## Favourite Field of Math?

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- MartianInvader
**Posts:**809**Joined:**Sat Oct 27, 2007 5:51 pm UTC

### Re: Favourite Field of Math?

To btilly and Kalak_z:

I've never seen anything more powerful than Euler characteristic used in graph theory, and Euler characteristic is, what? 200 years old? It's a really elementary piece of algebraic topology.

If you take an algebraic topology class out of Allen Hatcher's book, there is an exercise to prove that the complete graph on 5 vertices doesn't embed in the plane using Euler characteristic, so I guess that loose connection is cool enough to persist here and there.

On the other hand, if you mix in a little algebra, the field of geometric group theory basically studies graphs (the Cayley graphs of finitely generated groups) and uses a heck of a lot of topology. Geometric group theory is my favorite field of math, but I checked "topology" since I study it mostly from a topological viewpoint.

Also, for the record, I think calculus belongs in "applied".

I've never seen anything more powerful than Euler characteristic used in graph theory, and Euler characteristic is, what? 200 years old? It's a really elementary piece of algebraic topology.

If you take an algebraic topology class out of Allen Hatcher's book, there is an exercise to prove that the complete graph on 5 vertices doesn't embed in the plane using Euler characteristic, so I guess that loose connection is cool enough to persist here and there.

On the other hand, if you mix in a little algebra, the field of geometric group theory basically studies graphs (the Cayley graphs of finitely generated groups) and uses a heck of a lot of topology. Geometric group theory is my favorite field of math, but I checked "topology" since I study it mostly from a topological viewpoint.

Also, for the record, I think calculus belongs in "applied".

Let's have a fervent argument, mostly over semantics, where we all claim the burden of proof is on the other side!

### Re: Favourite Field of Math?

wait, I thought calculus and analysis were the same thing. can someone please explain the difference?

It rains on the enemy too!

- Cosmologicon
**Posts:**1806**Joined:**Sat Nov 25, 2006 9:47 am UTC**Location:**Cambridge MA USA-
**Contact:**

### Re: Favourite Field of Math?

It's quite simple: calculus is what's covered in a course called "Calculus", whereas analysis is what's covered in a course called "Analysis". Maybe these are the same thing, but it sure doesn't feel like it when you're doing it!

### Re: Favourite Field of Math?

This has been my experience so far: in Calculus you learn how to calculate things (like derivatives, integrals, areas, volumes, limits, etc.); whereas in Analysis, you learn how to prove things. I guess you could say the difference is in rigor. Or else you could say one is learning how to use the tool, the other is learning how to build the tool.

### Re: Favourite Field of Math?

pkuky wrote:wait, I thought calculus and analysis were the same thing. can someone please explain the difference?

Calculus is basically "how to do derivatives and integrals". And lots of applications of the same.

Analysis is about rigorously proving things about certain kinds of mathematical systems. Which ones is sort of a matter of feel, but if it involves epsilons, deltas and infinite series, it is probably analysis. (Could be topology - the boundary gets fuzzy in places.) The first of these systems happens to involve lots of things with relatively nice types of functions over the real numbers. Which is similar territory to calculus. But not the same. And analysis quickly winds up wandering over all sorts of interesting territory. (Like Hilbert spaces.)

Perhaps an example will show the difference. A typical calculus problem is to find the integrals or derivatives of a list of functions. A typical analysis problem might be to demonstrate that if x

_{0}< x

_{1}, f(x

_{0}) < g(x

_{0}), f(x

_{1}) > g(x

_{1}) and f and g are both continous functions, then there is an x with x

_{0}< x < x

_{1}with f(x) = g(x). Those two types of exercise have a very different "feel". No mathematician would mistake a problem you see in an analysis course for one you'd see in a calculus course. Or vice versa.

The more I think about it, the more I like my comparison with arithmetic and algebra. You're going to have trouble doing algebra if you don't know how to multiply and divide numbers. But how to do long division isn't really part of the subject of algebra. Well calculus and analysis are similar. If you can't do calculus, then you're missing a necessary foundation for analysis. But analysis is not about how to do calculus problems.

Some of us exist to find out what can and can't be done.

Others exist to hold the beer.

### Re: Favourite Field of Math?

Something I've been looking for for a while now... Does anyone know of a website (or book I suppose) that outlines some of the major problems that each field of mathematics is seeking to solve? This is partly out of my own personal interest, but also to present to people who see high school math as the end of what math has to offer, and don't understand how or why anyone would pursue a degree in math.

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### Re: Favourite Field of Math?

Qoppa wrote:Something I've been looking for for a while now... Does anyone know of a website (or book I suppose) that outlines some of the major problems that each field of mathematics is seeking to solve? This is partly out of my own personal interest, but also to present to people who see high school math as the end of what math has to offer, and don't understand how or why anyone would pursue a degree in math.

http://www.math.niu.edu/~rusin/known-math/ is a website that tries that. The best book that I can suggest is The Mathematical Experience.

Some of us exist to find out what can and can't be done.

Others exist to hold the beer.

- lastchance
**Posts:**7**Joined:**Sun Dec 02, 2007 11:59 pm UTC

### Re: Favourite Field of Math?

As of right now, algebra, hands down. I've never seen so many simple, yet powerful theorems in one place. Sure, analysis has its austere beauty, but I'd rather be studying groups any day.

### Re: Favourite Field of Math?

lastchance wrote:As of right now, algebra, hands down. I've never seen so many simple, yet powerful theorems in one place. Sure, analysis has its austere beauty, but I'd rather be studying groups any day.

That's the fundamental divide. Ask virtually any mathematician whether they prefer algebra or analysis and you'll usually get a firm answer. This gets correlated with all sorts of odd stuff.

The oddest that I've ever seen was when a group of 20 or so were eating corn on the cob. Someone noticed that every last analyst in the room ate their corn in spirals. (Constantly turn the corn.) Every last algebraist ate their corn in rows. (Start at one end, eat to the other. Turn slightly, repeat.) Made no sense to anybody in the room, but we had a statistically significant correlation. It still makes no sense to me but it has held up pretty well since. (For the record I like analysis, and I eat in spirals. My father-in-law is into algebra, and he eats in rows.)

In fact the most bizarre answer I've had was a logician who very unusually said his field was right on the boundary between algebra and analysis. I asked him how he eats corn and he was very sheepish. He said he ate it in loose spirals. He'd never seen anyone else eat it that way (nor have I), but it is halfway between what algebraists and analysts typically do!

Some of us exist to find out what can and can't be done.

Others exist to hold the beer.

### Re: Favourite Field of Math?

This is absolutely the most bizarre thing I have read in two weeks at least. I am highly tempted to survey the math people around here to see if it holds. Are there any other things I have to put on the survey?btilly wrote:That's the fundamental divide. Ask virtually any mathematician whether they prefer algebra or analysis and you'll usually get a firm answer. This gets correlated with all sorts of odd stuff.

The oddest that I've ever seen was when a group of 20 or so were eating corn on the cob. Someone noticed that every last analyst in the room ate their corn in spirals. (Constantly turn the corn.) Every last algebraist ate their corn in rows. (Start at one end, eat to the other. Turn slightly, repeat.) Made no sense to anybody in the room, but we had a statistically significant correlation. It still makes no sense to me but it has held up pretty well since. (For the record I like analysis, and I eat in spirals. My father-in-law is into algebra, and he eats in rows.)

In fact the most bizarre answer I've had was a logician who very unusually said his field was right on the boundary between algebra and analysis. I asked him how he eats corn and he was very sheepish. He said he ate it in loose spirals. He'd never seen anyone else eat it that way (nor have I), but it is halfway between what algebraists and analysts typically do!

EDIT: You said the opposite elsewhere, as can be found by Googling. I'm confused now -- which is correct? (Surely you know which way YOU eat corn.)

### Re: Favourite Field of Math?

++$_ wrote:This is absolutely the most bizarre thing I have read in two weeks at least. I am highly tempted to survey the math people around here to see if it holds. Are there any other things I have to put on the survey?btilly wrote:That's the fundamental divide. Ask virtually any mathematician whether they prefer algebra or analysis and you'll usually get a firm answer. This gets correlated with all sorts of odd stuff.

The oddest that I've ever seen was when a group of 20 or so were eating corn on the cob. Someone noticed that every last analyst in the room ate their corn in spirals. (Constantly turn the corn.) Every last algebraist ate their corn in rows. (Start at one end, eat to the other. Turn slightly, repeat.) Made no sense to anybody in the room, but we had a statistically significant correlation. It still makes no sense to me but it has held up pretty well since. (For the record I like analysis, and I eat in spirals. My father-in-law is into algebra, and he eats in rows.)

In fact the most bizarre answer I've had was a logician who very unusually said his field was right on the boundary between algebra and analysis. I asked him how he eats corn and he was very sheepish. He said he ate it in loose spirals. He'd never seen anyone else eat it that way (nor have I), but it is halfway between what algebraists and analysts typically do!

EDIT: You said the opposite elsewhere, as can be found by Googling. I'm confused now -- which is correct? (Surely you know which way YOU eat corn.)

Googling I find http://blade.nagaokaut.ac.jp/cgi-bin/sc ... -talk/9662 in which I said it backwards.

And yes, I know which way I eat corn - a spiral.

Some of us exist to find out what can and can't be done.

Others exist to hold the beer.

### Re: Favourite Field of Math?

That is fascinating. I haven't eaten corn in years - probably before I even knew what algebra and analysis were - but I definitely tend toward algebra, and I've got a feeling used to eat my corn in spirals. It's so long ago that I can't be absolutely sure that I actually did, so this could just be some weird psychological thing that I don't really understand.

This is a placeholder until I think of something more creative to put here.

- Torn Apart By Dingos
**Posts:**817**Joined:**Thu Aug 03, 2006 2:27 am UTC

### Re: Favourite Field of Math?

That means Donald Duck is an algebraist! By the way, I prefer analysis and I eat corn in spirals.

### Re: Favourite Field of Math?

btilly wrote:In fact the most bizarre answer I've had was a logician who very unusually said his field was right on the boundary between algebra and analysis. I asked him how he eats corn and he was very sheepish. He said he ate it in loose spirals. He'd never seen anyone else eat it that way (nor have I), but it is halfway between what algebraists and analysts typically do!

I think that's how I eat my corn too. In my first pass through a cob, I eat a helix that's congruent to its complement. Then I eat the complement. Is that what your father means by a "loose spiral"?

Also, I'm a combinatorialist who prefers algebra. It's possible, though, that I only dislike analysis because I haven't really done any of the cool stuff yet. We'll see.

### Re: Favourite Field of Math?

to btilly,

I was unclear earlier. I was not trying to say that all of graph theory can be considered topology, just that there are parts of graph theory that can be considered topology. And, since Nimz was unclear about what specifically he was working on, I was just trying to convince you that his comment may not have been as absurd as you had suggested. In fact there is enough interest in the intersection of graph theory and topology that there are books about it such as Gross and Tucker "Topological Graph Theory." Also, I have a friend who is working on planar embeddings of infinite graphs. This has far more to do with topology and geometry than discrete math, but still falls under the category of graph theory.

As for not seeing Euler's formula in a graph theory class, both of the Graph theory classes that I have taken have covered it. One was from Tucker's "Applied Combinatorics" and the other was from Godsil and Royle's "Algebraic Graph Theory," so I would say that it is a part of graph theory, and it does require topology to define a face (a face is a connected component of the compliment of the image of a graph in a two manifold).

I was unclear earlier. I was not trying to say that all of graph theory can be considered topology, just that there are parts of graph theory that can be considered topology. And, since Nimz was unclear about what specifically he was working on, I was just trying to convince you that his comment may not have been as absurd as you had suggested. In fact there is enough interest in the intersection of graph theory and topology that there are books about it such as Gross and Tucker "Topological Graph Theory." Also, I have a friend who is working on planar embeddings of infinite graphs. This has far more to do with topology and geometry than discrete math, but still falls under the category of graph theory.

As for not seeing Euler's formula in a graph theory class, both of the Graph theory classes that I have taken have covered it. One was from Tucker's "Applied Combinatorics" and the other was from Godsil and Royle's "Algebraic Graph Theory," so I would say that it is a part of graph theory, and it does require topology to define a face (a face is a connected component of the compliment of the image of a graph in a two manifold).

### Re: Favourite Field of Math?

Kalak_z wrote:to btilly,

I was unclear earlier. I was not trying to say that all of graph theory can be considered topology, just that there are parts of graph theory that can be considered topology. And, since Nimz was unclear about what specifically he was working on, I was just trying to convince you that his comment may not have been as absurd as you had suggested. In fact there is enough interest in the intersection of graph theory and topology that there are books about it such as Gross and Tucker "Topological Graph Theory." Also, I have a friend who is working on planar embeddings of infinite graphs. This has far more to do with topology and geometry than discrete math, but still falls under the category of graph theory.

As for not seeing Euler's formula in a graph theory class, both of the Graph theory classes that I have taken have covered it. One was from Tucker's "Applied Combinatorics" and the other was from Godsil and Royle's "Algebraic Graph Theory," so I would say that it is a part of graph theory, and it does require topology to define a face (a face is a connected component of the compliment of the image of a graph in a two manifold).

Points taken.

As for graph theory, my last graph theory course was so long ago that I don't remember what was in it. I don't remember seeing Euler's formula there, but it might well have been and I could have just forgotten it.

Some of us exist to find out what can and can't be done.

Others exist to hold the beer.

### Re: Favourite Field of Math?

Algebra. I have a real hard time being creative in analysis. Probably because I never did any of the suggested assignments in the courses.

Although I'm surprised no one (I think, I just read through real quick) has mentioned matrix theory at all. It has such rich theoretic structure, as well as lots of application potential, and links to many different parts of mathematics, ie algebra, analysis, combinatorics. I imagine you could define an interesting metric space topology using matrix norms. So far all the research I've done has been in matrices, so maybe I'm a bit biased here.

Although I'm surprised no one (I think, I just read through real quick) has mentioned matrix theory at all. It has such rich theoretic structure, as well as lots of application potential, and links to many different parts of mathematics, ie algebra, analysis, combinatorics. I imagine you could define an interesting metric space topology using matrix norms. So far all the research I've done has been in matrices, so maybe I'm a bit biased here.

### Re: Favourite Field of Math?

siclar wrote:\I imagine you could define an interesting metric space topology using matrix norms.

At least as far as finite dimensional (nxn) matrices are concerned, I'm pretty sure all the usual norms you can put on matrices give rise to the same old Euclidean topology on R

^{n^2}or C

^{n^2}. If you are talking about infinite dimensional matrices, then there are different topologies.

Cheers,

Mike

addams wrote:This forum has some very well educated people typing away in loops with Sourmilk. He is a lucky Sourmilk.

### Re: Favourite Field of Math?

MartianInvader wrote:To btilly and Kalak_z:

I've never seen anything more powerful than Euler characteristic used in graph theory, and Euler characteristic is, what? 200 years old? It's a really elementary piece of algebraic topology.

If you take an algebraic topology class out of Allen Hatcher's book, there is an exercise to prove that the complete graph on 5 vertices doesn't embed in the plane using Euler characteristic, so I guess that loose connection is cool enough to persist here and there.

On the other hand, if you mix in a little algebra, the field of geometric group theory basically studies graphs (the Cayley graphs of finitely generated groups) and uses a heck of a lot of topology. Geometric group theory is my favorite field of math, but I checked "topology" since I study it mostly from a topological viewpoint.

Also, for the record, I think calculus belongs in "applied".

where are you studying GGT?

http://www.cdbaby.com/cd/mudge <-- buy my CD (Now back in stock!)

### Re: Favourite Field of Math?

number theory

simply because it's so hard to realize that 8*7 = 56

(Oh me yarm it took more than 3 sec!)

simply because it's so hard to realize that 8*7 = 56

(Oh me yarm it took more than 3 sec!)

while moronic idle do something (stupid).

### Re: Favourite Field of Math?

btilly wrote:Kalak_z wrote:to btilly,

I was unclear earlier. I was not trying to say that all of graph theory can be considered topology, just that there are parts of graph theory that can be considered topology. And, since Nimz was unclear about what specifically he was working on, I was just trying to convince you that his comment may not have been as absurd as you had suggested. In fact there is enough interest in the intersection of graph theory and topology that there are books about it such as Gross and Tucker "Topological Graph Theory." Also, I have a friend who is working on planar embeddings of infinite graphs. This has far more to do with topology and geometry than discrete math, but still falls under the category of graph theory.

As for not seeing Euler's formula in a graph theory class, both of the Graph theory classes that I have taken have covered it. One was from Tucker's "Applied Combinatorics" and the other was from Godsil and Royle's "Algebraic Graph Theory," so I would say that it is a part of graph theory, and it does require topology to define a face (a face is a connected component of the compliment of the image of a graph in a two manifold).

Points taken.

As for graph theory, my last graph theory course was so long ago that I don't remember what was in it. I don't remember seeing Euler's formula there, but it might well have been and I could have just forgotten it.

What I said on the first page was merely that graph theory has bits of both fields in it. I didn't think such a small comment would end up having a life of its own! My current research deals with embedding graphs into host graphs and then looking at how many edges from the given graph must pass through the edges of the host graph. I'm trying to prove that one of the embeddings with the minimum of the maximum number of edges through a host edge is a particular embedding when the given graph is an n-dimensional cube (Q

_{n}) and the host graph is a cycle containing 2

^{n}vertices (C

_{2n}). With that particular embedding, the maximum number of edges through a host edge is floor(5/3 * 2

^{n-2}).

Being as this is the xkcd maths forum, I'm sure at least one person got more than just [insane gobbledy-gook] out of that.

LOWA

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