Math Interests

[DJBruce]

So what areas, and/or topics in mathematics are you all interested? Analysis, Algebra, Algebraic Geometry, Number Theory, Applied Maths, etc..

 

Personally I am not at a point where I am absolutely sure what I am interested in, but from my limited experience analysis, algebraic geometry, and number theory all seem to be towards the top of my interest list.

[ajb]

So what areas, and/or topics in mathematics are you all interested?

 

Super and graded differential geometry, higher geometric structures (analogues of Poisson bivectors for example), odd geometric structures on supermanifolds (Schouten and odd Jacobi structures for example), the applications of the aforementioned in physics. I have some interest in algebra, but this is mostly in using geometric ideas to re-express algebraic ideas and constructions.

 

Things I would like to learn more about include: noncommutative geometry, tropical geometry, Gromov--Witten theory and symplectic field theory, and other "stringy" inspired geometry. Also a little geometric representation theory maybe.

[khaled]

Analysis, Logic, and Discrete Mathematics

[DrRocket]

So what areas, and/or topics in mathematics are you all interested? Analysis, Algebra, Algebraic Geometry, Number Theory, Applied Maths, etc..

 

Personally I am not at a point where I am absolutely sure what I am interested in, but from my limited experience analysis, algebraic geometry, and number theory all seem to be towards the top of my interest list.

 

At this level it is all interesting. I am an analyst.

[ajb]

At this level it is all interesting. I am an analyst.

 

I once did a graduate course on analysis, never again. It was extremely difficult and a lot of hard work.

[Athena]

I find mathematics mystical and I love the sense of wonder it causes me to have. I am also disappointed that the discussions I attempt to have do not go well.

[mississippichem]

I once did a graduate course on analysis, never again. It was extremely difficult and a lot of hard work.

 

I picked up a book on analysis once. I read about three chapters, got mentally exhausted and gave the book to a friend. Talk about a gift that becomes a curse .

[DrRocket]

I picked up a book on analysis once. I read about three chapters, got mentally exhausted and gave the book to a friend. Talk about a gift that beco :)mes a curse .

 

Which book ?

[ajb]

I picked up a book on analysis once. I read about three chapters, got mentally exhausted and gave the book to a friend. Talk about a gift that becomes a curse .

 

My experience of analysis is that it is very hard and a lot of work. The deepest thing I use from analysis fairly regularly is the existence and uniqueness of solutions to first order differential equations!

 

I remember in the graduate course spending quite a lot of time on the wave equation in different dimensions. My impression, and I will say that it was naive, was that analysis was a lot of hard work to get at things that physicists take for granted. But maybe all of mathematics is a bit like that.

[mississippichem]

Which book ?

Introductory Real Analysis: Andreĭ Nikolaevich Kolmogorov, Sergeĭ Vasilʹevich Fomin

 

The book was decent from my humble vantage point. I think the student [myself] was the problem.

 

My experience of analysis is that it is very hard and a lot of work. The deepest thing I use from analysis fairly regularly is the existence and uniqueness of solutions to first order differential equations!

 

Part of the reason for me giving up on analysis was that I couldn't yet see any application for it. I'm alright with pure mathematics, but usually one ends up with an application at some point. I didn't see it happening. That's where the goals of scientists and mathematicians diverge [pardon the pun]. I use math as a means to an end where as mathematicians use science to create interesting math problems, or much of the time, they just use math to create interesting math.

 

I think we need mathematicians where they might not need us. Oh well, at least engineers and physicians need us. You ajb, you're somewhere in between the two right?

[ajb]

You ajb, you're somewhere in between the two right?

 

I think so, but probably closer to pure mathematics at the moment.

[DrRocket]

Introductory Real Analysis: Andreĭ Nikolaevich Kolmogorov, Sergeĭ Vasilʹevich Fomin

 

The book was decent from my humble vantage point. I think the student [myself] was the problem.

 

 

Not a bad book, but not a good introduction either. It is really a cross between what is taught in a real analysis course and a functional analysis course -- functional analysis usually follows real analysis. In my opinion it is too heavy for a real analysis course and way too light for a functional analysis course. I am not surprised that you gave up on it.

 

For your purposes a course that does analysis on [math] \mathbb R^n[/math] properly, rather than dealing with more abstract spaces, would be appropriate. Two of the best are Elements of Real Analysis by Bartle and Principles of Mathematical Analysis by Rudin. (Bartle is the easier of the two). The next level would be Rudin's Real and Complex Analysis or Folland's Real Analysis: Modern Techniques and Their Applications, and then Rudin's Functional Analysis. Before you tackle functional analysis a good course in point-set topology is needed and some abstract algebra is helpful.

 

You may observe a recurrence of the name Rudin. There is a reason for that. His three analysis books are classics and have been the benchmark in advanced undergraduate and graduate analysis texts for many years.

 

For chemistry I would think that things like Fourier analysis (for spectroscopy) and operator theory (for quantum mechanics) would be the major areas of interest. Those come under the heading of functional analysis, though you will find much good material in Rudin's second book.

 

 

Part of the reason for me giving up on analysis was that I couldn't yet see any application for it. I'm alright with pure mathematics, but usually one ends up with an application at some point. I didn't see it happening. That's where the goals of scientists and mathematicians diverge [pardon the pun]. I use math as a means to an end where as mathematicians use science to create interesting math problems, or much of the time, they just use math to create interesting math.

 

I think we need mathematicians where they might not need us. Oh well, at least engineers and physicians need us. You ajb, you're somewhere in between the two right?

 

Analysis is just the extension of calculus. There are LOTS of applications. Fourier series, Fourier transforms, ordinary differential equations, partial differential equations, Hilbert spaces, and operator theory are subjects in analysis.

 

However, most mathematics classes will not directly discuss specific applications. The specific applications are better discussed in classes dedicated to the application. The heat equation, for instance might receive some attention in a PDE class, but heat transfer would not be discussed in any detail.

'

I would avoid "applied mathematics" classes. In my experience so-called "applied mathematicians" are weak in two areas -- mathematics and applications. If you want to see good applications of mathematics look to the more theoretical chemists, engineers and physicists.

[Dave]

My interests are on completely the other end of the scale to you guys. For my PhD I've been focused on studying fluid dynamics (particularly flow through a pipe), and really the mathematics comes into play in terms of numerical analysis and getting decent schemes to solve Navier-Stokes. So at this point I guess I'm somewhere between a physicist, engineer and mathematician.

 

The thing is that I quite like that. The idea that I could focus purely on, say, analysis or topology and not be able to apply that to something which is a bit more physical isn't the kind of thing that I can easily get into, and is certainly not anything that I would enjoy.

 

So I have to take umbrage with your last comment DrRocket. In this (mathematics) department there is no shortage of 'applied mathematicians' who are in themselves excellent at mathematics and know the applications of their specific subfield inside out. It's worth noting that I don't consider myself to be one of them and I've always been more on the physics side of things, but the point still stands regardless.

 

I don't think this is where you were going with this, but to air a particular grievance of mine anyway: I really detest the snobbery that both pure and applied mathematicians seem to have for one another. Personally, I have the utmost respect for anyone that chooses to study mathematics (and the related subjects) at a higher level. I'm very grateful to work in a department which actively encourages collaboration, and this shows in the quality of the research that gets put out. At the end of the day, pure mathematics is pretty much useless to the wider population without applications. Likewise, applied mathematics relies on theoretical results in order to progress. Since it's pretty much a fact that the level of funding is largely reliant on the usefulness of the applications that come out of the work, in my mind at least there is nothing to lose by collaborating and an awful lot to gain.

[DJBruce]

So for those of you who have identified a specialty how did you come to find that specialty? Did you find an interesting problem that then lead you to that field? Did you notice that your skills were more applicable to a certain area? Or did you just take a course that really interested you?

[DrRocket]

My interests are on completely the other end of the scale to you guys. For my PhD I've been focused on studying fluid dynamics (particularly flow through a pipe), and really the mathematics comes into play in terms of numerical analysis and getting decent schemes to solve Navier-Stokes. So at this point I guess I'm somewhere between a physicist, engineer and mathematician.

 

The thing is that I quite like that. The idea that I could focus purely on, say, analysis or topology and not be able to apply that to something which is a bit more physical isn't the kind of thing that I can easily get into, and is certainly not anything that I would enjoy.

 

So I have to take umbrage with your last comment DrRocket. In this (mathematics) department there is no shortage of 'applied mathematicians' who are in themselves excellent at mathematics and know the applications of their specific subfield inside out. It's worth noting that I don't consider myself to be one of them and I've always been more on the physics side of things, but the point still stands regardless.

 

I don't think this is where you were going with this, but to air a particular grievance of mine anyway: I really detest the snobbery that both pure and applied mathematicians seem to have for one another. Personally, I have the utmost respect for anyone that chooses to study mathematics (and the related subjects) at a higher level. I'm very grateful to work in a department which actively encourages collaboration, and this shows in the quality of the research that gets put out. At the end of the day, pure mathematics is pretty much useless to the wider population without applications. Likewise, applied mathematics relies on theoretical results in order to progress. Since it's pretty much a fact that the level of funding is largely reliant on the usefulness of the applications that come out of the work, in my mind at least there is nothing to lose by collaborating and an awful lot to gain.

 

I have had a few decades of experience with both mathematics and real applications, including fluid dynamics. I stand by my statement. The good applied mathematicians that I knew left and went into the actual areas in which the applications themselves were found. The remainder, in their own words, neither prove theorems nor solve problems that anyone cares about.

 

I am sure that there exceptions, and exceptional schools. Courant and Friedrichs did good work on gas dynamics for instance. But in general the best applications of mathematics come from the physicists and engineers who work in the applied areas, and mathematics comes from mathematicians. This in no way contradicts your statement that collaboration is beneficial. I have seen productive collaborations between real mathematicians and physicists, chemists and engineers. I even know of pure mathematicians with substantial education in engineering.

[mississippichem]

Not a bad book, but not a good introduction either. It is really a cross between what is taught in a real analysis course and a functional analysis course -- functional analysis usually follows real analysis. In my opinion it is too heavy for a real analysis course and way too light for a functional analysis course. I am not surprised that you gave up on it.

 

For your purposes a course that does analysis on [math] \mathbb R^n[/math] properly, rather than dealing with more abstract spaces, would be appropriate. Two of the best are Elements of Real Analysis by Bartle and Principles of Mathematical Analysis by Rudin. (Bartle is the easier of the two). The next level would be Rudin's Real and Complex Analysis or Folland's Real Analysis: Modern Techniques and Their Applications, and then Rudin's Functional Analysis. Before you tackle functional analysis a good course in point-set topology is needed and some abstract algebra is helpful.

 

You may observe a recurrence of the name Rudin. There is a reason for that. His three analysis books are classics and have been the benchmark in advanced undergraduate and graduate analysis texts for many years.

 

Thanks for the suggestions. I'll look into some of these. I've only given up on formal analysis temporarily. I have a huge interest in spectroscopy as well as electronic structure theory so anything more I can learn about Fourier analysis in addition to the basics I have now will be appreciated in grad school.

[DrRocket]

Thanks for the suggestions. I'll look into some of these. I've only given up on formal analysis temporarily. I have a huge interest in spectroscopy as well as electronic structure theory so anything more I can learn about Fourier analysis in addition to the basics I have now will be appreciated in grad school.

 

Fourier analysis is a huge area. Unfortunately a lot of what is written on the subject in physics and engineering books is just wrong. The better books on the subject require a strong background in real and complex analysis and quite a bit of functional analysis..

 

Rudin's book on Real and Complex analysis has a lot of good information. That is not surprising since he also wrote the classic Fourier Analysis on Groups, which is required reading for every specialist in harmonic analysis. Another excellent book is An Introduction to Harmonic Analysis by Yitzhak Katznelson (a PDF copy was once avvailable at Katznelson's web site). I was also once available as an inexpensive Dover book -- you might find one on the used book market.

[DrRocket]

Thanks for the suggestions. I'll look into some of these. I've only given up on formal analysis temporarily. I have a huge interest in spectroscopy as well as electronic structure theory so anything more I can learn about Fourier analysis in addition to the basics I have now will be appreciated in grad school.

 

Fourier analysis is a huge area. Unfortunately a lot of what is written on the subject in physics and engineering books is just wrong. The better books on the subject require a strong background in real and complex analysis and quite a bit of functional analysis..

 

Rudin's book on Real and Complex analysis has a lot of good information. That is not surprising since he also wrote the classic Fourier Analysis on Groups, which is required reading for every specialist in harmonic analysis. Another excellent book is An Introduction to Harmonic Analysis by Yitzhak Katznelson (a PDF copy was once avvailable at Katznelson's web site). I was also once available as an inexpensive Dover book -- you might find one on the used book market.

Edited June 3, 2011 by DrRocket

[khaled]

I'm a computer scientist, so I'd say Logic, Discrete Mathematics, Analysis, Model Theory, Number Theory, Algorithms, with some interest in Geometry, and Quantum Theory ...

 

To me, while I was still a bachelor student in computer science, which was five years, and since the second year, I have started to write researches in Logic, Algorithms, and Model Theory .. I was in the third year, I had no idea about scientific scholarships and researchers around the world meeting in conferences .. I wrote tens of papers, most were no good, but considered a start of realizing that I need to be good in the field to start writing researches .. and that I need to meet specialized people to help me in, In my senior year, I published a paper, with not much notations, about one of my attempts to do Randomness Generation ...