What research basically is [in mathematics]

[faiziqb12]

I want to pursue a matametical career but all of us know that a good research career or even professorship at good universities needs a good amount of research papers.
so what basically is research , is it contributing new things or giving depper understanding of a known fact for example a mathametical equation?
please reply ASAP
Thanks

 

[rootone]

'Research' is generally a term used when talking about observational aspects of science.
Telescopes and test tubes and so on.
Mathematics doesn't really require research as the whole idea of it is built upon axioms like addition, 1+1=2.
Would be really hard to do an experiment which either validated or falsified a concept like addition without assuming addition is a valid idea to begin with.

 

[micromass]

Mathematics doesn't really require research

Wow. Talk about the most wrong thing I've seen all week. Of course there is research in mathematics. You seem to be under the impression that research is only experimental stuff. It's not.

 

[radium]

People seem to think theoretical physics and math aren't research since there may be no experimental data (this is not true for many fields of theoretical physics). However, anything that involves thinking about some unanswered question is research. From what I have observed in math (I have a good friend doing his PhD in math), the focus is on proving things as in making something completely rigorous.

 

[S.G. Janssens]

Post #2 is intolerable not only for the ignorance it displays about the field of mathematics, but mostly because it offends every mathematician on this forum and, indeed, an entire well-established and age old academic discipline.

Earlier today, I made an attempt to more briefly express my judgement, but that attempt failed.

 

[faiziqb12]

fine , guys please forget the wrong of post #2 and start focusing on my original questionn

 

[rootone]

I confess to my sin although I think it might have been misconstrued as well.
I didn't mean to imply that math isn't science, yes that would be truly ridiculous.
I meant it more in terms of 'research' being generally understood as investigating stuff with instruments.
My apologies to anyone offended by my misunderstanding of the word.

 

[micromass]

I
I meant it more in terms of 'research' being generally understood as investigating stuff with instruments.

Why would anybody understand research as that? I mean sure, a high schooler with no experience in academia might understand research in that terms. But it's clear the OP is not talking about that kind of research.

 

[Leonarte]

Hi,
I'm from computer science, which in general ways is a branch of math, and as well we have some misconception on What we research (specially on the more theoretical areas).

In math, so far as I know, you can make research on both "areas", the more practical (applied math) which can be used on computing, engineering, physics and stuff. Other one is the pure math research, which sometimes May seem useless today, but in the future can be used (like number theory, useless some years ago, but now is the base for Cryptography).

Anyway, you Will be proving equations, and testing some theories. In my university (in Brazil, and we are not famous for math publications) the math researchers public less arcticles than any other departament (I think because of the hard side on proving things in math).

Again, I'm not from math, my knowledge is from some friends I have in this area and on my own field (which is theoretical computer science ).

Best redards,
Leon.

 

[S.G. Janssens]

in Brazil, and we are not famous for math publications

The contribution of Brazilian mathematicians to the modern theory of differential equations and dynamical systems is quite substantial, see e.g. Jacob Palis and his students: https://en.wikipedia.org/wiki/Jacob_Palis

 

[Leonarte]

The contribution of Brazilian mathematicians to the modern theory of differential equations and dynamical systems is quite substantial, see e.g. Jacob Palis and his students: https://en.wikipedia.org/wiki/Jacob_Palis

Oh, I meant my university specifically. But I really didn't know about those contributions. Thanks for sharing :)

 

[faiziqb12]

Why would anybody understand research as that? I mean sure, a high schooler with no experience in academia might understand research in that terms. But it's clear the OP is not talking about that kind of research.

everytime somebody asks a good question , nobody gives a good answer , but loves critisizing other peoples answers

 

[SteamKing]

everytime somebody asks a good question , nobody gives a good answer , but loves critisizing other peoples answers

Research in mathematics can involve several things.

Someone earlier mentioned developing proofs of certain theorems, which does involve research.

At times, prominent mathematicians have made various conjectures about things mathematical. While initially informed speculation, a conjecture needs to be shown to be true or false to determine if it belongs in the body of mathematical work.

There are various unsolved mathematical problems, the solution to which may or may not have practical applications. Attempting to find these solutions often involves a great deal of research in the mathematical literature, to see what others have tried.

Some researchers may decide to take their mathematics research into previously unexplored areas. Chaos theory is one such recent example of a new branch of mathematics which developed out of older topics in math, like the solution of certain differential equations.

This is just a quick discussion off the top of my head. There may be much, much more to uncover, which you should try to do on your own.

 

[faiziqb12]

Research in mathematics can involve several things.
Someone earlier mentioned developing proofs of certain theorems, which does involve research.

SO , basically research means just investigating mathametics , and does not necesserily require a person to do some new discovery .
right , sir ?

and what do you need to do to get a promising research position anywhere , (not asking about being a professor)?

and does pure mathametics give you anything more than just adjusting you at a professors job?

 

[micromass]

SO , basically research means just investigating mathametics , and does not necesserily require a person to do some new discovery .

Research is basically synonymous with making new discoveries. But you can make a lot of new discoveries. You can of course invent (and prove) new result. Or you can give new proofs and insights to already known results. That is valid research too. But it does have to be new and original.

and what do you need to do to get a promising research position anywhere , (not asking about being a professor)?

Sorry, but unlike physics and engineering, it is almost impossible to do research in pure mathematics outside of academia.

and does pure mathametics give you anything more than just adjusting you at a professors job?

Probably not.

 

[MidgetDwarf]

One of my professors who is a mathematician, is working with an Alzheimers research group. Before that, years ago, she went to Canada regarding an issue with their maple trees.

My other professor, found a solution to a differential equation a while back.

 

[faiziqb12]

One of my professors who is a mathematician, is working with an Alzheimers research group. Before that, years ago, she went to Canada regarding an issue with their maple trees.

My other professor, found a solution to a differential equation a while back.

most probably your professor is a applied maths professor.but i think pure is better than applied , even if it doesn't sound like having a value nowadays

 

[micromass]

but i think pure is better than applied , even if it doesn't sound like having a value nowadays

Sorry, but that's a really dumb opinion. It's the kind of things I see from undergrads. Pure mathematicians are seen like geniuses, while people look down on applied mathematicians. This needs to stop. Sure, you can say that you enjoy pure better than applied. Me too. But saying that one is better than the other? That's a poisonous opinion.

Besides, pure math is nothing without applied math. A lot of pure math only makes sense after you know the applications. This kind of divorce between pure and applied is very much unhealthy. http://pauli.uni-muenster.de/~munsteg/arnold.html Knowing everything about Hilbert space is cool. Knowing also how it is applied in QM, there is where the miracles happen!

 

[S.G. Janssens]

Besides, pure math is nothing without applied math. A lot of pure math only makes sense after you know the applications. This kind of divorce between pure and applied is very much unhealthy. http://pauli.uni-muenster.de/~munsteg/arnold.html Knowing everything about Hilbert space is cool. Knowing also how it is applied in QM, there is where the miracles happen!

Yes, yes yes! You stole my post!

In reality, there is a continuous transition between pure and applied mathematics and I prefer a spot somewhere midway. To me, there is nothing more beautiful than solving concrete problems (from engineering mechanics, for example) using abstract methods, thereby discovering the mathematical structure that such problems may have in common.

Sometimes, people think that there is a trade-off between applicability and profoundness of a mathematical result. This is very much not true.

 

[faiziqb12]

A lot of pure math only makes sense after you know the applications. This kind of divorce between pure and applied is very much unhealthy.

thats what the thing comes in , i don't really care about applications of pure maths and i solely love maths without being concerned about its applications

 

[MidgetDwarf]

most probably your professor is a applied maths professor.but i think pure is better than applied , even if it doesn't sound like having a value nowadays

They were both studied pure math, and later were interested on how they can apply mathematics to physics/engineering or biology/medicine, respectively.

 

[Dishsoap]

I did research in mathematics for one year only, but I was surprised by the level to which new knowledge is constantly being contributed. I had taken a course in discrete math and loved graph theory, so I started doing research with a professor in the field. There was absolutely new discoveries being made - how to find an Euler-free tour in very particular types of graphs, and this was exciting to me because at first we were not concerned with "proving" things with strict formalism, but by using logic to think through graph structures. It didn't require a PhD; it was research that a kindergartner could do, if only they were highly intelligent. In doing this, I realized that even on a very fundamental level, great progress is being made in mathematics research, via new discoveries.

I also realized that mathematics research is not for me . Even though the concepts presented were very abstract, I was able to apply what I had learned even in physics research, doing bracketing theorem and such.

 

[micromass]

it was research that a kindergartner could do, if only they were highly intelligent.

Oh, but I must correct you there. It is true that such things don't require much prerequisites and are very easy to get into, especially compared with something like algebraic geometry which takes years of studying. True, any kindergartners could understand what you do. But that doesn't mean the research is trivial. It is actually very very difficult. Not every mathematician would be able to do this kind of graph theory and discrete mathematics. It really takes a very special kind of brain.

 

[Dishsoap]

Oh, but I must correct you there. It is true that such things don't require much prerequisites and are very easy to get into, especially compared with something like algebraic geometry which takes years of studying. True, any kindergartners could understand what you do. But that doesn't mean the research is trivial. It is actually very very difficult. Not every mathematician would be able to do this kind of graph theory and discrete mathematics. It really takes a very special kind of brain.

I never said the research was trivial, and I absolutely, positively did not mean to imply this. On the contrary, every meeting with the group I sat there open-mouthed and wide-eyed because the rest of the group was so far beyond me that I never had any hopes of catching up. Those were some of the most intelligent people I have ever met in my life, and by joining the group I was holding them back. I said that a kindergartner could do it to imply that the research was very fundamental and not, as you said, requiring much prerequisite knowledge.

 

[S.G. Janssens]

thats what the thing comes in , i don't really care about applications of pure maths and i solely love maths without being concerned about its applications

As I wrote in post #19, there is a continuous transition between pure and applied mathematics. To give just one (rather simplified) example,

• point-set topology is essential for functional analysis (e.g. via Baire's category theorem or weak topologies)
• functional analysis is the basis for the modern theory of PDE (e.g. via the formalism of Sobolev spaces)
• the theory of PDE is central to proving the validity of numerical approximation procedures for solving specific equations
• numerical procedures in turn are ubiquitous in day-to-day mechanical engineering (e.g. for simulating fluid flows)

Hence, what is an application to one mathematician might be "pure" to the other. In one of my textbooks on point-set topology, Baire's theorem was mentioned as "an application", but I'm not sure whether someone working in the theory of PDE would call it like that.

Of course, to do point-set topology one does not need to know all about the numerical intricacies of Navier-Stokes or vice versa, but no matter where you are positioned in such a "chain", being aware of the applicability of your particular field to adjacent fields often greatly helps you to make progress, even when you don't care about the actual applications.

 

[faiziqb12]

Hence, what is an application to one mathematician might be "pure" to the other.

so , how might one do simultaneouslty in pure and applied , or change between pure and applied. is that possible?
also can one choose to do bachelors degree in pure maths and then a postgraduate degree in applied maths ?

 

[S.G. Janssens]

so , how might one do simultaneouslty in pure and applied , or change between pure and applied. is that possible?

Surely one can do research in both pure and applied mathematics:

• By positioning oneself somewhere on the continuum between the "pure" and "applied" sides,.e.g. working on the theory of PDE, developing differential geometric methods for the control of nonlinear mechanical systems, etc. There are many examples.
• By simply pursuing multiple interests. It is not at all uncommon for mathematicians to enjoy both pure as well as more applied fields at the same time. Also, sometimes pure fields have very direct applications. Think of number theory and cryptography or graph theory and optimization.

Of course, the further apart fields are and the more advanced you are in the course of your studies, the more difficult it is to work in both of them or change from one to the other. For example, I would have a hard time changing my specialization to something like algebraic topology at this point. (Fortunately, I lack the appetite to do so.) It is never entirely impossible, though.

also can one choose to do bachelors degree in pure maths and then a postgraduate degree in applied maths ?

Certainly, see post #21 for example. In the preface of his book Analysis for Applied Mathematics, the well-known American approximation theorist Ward Cheney writes: A look at the past would certainly justify my favorite algorithm for creating an applied mathematician: Start with a pure mathematician, and turn him or her loose on real-world problems.