When did these math branches reach their peak of development?

[dijkarte]

Since when each of the following math branches stopped developing, so there have been no new established theories and topics added?

Abstract algebra

Set theory

Probability and statistics

Differential equations

Calculus and analysis

Fourier analysis

 

[homeomorphic]

Those haven't stopped developing at all, except maybe set theory. Set theory isn't very active. I'm not sure exactly when that happened.

 

[dijkarte]

Could you please give me an example where a new theory/method has been added since the past 10 year for one area? And I don't mean advanced/research active topics...
Maybe my questions was not clear. What I want to know if anything changed with new theories that replaced old ones, generalization, ...etc.

 

[micromass]

Those haven't stopped developing at all, except maybe set theory. Set theory isn't very active. I'm not sure exactly when that happened.

Uuuhm, set theory is still very active... It might not be as popular, but there is still quite some research in it.

 

[micromass]

Abstract algebra

http://arxiv.org/list/math.AC/recent
http://arxiv.org/list/math.RA/recent
http://arxiv.org/list/math.RT/recent

Set theory

http://arxiv.org/list/math.LO/recent

Probability and statistics

http://arxiv.org/list/math.PR/recent
http://arxiv.org/list/math.AP/recent

Differential equations

http://arxiv.org/list/math.DS/recent
http://arxiv.org/list/math.ST/recent

Calculus and analysis

http://arxiv.org/list/math.CA/recent
http://arxiv.org/list/math.FA/recent

Fourier analysis

http://www.springerlink.com/content/1069-5869

 

[dijkarte]

Yup they are active research areas, but do they rule out any theories we studied 10 years ago? In other words, if I'm studying from a math book dated in 60s, am I studying something outdated and invalid?
Are these research topics taught and listed in undergraduate syllabus?

 

[Angry Citizen]

Yup they are active research areas, but do they rule out any theories we studied 10 years ago? In other words, if I'm studying from a math book dated in 60s, am I studying something outdated and invalid?

Is that even possible? As I understand it, mathematics is built up successively using deductive arguments that must necessarily be true. I freely admit that I don't know, but it wouldn't make sense to me if a "proof" were suddenly found to be incorrect.

 

[dijkarte]

I freely admit that I don't know, but it wouldn't make sense to me if a "proof" were suddenly found to be incorrect.

Exactly that's what I mean, whether recent researches ruled out any previous theories, and then courses and texts have to be updated accordingly.

I know everything is an active research area and can develop, and not only applicable to math.

But let me reword my question a bit.

If I'm studying ODEs from a text that dates back to 1960s, and another student is studying the same subject ODEs from a different new text say 2011. And both are studying at the same level, say undergraduate. How this person knowledge will be different than mine in this subject?

 

[micromass]

Exactly that's what I mean, whether recent researches ruled out any previous theories, and then courses and texts have to be updated accordingly.

I know everything is an active research area and can develop, and not only applicable to math.

But let me reword my question a bit.

If I'm studying ODEs from a text that dates back to 1960s, and another student is studying the same subject ODEs from a different new text say 2011. And both are studying at the same level, say undergraduate. How this person knowledge will be different than mine in this subject?

It all depends on the topic at hand, but I don't think your knowledge will be very different. The texts will be different however, but this won't matter much.

My experience is actually that older texts are actually better (not always of course). Old texts usually care only about rigor. New texts are often dumbed down.

Here are a random example of 5 texts that I have:

Spivak - Calculus: 1967
Kelley - Topology: 1955
Rudin - Principles of mathematical analysis - 1953
Artin - Algebra - 1991
Arnold - Ordinary differential equations - 1978

These texts are all considered top notch and they are all (with maybe the exception of Artin) quite old.

The quality of a text has very little to do with how old the text is. Old texts can be good, or they can be bad. So don't look at the age very much when choosing a text.

Sometimes, you do need newer texts however. For example, if you want material on recently discovered things. But at undergraduate level, or beginning graduate level, this won't be an issue.

 

[Number Nine]

In other words, if I'm studying from a math book dated in 60s, am I studying something outdated and invalid?

Mathematics is a deductive science. Everything that was true 60 years ago is still true today.

 

[homeomorphic]

Uuuhm, set theory is still very active... It might not be as popular, but there is still quite some research in it.

Okay, but it is a fairly small area, I would say. I don't know of any math professors who work on set theory, but then, I'm fairly out of touch with that stuff.