English Language & Usage: Multi-Layered Discourse Room

Faheem Mitha

11:00 PM

And might occasionally be useful.

Cerberus

Of course.

I wish I knew about it.

Faheem Mitha

Actually, most people would probably benefit most from a short course in logical thinking.

Cerberus

How would such a course commence?

I did do logic when I was studying philosophy.

Faheem Mitha

@Cerberus I'm not starry-eyed about math. I think educators tend to overstate its advantages. As with much conventional education. You'd probably be better off taking courses in flirting with girls. Or other practical things.

@Cerberus Yes, logic is good.

Rand al'Thor

@FaheemMitha To be fair, if they're arts students, say, why should they know complex analysis? It's not taught until midway through undergrad maths.

@FaheemMitha rofl

Cerberus

11:02 PM

@FaheemMitha I don't think that would get me very far!

Faheem Mitha

@Randal'Thor I think the UC teams tend to be a mixture of arts and science people. Though they seem to be skewed heavily in the direction of arts. Would you agree?

Cerberus

What is complex analysis?

Rand al'Thor

Or a choice between courses in maths and courses in flirting, given that mathematicians are stereotypically incapable of social interactions.

Faheem Mitha

@Cerberus Analysis on the complex numbers. Analysis is probably better known as calculus. Which is the same thing to a first approximation.

Cerberus

Hmm I don't really know what any of those things are.

Faheem Mitha

11:03 PM

@Randal'Thor Well, social interaction, like other things, can be learned. People aren't born being able to read and talk, either.

Cerberus

I did like maths well enough in school.

Rand al'Thor

@FaheemMitha I haven't noticed any skew - seems like a fairly good mix to me.

Faheem Mitha

@Cerberus You must have run across calculus. They teach in in UK and US high schools. And certainly in European ones as well.

@Randal'Thor Ah. Possibly I'm mistaken.

Cerberus

Probably, but I have no idea what it is.

Faheem Mitha

@Cerberus Ok. Well, it's not a big deal.

Rand al'Thor

11:04 PM

There are some particular teams which are very skewed, like Imperial last year which was all scientists, but overall it's not bad IMO.

Cerberus

I think the toughest things we did in school were integrating goniometrical functions and stuff.

Faheem Mitha

@Randal'Thor Well, it certainly is lively. And impressive.

@Cerberus Integrating is part of calculus/analysis.

Rand al'Thor

@FaheemMitha What kind of maths were/are you most into?

Cerberus

OK.

Rand al'Thor

@Cerberus Differentiation? Integration?

Cerberus

11:05 PM

Yes, we did those.

Faheem Mitha

@Randal'Thor I once tried to be a grad student in Algebraic Geometry. That didn't go so well.

Rand al'Thor

@Cerberus Complex analysis is the coolest part of analysis :-D

Faheem Mitha

What kind of math do you do? Where do you work?

Cerberus

Oh!

What is it about?

Or that's probably too complicated to explain.

Faheem Mitha

Complex analysis is fun.

@Cerberus Not that complicated.

Rand al'Thor

11:07 PM

@FaheemMitha Yikes, algebraic geometry is hard. I always hated it (though part of that might be because I learned it badly from an awful lecturer).

Also, it should be called algebraic algebra.

Faheem Mitha

@Randal'Thor Yes, it's hard.

The problem with research math is that unless you're really really good at it, there is no space for you.

Rand al'Thor

@FaheemMitha I do analysis. Sorry, I'd rather not say exactly where I work :-)

@Cerberus You know complex numbers?

Faheem Mitha

Also, of course, you have to really, really like doing it. Like, all the time.

And you can't do anything else afterwards. Because you won't be qualified. It's a fairly drastic form of specialization.

Cerberus

@Randal'Thor I've just read about them.

Faheem Mitha

@Randal'Thor No problem. Are you a grad student?

Cerberus

11:09 PM

I know what i is.

> The Italian mathematician Gerolamo Cardano is the first known to have introduced complex numbers. He called them "fictitious" during his attempts to find solutions to cubic equations in the 16th century.

Faheem Mitha

@Cerberus Complex numbers come about naturally as the algebraic closure of the reals.

Cerberus

As in, when solving x²=-1?

Faheem Mitha

@Cerberus Right

Cerberus

Or is a purely imaginary number simple?

Faheem Mitha

@Cerberus Don't follow.

Rand al'Thor

11:11 PM

@Cerberus Right. Well, complex analysis is when you do analysis (differentiation and integration) on the plane of complex numbers (x+iy). Instead of considering functions mapping real numbers to real numbers, look at functions mapping complex numbers to complex numbers.

It seems as though you've just made everything much more complicated, but in a sense it actually becomes much simpler.

Faheem Mitha

Yes, because with some natural restrictions, you get much nicer analysis to work with.

Rand al'Thor

Because differentiability (existence of the derivative) for a complex function is a really REALLY strong condition. Which means that any differentiable complex function has loads and loads of other cool properties.

Faheem Mitha

In some sense, the complex plane is a natural place to do analysis.

Cerberus

@FaheemMitha The number i is purely imaginary, isn't it?

Rand al'Thor

On the reals, you can get a function which is differentiable everywhere and second differentiable nowhere, or infinitely differentiable everywhere but can't be written as a power series.

Cerberus

11:13 PM

@Randal'Thor Ah OK, I see.

Rand al'Thor

In the complex plane, every differentiable function is also infinitely differentiable and can be written as a power series.

Faheem Mitha

In other words, the

Cauchy–Riemann equations

In the field of complex analysis in mathematics, the Cauchy–Riemann equations, named after Augustin Cauchy and Bernhard Riemann, consist of a system of two partial differential equations which, together with certain continuity and differentiability criteria, form a necessary and sufficient condition for a complex function to be complex differentiable, that is, holomorphic. This system of equations first appeared in the work of Jean le Rond d'Alembert (d'Alembert 1752). Later, Leonhard Euler connected this system to the analytic functions (Euler 1797). Cauchy (1814) then used these equations to...

Cerberus

And they asked students to solve a problem about integrating complex numbers in the UC?

Faheem Mitha

@Cerberus Yes.

@Cerberus Oh, no. Much simpler than that.

At one point, Paxman described the Taylor series expansion of the exponential, and asked them to identify it. They had no idea what he was talking about.

Rand al'Thor

@FaheemMitha Oh yes, I remember that one!

Faheem Mitha

11:14 PM

He then asked for the radius of convergence and they were like - what???

Rand al'Thor

That's not even complex analysis.

Faheem Mitha

@Randal'Thor True.

Rand al'Thor

Taylor series are in A-level maths, I think.

Or at least A-level Further Maths.

Cerberus

@FaheemMitha I've never heard of one Taylor...

Faheem Mitha

@Randal'Thor yes, they are. It's actually a really easy question for a quiz, provided you know a little math.

Cerberus

11:16 PM

@Randal'Thor Hmm what kind of simple function can be differentiated only once?

Faheem Mitha

I thought they were pretty well up on biological terms, though. I can never keep those straight.

@Cerberus Those are typically pathological.

Cerberus

@FaheemMitha So, if it is only imaginary and not real, is it still complex?

Faheem Mitha

@Cerberus All real numbers are complex too.

Cerberus

Oh.

Faheem Mitha

Complex numbers are all numbers of the form a+ib, where a and b are real.

Cerberus

11:17 PM

But if you have just i, where is the a?

Faheem Mitha

That's an algebraically closed field.

@Cerberus a=0, b=1

Cerberus

OK then any number is complex?

Rand al'Thor

@Cerberus No simple functions. Things like the Weierstrass blancmange function are continuous everywhere and differentiable nowhere, but pathological and complicated to define.

Faheem Mitha

@Cerberus Any real number, yes.

Rand al'Thor

Unless you mean differentiable only once at a point, in which case think of a curve whose rate of curvature changes abruptly.

Cerberus

11:18 PM

@Randal'Thor Ah OK, interesting.

@FaheemMitha Then differentiating x=2 is also complex analysis?

Faheem Mitha

@Cerberus Taylor was a British 17th century mathematician, from memory.

Rand al'Thor

Or (which might be easier) think of a curve which is continuous but not differentiable at a point - i.e. has a sharp bend - and then integrate that function to get one which is differentiable exactly once.

Faheem Mitha

@Cerberus x=2 isn't a function. Do you mean x-2?

Rand al'Thor

@FaheemMitha Specifically English. As opposed to Maclaurin, who was Scottish.

Bit of rivalry there, I think ;-)

Faheem Mitha

Brook Taylor

Brook Taylor FRS (18 August 1685 – 29 December 1731) was an English mathematician who is best known for Taylor's theorem and the Taylor series. == Life and work == Brook Taylor was born in Edmonton (at that time in Middlesex) to John Taylor of Bifrons House in Patrixbourne, Kent, and Olivia Tempest, daughter of Sir Nicholas Tempest, Bart., of Durham. He entered St John's College, Cambridge, as a fellow-commoner in 1701, and took degrees of LL.B. and LL.D. in 1709 and 1714, respectively. Having studied mathematics under John Machin and John Keill, in 1708 he obtained a remarkable solution of the...

Cerberus

11:20 PM

@FaheemMitha Okay, then f(y)=2

Faheem Mitha

More accurately, 18th century.

Cerberus

Or just any very simple function.

f(x)=2x

Faheem Mitha

@Cerberus That's a constant function. Yes, you could trivially define it on the complex plane. Along with any other polynomial function.

Cerberus

Since 2 is a real number, it must be a complex number, right?

And so differentiating this function would be complex analysis?

Faheem Mitha

@Cerberus Right

@Cerberus Sure, I guess so.

Cerberus

11:22 PM

That sounds strange to a layman.

But I believe you.

Rand al'Thor

@Cerberus Yes, but a very simple case of complex analysis which is also real analysis.

Cerberus

So it is a simple complex analysis.

What paradox!

Rand al'Thor

Admittedly, every complex differentiable function is also real differentiable, so you could say that all complex analysis is also real analysis.

Then you might ask, what's the advantage of extending real functions to the complex plane? Why not just consider them as real functions, since imaginary numbers are, well, imaginary?

And this is where things get even more interesting. Many problems on the real line - for instance, calculating real integrals - can be solved by extending them to the complex plane, using the cool properties of complex differentiable functions, and then going back to the real line at the end.

Cerberus

I hear that some real functions are easier to differentiate/integrate in the complex plane?

Right, that.

Faheem Mitha

@Cerberus he's talking about definite integrals. Mostly.

Cerberus

11:25 PM

OK.

Faheem Mitha

Sometimes termed contour integration. Which is very cool, as math goes. And simple, as cool math often isn't.

Most research level math is, for better and worse, completely incomprehensible to the layman. And even to most math undergraduates, probably.

Cerberus

Yes.

Faheem Mitha

So its coolness is lost on most people.

Cerberus

Alas.

Rand al'Thor

@FaheemMitha Yes, one of the things I like about complex analysis is that you don't really need super-abstract ideas and structures in order to understand and even research it.

Faheem Mitha

11:28 PM

@Cerberus You could hang out in the Math SE. There are actually people there who like math, and aren't research mathematicians. Yes, they do exist.

Rand al'Thor

I mean, can you imagine trying to explain algebraic geometry to Cerb here?

What the hell is a scheme, in layman's terms?

Faheem Mitha

@Randal'Thor Yes, it's a nice concrete area. And possibly still an active area of research. Though I wouldn't know.

Cerberus

@FaheemMitha I've been there.

But all the math stuff is way over my heads.

Faheem Mitha

@Randal'Thor Well, it's not so bad. You can use cartesian geometry as an example. The conics are all schemes/varieties, of course.

@Cerberus Ok.

It's nice that there are people out there who like math, without doing it professionally. And who can now use the miracle of the net to find each other.

Cerberus

Yay!

Faheem Mitha

11:30 PM

But granted, cartesian geometry only gets one so far.

tchrist

Too much hyperbole. :)

Faheem Mitha

@Randal'Thor The trouble with most math areas is that there are no applications. So you can't do anything with it afterwards. Even in industrial research.

Though some math people like to pretend otherwise.

Rand al'Thor

@tchrist We're talking about hyperbolas now, not hyperbole :-)

Faheem Mitha

@Randal'Thor So are you a grad student? You can tell me, without compromising your anonymity. :-)

Rand al'Thor

@FaheemMitha I used to be a cloistered pure mathematician too. But certain mathematical cultures put too much emphasis on ultra-abstract things like algebraic geometry ;-) PDEs, for instance, can be studied in a fairly pure mathematical way but still have loads of applications.

@FaheemMitha Yep :-)

Faheem Mitha

11:34 PM

@Randal'Thor Ok. I thought that was probably the case. :-)

@Randal'Thor Only at the very applied end of the scale.

Though I don't know anything about PDEs.

Is that your area?

MetaEd

@Randal'Thor I love mathematics because of the menagerie lions.

Rand al'Thor

@FaheemMitha How about you? You said you "once tried" to be a grad student - does that mean you ended up leaving academia?

Faheem Mitha

@Randal'Thor I did, yes.

Or it left me. Whatever.

I guess I gave up. Though I'd still like to do research, given the opportunity.

As I mentioned above, I wound up getting a degree in statistics.

Rand al'Thor

@FaheemMitha I don't think of myself as a PDE theorist (PDE-ist? what's the word for someone who studies PDEs?), but I have got some results in that area.

Mitch

@KitZ.Fox I feel really sick now

Rand al'Thor

11:37 PM

@FaheemMitha Statistics is sometimes seen as part of maths, but then it goes all the way over to things like biostatistics, which AFAICU is not very mathematical?

Faheem Mitha

@Randal'Thor I see.

Mitch

@Randal'Thor numerical analysis if that's your application area. Just analysis otherwise

Faheem Mitha

@Randal'Thor No, statistics is definitely not part of math. Though it does use some math at the more mathematical end.

Though nothing super-fancy, for the most part. Afaik.

Mitch

wait that's the study area, not the person doing it. liek geometrician? Logician? Combinatorialist? Analyst just doesn't sound right

Faheem Mitha

@Randal'Thor I met at least one eminent biostatistician who clearly didn't know what a limit was.

So no, it's not very mathematical. They like their formulas, though.

Rand al'Thor

11:40 PM

@Mitch I think I'm gonna post a single-word-request.

Faheem Mitha

Statistics research can be quite interesting, I think. I did a bit.

Mitch

@FaheemMitha it's all math, just not covering the same subcategories as traditional mathematics.

Faheem Mitha

But it's very different from mathematical research.

Mitch

more applied math

Faheem Mitha

@Mitch What, statistics? No, it's not math. It's a different subject.

Mitch

11:41 PM

@FaheemMitha I don't see how it's not math.

tchrist

@Randal'Thor I am voting to close this single-word-request for lameness. Review Completed.

Faheem Mitha

@Mitch Um, that would involve some explanation.

Mitch

as would your point of view

Rand al'Thor

@tchrist MOD ABUSE!

Faheem Mitha

@Mitch It would be a bit hard to explain. There must be articles out there discussing it.

Unless someone else wants to take a crack at it. But you'd be better off talking to people in the stats SE chat room. If there is anyone there.

@Cerberus So, how are things going? Long time no see. Or type. Whatever.

Cerberus

11:45 PM

@FaheemMitha Even I know that!

@FaheemMitha Hah, not much new here, you?

Faheem Mitha

@Cerberus Not a lot. I'm doing an Airbnb rental thing now. It's humble, but I need the money. Too much work for the money, really.

Might be better off doing part time tech work...

@Cerberus It was quite shocking.

Rand al'Thor

0

Q: What's the word for someone who studies PDEs?

Mathematics has many words for the people who specialise in particular subdisciplines: geometers, who study geometry algebraists, who study algebra combinatorialists, who study combinatorics probabilists, who study probability statisticians, who study statistics analysts, who study analysis ???...

Downvote away!

Faheem Mitha

@Randal'Thor I nominate Podder.

Cerberus

@FaheemMitha Hmm but you've been doing that for quite a while, haven't you?

What does most of the Airbnb work consist in?

Is it the cleaning that takes the most time?

Faheem Mitha

@Cerberus No, not that long. Started in April. Got my first business in May. Spent ages getting it ready, though. And it's take me a while to recoup my investment.

@Cerberus Lots of online correspondence. Checking guests in and out.

Keeping them happy when they are here.

Unfortunately, one cannot charge much here. The Americans and Europeans do a bit better. Though they have to do their own cleaning - yuck.

Cerberus

11:53 PM

@FaheemMitha Oh, but I remember you've been talking about letting this room to people for ages?

Or was that not through Airbnb?

Ah OK.

Rand al'Thor

> You could, I don't know, call the field partial differentiation, and call the people partial differentiators?

@Cerberus That actually sounds pretty good!

Cerberus

So you never clean?

Faheem Mitha

@Cerberus Yes, I'd been talking about it for ages. Hadn't done it till recently, though.

Cerberus

@Randal'Thor Oh, good.

Faheem Mitha

@Cerberus No, only via Airbnb. I've not used anything else.

Cerberus

11:54 PM

@FaheemMitha Ah OK.

Faheem Mitha

Like I said, it took ages to set up.

India isn't a good place for getting things done. Unless you've got a strong support system. Which I don't.

@Cerberus Other people clean. I'd hate to do it. And I don't have tons of time and energy anyway.

Tonepoet

Every time I see people ask if something is an oxymoron on this website, the subsequent answers depress me. Does anybody else feel that way? >_>

Kit Z. Fox

@Randal'Thor we could just call them mathematicians.

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