $[a,b)$

Le digo $A$ de Arquimedeana mejor, para evitar disgustos.

Munkres le dice $K$

Mariano Suárez-Alvarez

la primera, ninguna

es solo un ejemplo

La segunda da la nocion de continuidad superior (o inferior, o a izquierda, o a derecha... alguna de ellas)

Peter Tamaroff

@MarianoSuárez-Alvarez Parece que solo las menciona para mostrar que son mas finas que la topologia estandar pero no son comparables.

@MarianoSuárez-Alvarez Es algo complicado?

Mariano Suárez-Alvarez

que cosa?

Peter Tamaroff

@MarianoSuárez-Alvarez La nocion de continuidad superior.

Mariano Suárez-Alvarez

4:15 AM

para nada

debe estar en wikipedia

Peter Tamaroff

@MarianoSuárez-Alvarez Aparece en Teoría de la Medida (Measure Theory), puede ser?

Mariano Suárez-Alvarez

puede ser

buscá semicontinuidad

Peter Tamaroff

@MarianoSuárez-Alvarez Bullseye

Mariano Suárez-Alvarez

una funcion f:R\to R es continua sii para cadaa sucesión convergent $x_n\to x$ es $\lim f(x_n)=f(x)$.

Es superiormente continua si $\lim f(x_n)\leq f(x)$.

(o $\limsup$, en realidad)

Peter Tamaroff

@MarianoSuárez-Alvarez Claro.

No suena muy interesante =P

Mariano Suárez-Alvarez

4:18 AM

es útil

hay muchas funciones que son semicontinuas pero no continuas

si interpretás el valor $f(x)$ como una medida de la bondad de $x$, entonces la semicontinuidad te dice que «pasar al limite no empeora los puntos»

Peter Tamaroff

@MarianoSuárez-Alvarez En este caso, un funcion continua podria redefinirse y para que sea continua continua?

Mariano Suárez-Alvarez

esa idea es extremadamente útil

no

Peter Tamaroff

@MarianoSuárez-Alvarez Hehehe OK.

Mariano Suárez-Alvarez

y en esa forma es que aparecen muchas funciones semicontinuas: como medidas de bondad de cosas

Peter Tamaroff

@MarianoSuárez-Alvarez "bondad" en que sentido? (no tengo tanta empatía con mi teoría =P)

Mariano Suárez-Alvarez

4:23 AM

sabes lo que es el rango de una matriz?

Peter Tamaroff

@MarianoSuárez-Alvarez Ahora lo se. =)

Mariano Suárez-Alvarez

ok :-)

el rango es una función semicontinua de las matrices a los enteros

leo

@PeterTamaroff ha ha ha ha ha :D

Mariano Suárez-Alvarez

si todas las matrices de una sucesión tienen rango$\leq k$ y convergen a una matriz, entonces la matriz límite tiene rango $\leq k$.

(pero puede ser que el límite tenga rango estríctamente menor que $k$)

en este ejemplo, una matriz es buena si tiene rango grande, y la semicontinuidad de la funcion rango dice que al tomar un límite las matrices no mejoran

Peter Tamaroff

@MarianoSuárez-Alvarez Hmmm. Justo el 17 empiezo Algebra Lineal. Pero me interesa mas el Algebra Abstracta. Tengo un libro sobre Algebra Lineal que tiene 10 caps, estudie del primero y no segui. Todavia no le agarre gusto, supongo.

Mariano Suárez-Alvarez

4:28 AM

no hay ninguna diferencia

es IMPOSIBLE aprender "algebra abstracta" sin saber algebra lineal

de hecho, es imposible hacer matemática sin saber algebra lineal

Peter Tamaroff

@MarianoSuárez-Alvarez Hehehe me imagino. Por eso empece por algebra lineal.

@MarianoSuárez-Alvarez Se que es una pregunta un poco estupida, pero "imposible" en que sentido?

Este es el indice
1 Espacios vectoriales
2 Matrices
3 Transformaciones lineales
4 Espacio dual
5 Determinantes
6 Diagonalizaci¶on
7 Forma de Jordan
8 Espacios vectoriales con producto interno
9 Variedades lineales
10 Formas bilineales

Mariano Suárez-Alvarez

en que es imposible :-)

trust me

Peter Tamaroff

@MarianoSuárez-Alvarez I'm a mathematician.

Nathan Bouscal

I completely disagree that it's impossible to learn abstract algebra without knowing linear algebra.

Linear algebra is really important, yes, but you can still learn a great deal of abstract algebra without it.

Peter Tamaroff

**Defnicion 3.32** Sea $A \in K^{n\times m}$ Se llama rango columna de $A$, y se nota ${\rm rg}_C(A)$, a la dimension del subespacio de $K^n$ generado por las columnas de $A$, es decir, si $A = (C_1 \mid\cdots\mid C_m)$,
entonces ${\rm rg}_C(A) = \dim< C_1; \cdots ;Cm >$
Mediante el calculo del rango columna de una matriz A es posible obtener la dimension del subespacio de soluciones del sistema lineal homogeneo asociado a $A$.

@MarianoSuárez-Alvarez Esto esta en el libro. La definicion de Wikipedia parece ser un poco mas "light".

Es mas formal la definicion del libro, supongo.

Es bastante riguroso el libro. Antes de definir un $K$-esp. vectorial define una operacion binaria, un grupo, un anillo, un cuerpo, una accion, y despues un $K$-ev. Por eso me interese un poco mas con este libro.

leo

4:54 AM

bueno señores, buenas noches!

David Wheeler

5:13 AM

did i walk into the wrong country? i meant to turn right at Cuba.....

@NathanBouscal i would argue that in some respects linear algebra makes more sense if you learn a little abstract algebra first...but perhaps that is asking too much of people

Nathan Bouscal

@DavidWheeler Yeah. I can definitely say that once I learned some abstract algebra, certain aspects of linear algebra made significantly more sense.

David Wheeler

what i think is "neat" about linear algebra is it's very easy to see how the abstract and the concrete inform each other