Should an Analyst feel inferior?

[dreamtheater]

Hi everyone,

I have a general question on choosing a subfield within pure mathematics.

I personally find Analysis (more specifically, functional analysis) interesting, engrossing, and fun. I also find that I am better at it than, say, algebra.

On the other hand, I can't help but feel like (algebraic) topology, algebraic geometry, and even differential geometry is much more abstract and "advanced". That makes me feel a bit inferior. I am now getting into quite "advanced" material in Banach space theory at the graduate level, but all of it still seems quite simpler than, say Ext and Tor functors in algebra and topology. But is that really objectively the case? Or do I just "feel" like Ext and Tor functors are more complicated because I don't understand them as well?

(EDIT: In particular, I'd like to know if any students of algebra feels like anything in analysis seems "more complicated" to them than anything in algebra.)

I know that there are still many unanswered questions in functional analysis. But am I just suffering from a "grass always looks greener on the other side" phenomenon, or is there some objective way to justify the feeling that somehow topology or geometry is more "complicated" than analysis?

I also notice a cultural difference between analysts and other mathematicians. For example, I found that analysis professors are more likely to dress well, and behave more like a normal person, whereas algebra/topology profs are more likely to be wearing torn tshirts and act like the stereotypical absentminded professor.

 

[Tom Gilroy]

I've just begun a Ph.D in algebra (more specifically, vertex operator algebras). I did consider quite a lot of different areas before I came to a decision.

I used to be really interested in analysis and there are still some aspects of analysis I'm interested in, infact I preferred analysis to algebra for a long time. However in my final undergraduate year I had to take functional analysis formally, and the professor (who does not normally teach the course) made the course significantly more difficult than previous years, roughly on par with graduate courses I've seen. He also placed emphasis on areas I didn't find interesting or exciting (mostly fixed point theory), nor was he approachable or friendly. I'd read most of Kreyzig's book before the course, so I was quite comfortable with functional analysis and did very well, but the course left a bad taste in my mouth.

With algebra on the other hand, I really began to take to it when I had to take a second course in group theory that was pitched to a very high level. I worked through all of Robinson's book, I understood every line and finished just about every excercise, and I still only just managed to get an A. The most difficult course I've ever taken, and I really enjoyed it. In particular, there was a discussion on the sporadic groups in one lecture that was the first step in finding my way to VOAs.

After that course, algebra really seemed much easier to me than it ever had before, and indeed easier than a lot of analysis.Either way, I just chose what I wanted to study based on what I was most interested in, rather than what I felt would be "easy."

I would imagine though that it's probably just because you aren't as comfortable with algebra, topology and geometry that you feel it's more advanced than analysis. I would however agree that algebraic geometry is difficult. I've been told that it is extremely rare for a person to be ready to do research in algebraic geometry without having worked hard for at least five years learning the field.

I would agree that the analysts (and numerical analysts) I've met were always quite well dressed and very "normal." Statisticians too, I would say. Most algebraists I've met have been quite normal too, though a little less concerned with dressing to impress. The strangest mathematicians I have met have all been algebraists, mind...

 

[mathwonk]

I am an algebraic geometer. I always feel that analysts are much smarter than I am, but I try to not let them know this. I go to the men's room now and then and mess up my hair so I will look properly unkempt, and whenever an analysts walks by I say something about functors, moduli spaces or derived categories, provided of course it is an analyst who does not know what these are.

Recently when one of my analyst friends announced a course on Fourier transforms, I took advantage of the chance to ask him if it would cover Fourier - Mukai tranforms on principally polarized abelian varieties, confident he would not know what this was (neither do I).

Hang in there, if you can do analysis, the hardest and most substantive of all mathematical subjects, you have nothing to worry about.

 

[Mathnomalous]

Recently when one of my analyst friends announced a course on Fourier transforms, I took advantage of the chance to ask him if it would cover Fourier - Mukai tranforms on principally polarized abelian varieties, confident he would not know what this was (neither do I).

 

[Tedjn]

See, I find analysis extremely challenging, just about as much as algebra, so your grass is looking greener from my side.

 

[snipez90]

I don't know I've never actually asked an analyst that question, probably because he or she would kick my ***.

 

[MathematicalPhysicist]

I wonder where do you place number theorists?

How about combinatorists?

 

[mathwonk]

Well Gauss was a number theorist. And Barry Mazur left a successful career as a topologist to become a number theorist. It is hard to generalize about number theorists. Some I have met are very narrow, while others know lots of other stuff. The subject is extremely deep. However to come back to analysis, I suspect number theory got even deeper when Riemann connected it to analysis via his zeta function. Or perhaps Euler began the trend with his infinite product expansion. Gosh..., Euler, Gauss, Fermat, Hilbert, Artin, Dedekind,... the subject certainly has a brilliant list of participants. I myself don't know much about it.

I am less confident if you really want a serious answer. I know even less about combinatorics, but I have seen it used crucially in my own subjects. A really smart person can go into any field, and a dunce can try even the deepest subject. When say I am an algebraic geometer that may invoke an association with people like Mumford and Deligne and Grothendieck, but I don't even understand most of what they did. I hope to learn more of course and I enjoy following in their footsteps to some extent.

Probably that holds in other fields. Each field has its brilliant lights, and the rest of us largely follow them, or strike out on our own as well as we can. Something that seems to trouble a lot of questioners here is that of reputation. But professionals I know mostly do their thing because they love to do so. It matters what peers think of you, and how you feel about your work, but mostly it is a matter of personal satisfaction to understand something after hard effort.

 

[Poopsilon]

I think this is an excellent topic that the OP has decided to broach, and as I myself am now beginning to enter that stage of trying to evaluate the various levels of delight that I derive from the main subfields of mathematics, I personally would like to hear more about how the mathematicians, or budding mathematicians, on this forum came to their particular specialty or subfield, and what about it especially entrances them or what about it they find especially joyful or interesting as maybe opposed to other areas of mathematics.

 

[discrete*]

I think this is an excellent topic that the OP has decided to broach, and as I myself am now beginning to enter that stage of trying to evaluate the various levels of delight that I derive from the main subfields of mathematics, I personally would like to hear more about how the mathematicians, or budding mathematicians, on this forum came to their particular specialty or subfield, and what about it especially entrances them or what about it they find especially joyful or interesting as maybe opposed to other areas of mathematics.

I'll second this. I'm just finishing up my undergrad courses and am giving serious thought on what specialty I'll work in in the future -- and frankly, it's intimidating. I know where my interests and strengths lie, but only at a pretty general level. Like for instance, I'm interested in theoretical computer science, but that seems to be a wide net to cast. I've been told to just give it time and do what feels natural. One Professor even told me he could "see me working in combinatorics". Which I, as with all odd statements, took with a grain of salt. Again, it's just pretty intimidating and I'd love to hear some stories on how mathematician's chose their fields.

 

[MathematicalPhysicist]

Well Gauss was a number theorist. And Barry Mazur left a successful career as a topologist to become a number theorist. It is hard to generalize about number theorists. Some I have met are very narrow, while others know lots of other stuff. The subject is extremely deep. However to come back to analysis, I suspect number theory got even deeper when Riemann connected it to analysis via his zeta function. Or perhaps Euler began the trend with his infinite product expansion. Gosh..., Euler, Gauss, Fermat, Hilbert, Artin, Dedekind,... the subject certainly has a brilliant list of participants. I myself don't know much about it.

I am less confident if you really want a serious answer. I know even less about combinatorics, but I have seen it used crucially in my own subjects. A really smart person can go into any field, and a dunce can try even the deepest subject. When say I am an algebraic geometer that may invoke an association with people like Mumford and Deligne and Grothendieck, but I don't even understand most of what they did. I hope to learn more of course and I enjoy following in their footsteps to some extent.

Probably that holds in other fields. Each field has its brilliant lights, and the rest of us largely follow them, or strike out on our own as well as we can. Something that seems to trouble a lot of questioners here is that of reputation. But professionals I know mostly do their thing because they love to do so. It matters what peers think of you, and how you feel about your work, but mostly it is a matter of personal satisfaction to understand something after hard effort.

Well I don't seriously believe there's some rankings between the fields.

I myself really interested in number theory, geometry and topology and analysis of DE (and their applications in physics and engineering).
But algebra for its own sake seems to be like mathematical logic, very formal and a lot of structure but without "meat".
Even when you get specalise you need to know a little bit what goes outside your discpline cause you never know what connections you might find between different discplines.

 

[Klockan3]

But algebra for its own sake seems to be like mathematical logic, very formal and a lot of structure but without "meat".

This isn't true, algebra is really important in physics as well. It might not seem like it at a first glance but just about everything in maths can in one way or another be used in physics. Also you can't really do a good treatment of topology without algebra, if you want to go into differential geometry you would have to work with algebraic varieties and even the solutions to differential equations follows a group structure which is useful to map which is also used a lot in physics.

I'd say that if you really want to be a good mathematical physicist then you should take every maths course available at an undergraduate level, don't skip anything thinking that it won't be useful because that isn't true.

 

[MathematicalPhysicist]

This isn't true, algebra is really important in physics as well. It might not seem like it at a first glance but just about everything in maths can in one way or another be used in physics. Also you can't really do a good treatment of topology without algebra, if you want to go into differential geometry you would have to work with algebraic varieties and even the solutions to differential equations follows a group structure which is useful to map which is also used a lot in physics.

I'd say that if you really want to be a good mathematical physicist then you should take every maths course available at an undergraduate level, don't skip anything thinking that it won't be useful because that isn't true.

You can't humanely take every course offered from the maths department.
I said "Algebra for its own sake...", I know there are applications of discrete and continuous (and differentiable) groups in particle physics and condensed matter, and this is why I said its own sake doesn't interest me much.

But taking a course in the math department won't stress these links for me, it would be rigoruos nontheless.

 

[Klockan3]

You can't humanely take every course offered from the maths department.

Of course you can, as long as you skip courses with mostly overlaps, there isn't that much maths at the undergraduate level. Taking intermediate level courses in probability, topology, differential geometry, real+complex analysis, integration/measure theory, functional analysis, algebraic geometry + the usual courses you do in a physics major + some higher level physics courses in the more mathematical inclined fields should be more than possible within 4 years. Then during your grad studies you should either self study or take the small set of courses that are left, things like some number theory, optimization etc, at this point you should be broad enough that you can do each remaining course in a week or so.

The funny thing with maths is that the more you study the easier it gets since most fields within maths are a lot like each other.

 

[MathematicalPhysicist]

Are you taking labs with this curriculum?

I tried once taking 9 courses (with a lab), it's not something I'd recommend anyone.
There are more courses than you have listed.
And besides, if you really motivated you don't need a course to learn new stuff, if any, a course will limit you and not broaden your knowledge.

The last comment of yours I ofcourse accept, I do see resemblance between different approaches.

P.S
What have you decided eventually with your Msc and Phd?

 

[mathwonk]

Well i could go on about my math education to answer how i chose my specialty. But it seems to relate to how it was taught to me plus how much I enjoy thinking about it. I.e. thinking about geometry and topology stimulates my intellectual pleasure centers in a way that analysis and combinatorics and algebra do not. So when I was in graduate school I tried to be a several complex variables analyst but it was hard and painful. Still the experience benefited me when I did choose my field.

On the other hand I chose algebraic geometry partly because it was hard for me. It belonged to geometry and so I liked it, but it was a special class of geometry that was hard for me to visualize so I was challenged by it. This is silly of course, but it seemed to me that topology and differential topology, both of which I initially loved, were relatively easy, because I thought I could picture the difference between a discontinuous and a continuous map, or a continuous and a smooth map. Plus there are lots of these maps around, and you can change a smooth map slightly and keep it smooth.

On the other hand it seemed very hard to visualize an algebraic map, and if you change an algebraic map slightly it becomes non algebraic, by the analytic continuation principle. So algebraic geometry combined a subject I felt a natural talent for with a subject I found challengingly difficult, so this was an appealing combination.

Then finally I really enjoy talking about mathematics and the algebraic geometers I knew were greatly stimulating and fun to talk to. Young and energetic and inspiring.

So a combination of natural ability, challenge, and inspiring teaching drew me to my subject.

I never even had a number theory course nor a deep understandable analysis course. I had only two analysis courses. One was very formal, clear lectures but not insightful, and just memorizing proofs that did not connect to anything else in mathematics. The second course went way too fast, one month on real analysis one month on complex analysis, and then right into Riemann surfaces. Zooomm...

I only had one algebra course, from M. Auslander, and I loved it, but when he heard me say I was attracted to algebraic geometry he correctly declined to advise me, referring me instead to an expert in that field. When I did wind up in algebraic geometry, my background in algebra, topology, functional analysis and complex analysis helped a lot, as all those subjects are used there.

 

[Klockan3]

I want to get a really broad picture of maths, to me every major piece of maths I lack eventually comes back to bite me in the *** when I am studying physics courses so I really want to take just about all maths you can take that are still courses and not projects. Originally I started to study physics because I like to understand things and physics is the most closely related to understanding the foundations of our world. When I studied physics I realized that understanding maths is crucial to understanding physics so I started to take advanced maths courses. Ultimately I want to understand everything, it isn't possible but life isn't about having goals you can reach and I would undoubtedly get to understand a lot trying to reach this goal so it isn't a waste.

To me the maths is about making my mind more versatile, getting more ways to look at the world and thus getting more intuition for what the world really is. To me physics without maths is like a toy for people to play with, I have no interest playing around with things I just want to understand. I don't want people to describe what happens if we put together X and Y, I want to understand what, how, why as deeply as possible. To me knowing what happens is worthless except as a standard which models have to follow.

Maybe down the road I will narrow down a bit but right now I want to spread and just absorb everything that mathematics can throw at me. I will probably end up in something like string theory, but I have never really been interested in it so far. Most likely since I don't understand why. I didn't like quantum or relativity either till I really saw all the connections but then seeing those connections are some of the best moments I have had in my life.

Are you taking labs with this curriculum?

I should perhaps add that I am not studying in the US and where I go we don't focus that much on labs. You can do it if you want but you don't need to do more than the basic labs that just shows you situations where the formulas do and don't work in the real world. Roughly 70 hours worth of labs in total are mandatory I think.

What have you decided eventually with your Msc and Phd?

I am just about to finish a master in both maths and one in theoretical physics, I will probably go on for a phd in maths. I want to do mathematical physics and I had more freedom with the maths department if I wanted to go that route.

I would probably not had the freedom to chose courses like I did if I went to a college in the US so maybe it isn't fair to compare like that, I just mean that it is possible to study a lot faster than many imagines. One thing we don't have as well is the large amount of hand ins in the easier courses that are just there to keep the students occupied, so if you learn quickly you can easily take extra courses, it is even encouraged with many of the extra courses being made to fit the standard schedule.

If you just study the standard extra courses you would get roughly an equivalent to a maths and physics double major in 3 years but I took a bit extra on top of that as well. And I am not a freak case, there are many each year who studies extra like that, maybe 20% of the students even though just a few do as much or more than I do. Then we have a large chunk which can't keep up and thus eventually needs to retake years, this is more than half of the students so maybe my time plan is a bit optimistic for most students.

 

[Shackleford]

I recently decided to swap my major and minor. Now, I'm going to major in math and minor in physics (rather than major in physics, minor in math), so the mathematical physics route is piquing my interest. I'd like to learn more about it.

But on the pure math side, to me, it seems the most challenging and exciting part is analysis, proving things, proving why something is. You can prove in mathematics but not "prove" in physics. So, I'm not sure if I want to pursue pure or applied maths at the graduate level or pursue a mathematical physics route. One caveat though is that I don't know or care to learn any programming, but if I did any kind of applied maths I'd have to learn, or at least learn how to use Matlab, Mathematica, etc.

 

[Tom Gilroy]

Of course you can, as long as you skip courses with mostly overlaps, there isn't that much maths at the undergraduate level. Taking intermediate level courses in probability, topology, differential geometry, real+complex analysis, integration/measure theory, functional analysis, algebraic geometry + the usual courses you do in a physics major + some higher level physics courses in the more mathematical inclined fields should be more than possible within 4 years.

Just curious, you haven't mentioned courses in group theory, or ring theory or field theory, or other algebraic topics. Are they not available at undergraduate level where you studied, or do you not feel they're as important? Abstract algebra was emphasised hugely in my university.

 

[Klockan3]

Just curious, you haven't mentioned courses in group theory, or ring theory or field theory, or other algebraic topics. Are they not available at undergraduate level where you studied, or do you not feel they're as important? Abstract algebra was emphasised hugely in my university.

I didn't mention that since it is required for algebraic geometry and they aren't really advanced topics... But I realize that I wasn't really consistent with this since I mentioned real/complex analysis etc, but yes it is of course. I took most of that my second year.

 

[Tom Gilroy]

I didn't mention that since it is required for algebraic geometry and they aren't really advanced topics...

Well, that depends on the specific courses I suppose (that could be said about real/complex analysis too). I had to take a two semester course in group theory as an undergraduate which was very deep and advanced, and a two semester course in ring theory and field theory which was also quite advanced (though not quite as much so as the group theory course). Most universities over here don't put the same emphasis on algebra at the undergraduate level, so I was wondering about what the case is elsewhere. On the other hand, the topology course I had to take as an undergraduate was quite basic, so I guess it all evens out.

I suppose it's no wonder I'm studying algebra...

But I realize that I wasn't really consistent with this since I mentioned real/complex analysis etc, but yes it is of course. I took most of that my second year.

Thanks for clarifying. Just out of interest, were the algebraic topics (groups, ring and fields, maybe Galois theory) studied as part of an "abstract algebra" course (which seems to be the standard), or were they courses in their own right?

 

[Klockan3]

Thanks for clarifying. Just out of interest, were the algebraic topics (groups, ring and fields, maybe Galois theory) studied as part of an "abstract algebra" course (which seems to be the standard), or were they courses in their own right?

They are a part of many different courses, but the abstract algebra course consisted of the first 9 chapters of Dummit and Foote, but then we had already gone through the general notions of groups, rings and fields in the basic discrete maths course given in the first year. I think that the focus can vary greatly depending on where they are giving the courses. Where I come from analysis is the most popular so we got much courses relating to that and the plethora of physics courses eats a lot of time.

 

[MathematicalPhysicist]

I want to get a really broad picture of maths, to me every major piece of maths I lack eventually comes back to bite me in the *** when I am studying physics courses so I really want to take just about all maths you can take that are still courses and not projects. Originally I started to study physics because I like to understand things and physics is the most closely related to understanding the foundations of our world. When I studied physics I realized that understanding maths is crucial to understanding physics so I started to take advanced maths courses. Ultimately I want to understand everything, it isn't possible but life isn't about having goals you can reach and I would undoubtedly get to understand a lot trying to reach this goal so it isn't a waste.

Well, if your'e going through PhD route you'll become knowledgeable about particular narrow field, so I don't see how can you expand your horizons in math and physics if you're going through this route.

I should perhaps add that I am not studying in the US and where I go we don't focus that much on labs. You can do it if you want but you don't need to do more than the basic labs that just shows you situations where the formulas do and don't work in the real world. Roughly 70 hours worth of labs in total are mandatory I think.

That's a big difference, in my degree of math and physics combined I had (this semester is my last lab) three yearly courses each year for three years, it accumulated more than 70 hours, though I don't think they have contributed to me much so.
In retrospect I should have perhaps taken the pure maths route and add additional theoretical physics courses alongside.

I am just about to finish a master in both maths and one in theoretical physics, I will probably go on for a phd in maths. I want to do mathematical physics and I had more freedom with the maths department if I wanted to go that route.

What your topics for theses were?

I would probably not had the freedom to chose courses like I did if I went to a college in the US so maybe it isn't fair to compare like that, I just mean that it is possible to study a lot faster than many imagines. One thing we don't have as well is the large amount of hand ins in the easier courses that are just there to keep the students occupied, so if you learn quickly you can easily take extra courses, it is even encouraged with many of the extra courses being made to fit the standard schedule.

If you just study the standard extra courses you would get roughly an equivalent to a maths and physics double major in 3 years but I took a bit extra on top of that as well. And I am not a freak case, there are many each year who studies extra like that, maybe 20% of the students even though just a few do as much or more than I do. Then we have a large chunk which can't keep up and thus eventually needs to retake years, this is more than half of the students so maybe my time plan is a bit optimistic for most students.

I don't think your'e a freak, I myself can take several theoretical courses at once, but taking also labs with it, is an overload and from your account I understand that you haven't taken a lot of labs.

N.B
I myself not from US as well.

 

[Klockan3]

Well, if your'e going through PhD route you'll become knowledgeable about particular narrow field, so I don't see how can you expand your horizons in math and physics if you're going through this route.

Well, it is hard to live in academia without a PHD so I need to get it and it is not like you can't broaden yourself during the PHD. At least I get a lot of inspiration when reading loosely related subjects.

What your topics for theses were?

I am in the process of deciding that, I am not going to write two theses since there is no real point. But I got courses for writing a thesis in either subject and the subject I am going to chose is probably related enough with both so that I could have done it with either side anyway. It will probably be related to something like lie algebras, representation theory or differential geometry.

I don't think your'e a freak, I myself can take several theoretical courses at once, but taking also labs with it, is an overload and from your account I understand that you haven't taken a lot of labs.

Labs are quite worthless unless you are going to work in one some day, in my opinion. Sure you should have done a few experiments so that you don't lose touch of reality but it is not like they provide more understanding of already known facts than just theoretically explaining what setup they used for the lab, what the results were and what conclusions were drawn.

 

[MathematicalPhysicist]

Well, it is hard to live in academia without a PHD so I need to get it and it is not like you can't broaden yourself during the PHD. At least I get a lot of inspiration when reading loosely related subjects.

I am in the process of deciding that, I am not going to write two theses since there is no real point. But I got courses for writing a thesis in either subject and the subject I am going to chose is probably related enough with both so that I could have done it with either side anyway. It will probably be related to something like lie algebras, representation theory or differential geometry.

Labs are quite worthless unless you are going to work in one some day, in my opinion. Sure you should have done a few experiments so that you don't lose touch of reality but it is not like they provide more understanding of already known facts than just theoretically explaining what setup they used for the lab, what the results were and what conclusions were drawn.

Ah, so basically you have taken Msc courses in both math and theoretical physics, but writing just one thesis.

I agree, for someone who wants to go theoretical it doesn't help to do more labs, I tried asking the academic advisor to substitue this last lab of mine with graduate courses in theoretical particle physics, he didn't approved.