Riemann Roch Theorem: A Topological Perspective

[JFo]

What is it like to be a mathematician after completing grad school?
I know many go on to teach, and while that is all that is desired by some, others pursue their creative interests in research or do both. For those of you "proffesional" mathematicians, what is your daily life like? Do you have to have a second job to pay the bills so you can pursue your math interests on the side? Is it possible to just essentially work in a think-tank like environment where all you do is pursue the advancement of mathematical theory?

Do you collaborate with other mathematicians? If so, how? Do you braisntorm ideas together, or do you do most work independently coming together later for proof checking and second opinions?

Do you concentrate your attention on one particular field of math, or do you work on several branches?

I am very much fascinated by the lifestyle of real mathematicians, and welcome any thoughts or insights on the matter. Any insight to a day in the life of "your name here" as a mathematician is also appreciated.

-JFo

 

[mathwonk]

there are extremely few jobs available where you only do math research. the normal thing is to get a job at a university where you must do both teaching and research.

even these jobs are very hard to get, and many universities are trying, successfully, to exploit young mathematicians by offering them only temporary jobs with high teaching and often no benefits.

still some jobs do exist for the gifted and the lucky.


even if you do get such a job,
many people, including me, find it very very hard to do all the tasks associated with it. i.e. actually you have three jobs, research, teaching, and administration, and all three are potentially infinite. i.e. you could spend all your time trying to do anyone of them better.

but you are finite and have only finitely much time, so you must manage them somehow. since the students are always there, and class meets inexorably, the harder one to maintain is your research, which is your lifeline to intellectual stimulation, and also to your professional future.

i.e. at most schools, you cannot be promoted without doing research. in fact many schools expect you to generate outside research grant money, and are not satisfied with only research, if it does not generate money. since math and physical sciences have available to them only about 1/30th the amount of grant money available in biological, sciences (this was once true anyway), ana math is on the low end of that group, it is very hard to produce grant money. I think only about 1/3 of the proposals NSF receives which are deemed worthy of funding, are actually funded.

this past year, after pledging to raise the money available to NSF, congress actually cut it instead, while funding such things as the rock and roll hall of fame museum.

this means it is very hard to find funding for summer research, and often one must fund it one self, or teach in summers and try to get by.

Intellectually, I found it crucial to maintain a working seminar, with like minded people, in which we regularly met together to teach each other and learn material that hopefully would translate into research ideas and opportunities for us.

one must keep working at doing mathematics, rather than merely learning it as well.

The most productive period in my post grad school career was a postdoc at Harvard, where I had no teaching duties at all, and spent all day every day doing research and talking to the best people in the world in my area.

these positions are very competitive but NSF gave at that time about 10 a year? in mathematics in the whole country. unfortunately the stipend was so small I had to sell my car for food, and give up a whole year of the award to return to teaching. partly as a result of my experience, they later raised the stipend.

at one point my family moved back home, and i tried to live out the last month of my stay alone, with only $50 to live on for a whole month.

that was a fabulous intellectual experience while it lasted, and I benefited enormously. As a result I met many people around the world and received many national and international speaking invitations, and learned enough to last for a long time.


in ordinary teaching life, one must find a way to keep it interesting. how do you find joy in explaining the same simple facts about calculus or algebra or whatever to people year after year, often to students who show little interest in the material? this is a challenge.

some people try to perfect the routine so they have it down pat, and need spend little thoguht on it.

i do the opposite, i always rethink everything from scratch to try to make it new. the advantage to my way is i continually learn something new. the down side is ti takes a lot of time,

well that's enough from one person.

 

[matt grime]

You get employed by some institution on a post-doc if you want to do research. There are some other areas where they employ you too. I believe the NSA is the single biggest employer of mathematicians in the world, and you'll get to do research for them. There are some companies that do what is considered academic research such as the computer and communications companies.

If you think stats is maths then there are lots of agencies to work for coming up with new things, and investment banks like people with PhDs in maths to develop better ways of solving partial differential equations.

If we presume you mean pure maths research then it is some academic institution really that will let you do maths and pay you for it (quite handsomely in some cases). I know of exactly one person who is a research mathematician and not employed to be so, although that is a small lie. This person is independently wealthy and is a professor in a university but took a notional salary to avoid teaching. So, no, maths is never somethnig you do in your spare time (unless you're a writing up PhD student whose funding has run out... this is a UK thing where you only get 3 years of funding to do a PhD)

The most famous "unattached" mathematician, and about the only one i can think of to be honest, was Erdos. Besides that everyone I can think of is attached to some institution since you need to talk to other people, keep up with the current trends and have the benefits that being employed by some department brings. For instance, without being a member of a department you'll find it impossible to afford the cost of journal subscriptions to make sure you can access the latest papers.

Most mathematicians work in several areas, though they are often related, or they apply techniques across the spectrum. For instance Gowers got his Fields medal for solving many open problems in banach spave theory by developing combinatorial graph theory techniques.

I work on the overlap of category theor, representation theory, algebraic geometry and algebraic topology. What this actually means is not that I am a polymath (no pun intended) but that labels aren't actually useful: I gave a talk to some other postgrads at oxford in a group theory conference on some of the ways triangulated categories (things in algebriac topology) can be used to work with groups' that completely baffled an audience who were more used to getting bounds on the number of conjugacy classes of p-groups - all depends on what you think algebra is.

Personally, at the moment, I work independently on some small things then talk to other people to check I'm right and to generate more ideas to work on. But that is because I'm at the beginning of my career and very uninformed. Others will work together as in sit and talk the problem through a lot, some groups work together, but each person works on a small aspect of the proof: it all depends on what you're trying to prove - is it a big classification theorem or is it like me at the moment - small results trying to get an understanding of the way something works so we can later make bigger conjectures. Again, it all reflects on who you are, what you're doing at that time, and to some degree how you've been influenced by your colleagues whilst you learned your trade.

In short there is no 1 way you do maths.

 

[Icebreaker]

What about grant funding in Canada? Any word on it?

 

[matt grime]

The best thing to do is do a job search for post doctoral positions and see the pay scale and requirements. It appears in the US that most postdocs are expected to teach. In the UK postdocs are expressly not supposed to teach. The pay is better in the US, though. Starting maths postdoc in the UK varies from 18,000 UKP to 28,000 UKP, approx, but it is very rare to get more than 22,000, UKP which allowing for exchange rates and "relative cost of living" is approx. 23-36,000 USD at the moment, with the bulk of them below 30,000USD (current exchange rate is about 1.8 USD to the Pound).
All this is pre tax, and "back of the envelope" maths and very approximate, and is defintely to be taken with a pinch of salt.

It appears not unreasonable to expect 35-40K USD starting salary for a post doc in the US from my very limited experience (ie the 3 postdocs I know and a quick google)


In fact the teaching was why I quit America and chose to restart my PhD in the UK.

 

[Icebreaker]

But why did you restart a PhD? Could you not simply apply for post-doctorial research in the UK?

 

[JFo]

Wow, Thank you mathwonk and matt grime for your personal glimpse into the activities of real-life mathematicians . I think anyone interested in pursuing math should read these posts to get a view of what lies ahead.

I found this to be invaluable since I'm currently a first year (undergrad) student, and its hard to gauge what lies ahead. My only view of what its like to be a mathemtician at this point comes from my time involved in studying the various texts of mathematics, and my only experience in research comes from trying to make conjectures on my own and see what I can discover for myself. Of course, these are all just excercises to prepare for the future as anything I come up with is sure to be found in a book.

But still, its hard to get out of the mindset of being a student, where your major focus is on comprehension and regurgitation rather than independent thought, and not much attention is given on what is expected of you as you near the end of your education.

I didnt know that some problems are broken up, and people work on different parts of a proof. I guess it makes sense, after all two heads are better than one. I had always viewed that mathematician as a lone wolf that hunts alone, and only comes out when he's caught something.

Anyways, from these posts I have learned so much that can't be taught from a book. Thank you both very much.

-JFo

 

[mathwonk]

there is a lot more to say. the one thing that caught me by surprize was the fact that my dept head expected me to get a grant the first year i arrived at my new job. i had never heard anything about grants at grad school, so it was a completely new game to learn.

nowadays some schools teach grant writing to students since it is considered so important. if you have a grant you have time to do your research and if you do not have one you don't. it is almost as simple as that. so you are well advised to find out what grants are available and start applying for them as soon as possible.

another thing that makes teaching life more fun is to be allowed to teach honors courses. in honors courses the students are more likely to appreciate what you are trying to do for them by casting the material in an intellectually sound way. many non honors students are more interested in getting through, and may even resent any attempt to present higher quality material, since they may feel it just makes it harder to learn.

the conflict between your own love of the subject and desire to do things "right" must be measured and balanced with the average student's fear of math, and the desire only to survive, and get a good grade.

within the last three years i have received student teaching evaluations such as "one of the most challenging teachers I have ever had, certainly the most challenging math teacher." "He was willing to do whatever it took to help us learn, and had the highest standards both for us and himself." or "thank you for what you gave us, a few places in the world there are teachers like you."

that from my honors class,

but evaluations like : "the worst math class i have ever had, bar none", and others too painful to recall, from a non honors class that felt overworked.


When I was a college student I once audited a course on introduction to proofs in elementary set theory and topology and analysis, that started from a construction of the integers, went through metric space theory and through one variable calculus, went to every class, learned it all, and then never bothered to take it for credit.

At my university some students do the opposite, they take courses for credit that they already know just to try to get an easy A. this contrast between a desire to learn the subject, or to get high grades, can make it hard to identify with and hence to teach such students.

it is important to try to love the students, and not be too much in love only with the math. still to me that means helping them become what they are capable of, not just indulging bad habits further.

research has its own magnetic appeal. there is nothing so exhilarating as realizing that you have solved a problem that probably no one has ever solved before. my thesis cracked open a problem left untouched for over 80 years, and of interest at least for a little of that time to some extremely bright people.


you already seem to know how to do research, but for me it was an opening of a new world. my advisor helped me see that doing math was like taking a walk in a forest and observing the fauna and flora. i had always thought it was a stroll through a museum where everything was in glass cases, and bottled or pre-packaged.

i didn't know you could pick the stuff up and play with it, and turn it upside down, and haunt it or live with it until it told you secrets no one had noticed.

once i became a researcher i realized the whole subject is up for grabs. any similarity you can see between things is potentially useful, and should be explored.

i am mostly teaching at the moment, but i subscribe to algebraic geometry eprints and still read the latest postings of new results there every day.

i also try to attend the research seminar faithfully, and I speak in it occasionally. even if i only have a little to say myself, i can take the opportunity to learn something. giving a talk makes me realize that I have much to learn even about subjects i am "expert" in.

people just want to learn something, they do not mind if i am teaching them someone else's work, at least most do not, if i do it well.

when i sought my first job, i wanted a research oriented university. another option that might also appeal is at ateachign oriented college, where the teaching load is still low enough to allow and encourage some research. one advantage could be better students.

the top research universities have strong undergraduate students, but good teaching colleges, like Harvey Mudd or Haverford, or Swarthmore, have much stronger students than most state universities. the classes are smaller too.

so at some big state schools there is almost a total disconnect between the level of your research life and your teaching life. it can be hard to get used to trying to do world class research yourself, and yet teach on a decidedly low level.

 

[matt grime]

But why did you restart a PhD? Could you not simply apply for post-doctorial research in the UK?

Emphasis on the fact I quit my PhD in the US fed up at the teaching, and other things too. To do a postdoc one needs a doc, so I did one in the UK (with the intention of returning when I was able to better negotiate a teaching load that wasn't tedious inthe extreme).

See mathwonks bit about teaching non-honors students. And the evaluations thing. One of mine lamented the fact that I couldn't speak English properly, which came as a small surprise to me.

And if you do opt for grad school in the US then that is what you'll get too. Though possibly not the problems with having an English accent. Pcik carefully. Some take far better care of you than others, and don't expect to see a mathematics student in your class. Anyway, stop there before going off to another topic entirely.

Back on topic.

Some mathematicians do do the lone wolf thing. I've done it a little too much in the last couple of years, but it tends to be counter productive. After all you'll probably want to get a job working with/for some of these people in a couple of years time.

 

[CrankFan]

Interesting stuff. To take this in a slightly different direction, I was curious, what does being a graduate student in mathematics entail?

In general (if possible) what does being a graduate student, of any type, entail?

I've heard some accounts of graduate students which describe them as almost completely free to do their own research, and yet others describe how they are required to attend classes much like undergrad students. I was wondering if these differing accounts are due to students being at different levels in their development (maybe one is starting and the other is finishing), or perhaps the differences are a result of the unique philosophy of the institution where they study? Or perhaps what I've heard is just false. In any case I wanted to get a better picture of what the graduate studies process, for a mathematicians, is like in general.

Also I was curious about any more of the US/UK differences.

Thanks.

 

[matt grime]

I'll take this as someone who's been in both systems, I guess.

The US:

You will probably need to pass qualifier exams, usually in three subjects out of: algebra, analysis, topology, statistics, applied maths, depending on your preferred areas, some other places may not offer some of these, some may have, say, discrete maths instead . Depends on the institution.

These will need to be passed inside the first 2-3 years, probably 2 in the first, the 3rd inthe second or so, with some resits allowed.

These will be taught courses with exams at the end.

In addition to these, and in general, you will be expected, unless you can provide some reason otherwise (such as excellence in research already), to attend a certain numbers of lectures each week. Usually there will be no homework in these that is marked and as long as you attend you will be given an automatic A. Often, there will be some practical courses you should attend, such as "grant writing", "teaching skills", "latex for beginners", or "graduate seminars (present talks to grad students to get used to it)".

The number of hours will be variable upon the place and year your in. (Beginning students should attend more.)

After passing the qualifiers, you should expect to take chose, be given, a supervisor, and start wokring with him/her on more specialist areas. Some oral exam in the first few years to check on your more specialized progress in the areas you're looking at with a view to research, usually called a comprehension exam or something. This should be within 5 years of starting.

Finally, after the research comes the viva, usually within 7 years and no more than 10 from the starting date.

Whilst all this is going on you will be expected to teach undergraduates. The nature of this will vary as will the hours. Usually US nationals get preferential treatment as well as pay (this is a national movement to redress the balance of too few US students going into grag school and not a reflection on the actual universities). I had to teach 90 huors in my first year, rising to 120 in other years (the time off was to dtudy for qualifiers). The US vigre students had approx half that to teach, though not all home students are given this treatment and the awarding of them seems little to do with potential to be honest. The best 2 US students (one is now a PhD, the other left to become a Lawyer fed up with several aspects of the program he was in) I knew weren't given Vigre's and three who were quit with a masters unable to handle the maths.

The pay usually only lasts over 8 months with an expectation that you will either take extra teaching over the summer, or will be given funding by your supervisor. This varies from place to place. It depends on whether or not the university sees you as potential mathematics genii or teaching staff to shoulder the work load.

Now, let's look at the UK. PhDs are funded for 3 years, or possibly 4. The level of funding has improved dramatically over the last 4 years. I started on 7,500 UKP, which I believe is officially 500UKP below the poverty level, though now you can expect to start on 13,000UKP, more in applied areas.

4 years funding is offered only if you agree to teach. This is usually reserved for people who didn't obtain 1st class honours as an undergraduate. I think only Cambridge, Oxford and Warwick have competitive entry, usually to do with obtaining a distinction in part III at Cambridge. Quite a lot of people who go on to do research have done part III. There aren't that many people who carry on to do a PhD, but then the level of the undergraduate course is higher than in the US. For instance, I would expect anyone *entering* a PhD program in the UK in pure maths to know the basics in 3/4's of the following:

Galois Theory, Homology (simpicial) and homotopy theory, Representation theory, spectral theory, Riemann surfaces, hilbert spaces, algebraic geometry, lie algebras, measure theory.

to a level beyond anything in a qualifier course (some exceptions perhaps, say at Berkley), though I may be overstating the case based upon the graduates I know, and the areas in which I work, and this isnt' typical of all undergraduates, you must understand. We turn out more than our fair share (the vast majority) of "mathematicians" who wouldn't know a toplogical space if it bit them on the arse.

You must finish you PhD within 4 years or face some serious problems (up to 6 is allowed, but you face financial penalty, and the university will lose funding if you take more than 4).

The university may offer some courses for you to take, but usually not.

However, the graduate programs over here are in need of radical overhaul in my opinion. Especially in educating our graduates in the basics (to the previous generations).

My impression is that the US system turns out uniformly more rounded mathematicians than the UK. However the ability for the very clever to finish in 3 years very easily and move in means that we have a few very good mathematicians that we get out into the research world quickly.

If you like a slogan, a UK PhD will know a lot about little, but a US one will know a little about a lot.

 

[bfd]

JFo: I just read your thread and am really glad you started it. I'm looking for much of the same information. Fortunately there is a ton of good information out there on this topic. Some of the best resources I've used so far is the internet and also the college library by my house.

So far I have found (but not limited to) the following items:

- A Mathematician's Survival Guide by Steven G. Krantz (Excellent information regarding everywhere from: admission to graduate school to early career development)

-the online ams.org bookstore

- www.youngmath.net

- All the Mathematics You Missed (But Need to Know for Graduate School)
by Thomas A. Garrity (I just ordered this so I don't know about how exactly good it is just yet but it was recommened by the MAA and by a few other places)

If anyone needs more information let me know. Thanks to everyone who helped contribute to the thread.

 

[Haelfix]

It seems to me mathematicians have the hardest time getting good jobs after their PhDs. So many of my friends in that division who were clearly much smarter than me, struggled trying to get spots. Often simply b/c their PhDs was in a arcane subject that wasn't the current political mode of the day.

Its easier in physics and astrophysics of course, but as a postdoc I can attest to the horrendous task of trying to get funding. I'm lucky in the sense that I have very gifted people who help me and know where too look for, and I also have some connections now that often seem to be interested in what we are doing.

AFAICS, teaching is the easy part. Research the second hardest. Administration errata the hardest.

The only hard part about research is constantly finding new things that
a) Interest you
b) Are relevant
c) Doing something new enough to be deemed acceptable to the community, which leads to the administration part.

Once you are on the project, 95% is just application of what you've been taught and of course intellectually stimulating which makes all the pitfalls seem easy in comparison. Finding that particular project.. that's the hard thing. Theres always the stress in the back of your mind that you may be shooting down a blind alley, or are horribly wrong or just reinventing the wheel.

 

[bombadillo]

It's been interesting reading the posts of mathwonk and matt grime. Just a few stray comments from me:

I don't think the British undergrad degree is presently strong enough to allow one to do a PhD, except in the case where the student is a potential star, i.e. someone of the calibre of a Donaldson, Segal, or Atiyah. It's true that the British degree is much stronger than its American counterpart, but still standards have fallen a bit over the last few decades. Furthermore, the frontiers of research have moved forward quite a bit. The Part III addresses this problem, but it's open to only about a hundred students a year, and some of them are overseas students. Furthermore, the Part III is demanding: it's an intensive one-year course meant for very clever students (I think the majority of British students are coming with exemplary first class honours from Oxbridge). I think there's a decent Master's programme at Warwick, and maybe they've beefed up the M.Sc programme at London, but I don't know enough to comment on them. I do know that they generally lack the specialised advanced graduate-level courses one can see in American powerhouses like Princeton and Chicago.

Because funding is for a limited duration in the UK, there's tremendous pressure to come up with an acceptable thesis. This often means that doctoral candidates stick with safe areas -- e.g. cohomology groups of finite groups -- rather than areas that demand very solid backgrounds (e.g. arithmetic geometry). I recall Richard Borcherds telling me once how he was thinking of quitting his PhD halfway through, and becoming a C programmer (no progress with his research): the pressure to produce a thesis in tight time constraints is tremendous.

Here in the US, PhDs drift from one post-doc to another; two three-year postdocs back-to-back is not uncommon. Too many qualified people chasing too few jobs. I remember that the University of New Mexico -- hardly a top-flight school -- put out an ad for a non-tenure track assistant professorship: the ad attracted 700 applications. I don't expect any amelioration in this generally dismal situation. It's not easy producing good research as a post-doc: one has just cut the umbilicial cord with one's advisor, one's research thesis may be a dead end, a blind alley leading nowhere, one may not have the breadth and depth of background to create a research programme for oneself, and one has a number of onerous and thankless administrative and teaching duties ("teaching calculus to morons")

Some mathematicians have drifted towards finance (derivative and options pricing and the like); more have gravitated towards computing in one form or another. I suspect it's the same in the UK and Europe generally.

From a career point of view, I'm not sure a math PhD makes much sense: if you decide to do it, it has to be as a labour of love. Though it doesn't make sense, there's nothing else I'd rather do, though I haven't an ounce of talent, and I have to agonise over material that is trivial and blindingly obvious to more nimble-witted people.

 

[matt grime]

Standards aer certainly dropping in the UK. And I agree that anyone who hasn't done part III (or equivalent) is going to struggle to produce a strongly research driven PhD. For instance, one could graduate one of the Universities just outside the elite (Oxford, Cambridge, Warwick, Imperial et al, though I've reservations about Oxford too [slight joke]) with a 4 year MMath and the only algebra you know is some very simple representation theory, a smattering of galois theory, and a hint of algebraic topology.

Certainly, if you did that course (thinking it was high level, and the course doesn't advertize itself as easy), then a PhD for three years, then you would be nowhere near as well educated as someone who'd done a less demanding degree in the US but then been to grad school at Chicago, Harvard, Princeton, Berkley...


Europe I'm not so sure about. WHen I did part III there were about 200 people on the course (both pure and applied, which is actually quite pure in parts - non-linear stuff and QFT etc are applied there, this was about 5 years ago) and many of the German students were far better prepared for the course than me (though they all tended to be older, but takes more than 4 years to get a degree there).

There are two distinct kinds of thesis in pure maths. Those that display talent and those that are tedious. A lot of theses contain "just calculations" and little theory. Going to post grad conferences made me glad I was doing something theoretical, even if it is unclear what I need to do to make progress most of the time.

I must disgree that group cohomology is necessarily easy, or doesn't require a solid background (though there are lots of bad theses written about using combinatorics to prove yet another result about the symmetric group, mind you the same could be said about the logistic map in non-linear dynamics). Of course if you mean just "working out some cohomology groups", then it isn't very interesting a lot of the time. And of course comparing group cohomology to arithmetic geometry is a little like comparing algebraic topology to elliptic curves. Elliptic curves is the "safe" bet here, just as is group cohomology, yet it is one aspect of arithmetic geometry. It's not fair to compare a whole swathe of subjects to one aspect of one part of algebra. Having said that I would recommend poeple look at the arithemetic geometry path in preference to other parts of algebra if only because it is the more research intensive area right now with lots of interdisciplinary collaboration, and hence has much more opportunity for funding at post-doc level and after.


The n'th cohomology group H^n(G,M) is the n'th derived functor of Hom(M,?) applied to M. And anyone doing group cohomology ought to know all about derived categories, and thus algebraic geometry and topology, which I think counts as a solid background, but I am biased, admittedly, and should declare an interest: I'm currently working in homological algebra, though I occasionally think about parts of arithmetic geometry. I suppose it all depends on what you do with it. I think working with tiliting complexes, bousfield localizations, and homotopy colimits is demanding enough of a soul like me.

I shoule probably explain the very small set of experiences i have to justify my position: the one PhD student I've heard talking in an area that comes (vaguely) under arithmetic geometry wasn't very strong, and their PhD was essentially working out coefficients of a family of L-series. Mind you I have heard some (many in fact) shocking talks from people working in group theory (I don't classify cohomology theory as group theory - it is part of representation theory, and the one type of algebra we often use for exposition and ease is a (finite) group algebra, over some suitably large field: it is frobenius, symmetric, finite dimensional, a hopf algebra... taking instead a p-modular system allows us to do some interesting things, and it possesses a nice set of sub algebras whose representation theory we try to lift to the larger one. Not to mention flat modules are projective, and the stable module category is compactly generated. and then there are nice associated derived categories. The other algebras we like are obivously hereditary, and no group algebras are hereditary unless they are semisimple.)


NB. When i said that most places in the UK weren't competitive entry, I shold point out that that is based upon the assumption that you have a first class degree (or perhaps a good 2:1) from a good degree program.

 

[bombadillo]

Let me just add my own observations here and there; of course I'm not quibbling with you, and I've been away from the action so long that I'm scarcely in a position to do so.

First of all, I don't want to sail under false colours, so I should state that I'm a drop-out grad student. Mostly because I have no real ability, partly because the British first degree doesn't provide enough of a basis (anymore), and partly because my research advisor was incompetent. If I had to classify myself, it would be as a muddler and plodder who can usually be found at second-tier schools: no original ideas, no razor-sharp intellect, but can do a few calculations, and with careful nurtutre of a supportive advisor, can eventually turn out a pedestrian thesis.

Standards aer certainly dropping in the UK. And I agree that anyone who hasn't done part III (or equivalent) is going to struggle to produce a strongly research driven PhD. For instance, one could graduate one of the Universities just outside the elite (Oxford, Cambridge, Warwick, Imperial et al, though I've reservations about Oxford too [slight joke]) with a 4 year MMath and the only algebra you know is some very simple representation theory, a smattering of galois theory, and a hint of algebraic topology.

I don't think Oxford has anything comparable to the part III; they just have a D.Phil programme after the Bachelor's. I was admitted to the Imperial Master's programme; it's an "intercollegiate programme", which means the courses can be taught and taken at any of the London colleges: King's, Imperial, QMW, University, LSE. The programme was a joke, and not even a pale imitation of Cambridge (I wasn't good enough for the part III). Courses like representation theory could be taken either as a third-year undergrad or first-year grad student (usually the same course, or a bit more appended for the grad students). I don't think it goes any further than group characters (i.e. no modular reps). The Galois theory course I took was delivered by the lecturer verbatim from Stewart's book, which like most Galois theory books, is a clone of Artin'e original notes (there are better books out today). Algebraic topology moved so slowly, I quit in disgust, and the same for the course on knot theory (given by the same lecturer). The grad algebraic number theory course was based on Stewart and Tall, hardly a grad level text. I could go on in a similar vein, but I'm just providing detailed corroboration to your contention that UK grad programmes are in desperate need of radical overhaul: these programmes are just routine extensions of the undergrad programme, and not a preparation for research students.

Certainly, if you did that course (thinking it was high level, and the course doesn't advertize itself as easy), then a PhD for three years, then you would be nowhere near as well educated as someone who'd done a less demanding degree in the US but then been to grad school at Chicago, Harvard, Princeton, Berkeley...

Concur entirely. These grad programmes are designed for research students. The courses on homological algebra, commutative algebra, representation theory, algebraic geometry, differential topology, algebraic topology, and so on, are designed to provide a solid background to a prospective research student.

I must disgree that group cohomology is necessarily easy, or doesn't require a solid background (though there are lots of bad theses written about using combinatorics to prove yet another result about the symmetric group, mind you the same could be said about the logistic map in non-linear dynamics). Of course if you mean just "working out some cohomology groups", then it isn't very interesting a lot of the time. And of course comparing group cohomology to arithmetic geometry is a little like comparing algebraic topology to elliptic curves. Elliptic curves is the "safe" bet here, just as is group cohomology, yet it is one aspect of arithmetic geometry. It's not fair to compare a whole swathe of subjects to one aspect of one part of algebra. Having said that I would recommend poeple look at the arithemetic geometry path in preference to other parts of algebra if only because it is the more research intensive area right now with lots of interdisciplinary collaboration, and hence has much more opportunity for funding at post-doc level and after.

Didn't say it's easy, but in arithmetic geometry or algebraic geometry, one can struggle for three or four years (easily) to build a basic background, and still be completely at sea regarding a possible and tractable research problem. As I understand it, some grad students have struggled with Hartshorne's Algebraic Geometry for a couple of years, only to find they're still at square one with regard to a thesis problem.

The n'th cohomology group H^n(G,M) is the n'th derived functor of Hom(M,?) applied to M. And anyone doing group cohomology ought to know all about derived categories, and thus algebraic geometry and topology, which I think counts as a solid background, but I am biased, admittedly, and should declare an interest: I'm currently working in homological algebra, though I occasionally think about parts of arithmetic geometry. I suppose it all depends on what you do with it. I think working with tiliting complexes, bousfield localizations, and homotopy colimits is demanding enough of a soul like me.

Dunno. You've been through the part III, so it behoves me to doff my cap, and be deferential, but I'd have thought it's possible to do homological algebra in general, and cohomology of groups in particular, strictly from an algebraic point of view (though I readily concede that the topological motivation would be nice, maybe even handy at times).

I shoule probably explain the very small set of experiences i have to justify my position: the one PhD student I've heard talking in an area that comes (vaguely) under arithmetic geometry wasn't very strong, and their PhD was essentially working out coefficients of a family of L-series. Mind you I have heard some (many in fact) shocking talks from people working in group theory (I don't classify cohomology theory as group theory - it is part of representation theory, and the one type of algebra we often use for exposition and ease is a (finite) group algebra, over some suitably large field: it is frobenius, symmetric, finite dimensional, a hopf algebra... taking instead a p-modular system allows us to do some interesting things, and it possesses a nice set of sub algebras whose representation theory we try to lift to the larger one. Not to mention flat modules are projective, and the stable module category is compactly generated. and then there are nice associated derived categories. The other algebras we like are obivously hereditary, and no group algebras are hereditary unless they are semisimple.)

You're bang on target. Again, it's the weakness of the UK system. Narrow specialisation begins early. The problem with this is that once the newly-minted PhD is on his own, he lacks the broad rigorous grad education that would allow him to construct an independent and meaningful research program. So for the rest of his sorry career, he's stuck with developing the theory of topological quasi-thingoids. Walk around British universities, if you don't believe me, and look at what the faculty are doing (if anything).

 

[matt grime]

Hartshorne was a recommended undergrad textbook. And we (well, me at any rate) certainly did modular representation theory in part III.

I'd be interested to hear what it is you think you "do" with group cohomology in a thesis, as opposed to arithmetic geometry. I was trying to indicate you're comparing chalk and cheese. Doing something with group cohomolgy is no harder than doing something with abelian varieties, surely? Incidentally, the ring H(G,k) is graded commutative and finitely generated thus yields and algebraic variety, the dimension of whcih is the rate of growth of a, well, never mind. But you see that actually there is a lot more to group cohomology than you first think.

Oh, the L-series person was not from the UK. But there is a distinct tendency to specialize here far too soon. If I ever get to be in such a position I would only ever accept PhD students who'd done part III (if I were allowed to make such demands). However, there is a brain drain across to the US, and many universities who don't see massive competition for places for a great number of reasons need to fill places or they lose funds. CAmbridge, with its excessive demands of a distinction in part III to obtain a PhD place there has successfully driven a lot of students away to places where such stresses aren't placed on them.

 

[bombadillo]

Hartshorne was a recommended undergrad textbook. And we (well, me at any rate) certainly did modular representation theory in part III.

Tell me you're pulling my leg: Hartshorne's Algebraic Geometry as an undergrad text? Where? Even at Cambridge they were only using the Shaferevich book for Part II; for sure no undergrads I know have enough commutative algebra under their belt to tackle Algebraic Geometry. For a Part III course, maybe. Perhaps you mean Hartshorne's book on projective geometry?

I'd be interested to hear what it is you think you "do" with group cohomology in a thesis, as opposed to arithmetic geometry. I was trying to indicate you're comparing chalk and cheese. Doing something with group cohomolgy is no harder than doing something with abelian varieties, surely? Incidentally, the ring H(G,k) is graded commutative and finitely generated thus yields and algebraic variety, the dimension of whcih is the rate of growth of a, well, never mind. But you see that actually there is a lot more to group cohomology than you first think.

I retract my foolish and uninformed comments. I have Brown's book on the cohomology of groups, which I actually think I could comprehend if I put some work into it (and had 20 extra IQ points), and Mumford's book on abelian varieties (plus loads of other stuff on arithmetic geometry) which seems of another order of difficulty altogether. And don't use all this fancy lingo on me: I have a deep-seated inferiority complex as it is.

Oh, the L-series person was not from the UK. But there is a distinct tendency to specialize here far too soon. If I ever get to be in such a position I would only ever accept PhD students who'd done part III (if I were allowed to make such demands). However, there is a brain drain across to the US, and many universities who don't see massive competition for places for a great number of reasons need to fill places or they lose funds. CAmbridge, with its excessive demands of a distinction in part III to obtain a PhD place there has successfully driven a lot of students away to places where such stresses aren't placed on them.

Yes. I wasn't aware that Cambridge required a distinction in Part III. I think it just serves as one more filter. I think virtually anyone accepted for part III has enough wit to complete at least an ordinary doctorate.

 

[matt grime]

The undergrad course has changed at Cambridge since I did part IIB so I can't check, but I could have sworn that someone did a course on algebraic geometry or actually commutative algebra, but I could well be misremembering it. As I didn't take any courses on Algebraic geometry I am not a reliable source.

If you want to see what cohomology of groups looks like as a research topic then I recommend Dave Benson's books on it, both volumes (mathwonk is at UGA with Dave, I think). Now that is a demanding pair of textbooks. Not the most demanding I've raead, though. That is Peter May's Algeberaic topology, which is about 250 pages long and covers all that you do in any graduate course on topology in the first 10 pages.

The competition for funded places has been so high that Cambridge, as is often its wont, simply asks people to get part III distinctions. Oxford and even Warwick have been known to ask for merits too (pass, merit and distinction are the 3 grades now [and fail]). However a couple of years ago this attitude had backfired, with a shortage of candidates in some areas.

 

[bombadillo]

The undergrad course has changed at Cambridge since I did part IIB so I can't check, but I could have sworn that someone did a course on algebraic geometry or actually commutative algebra, but I could well be misremembering it. As I didn't take any courses on Algebraic geometry I am not a reliable source.

I believe you. Even at my second-string school, I took undergrad courses in commutative algebra and algebraic curves. About 20 years ago, the book being used at Cambridge for the part IIB was Shafarevich's Basic Algebraic Geometry, which is also not easy for an undergrad. For commutative algebra, the course would probably be pitched at the level of Atiyah and Macdonald's Introduction to Commutative Algebra.

If you want to see what cohomology of groups looks like as a research topic then I recommend Dave Benson's books on it, both volumes (mathwonk is at UGA with Dave, I think). Now that is a demanding pair of textbooks. Not the most demanding I've raead, though. That is Peter May's Algeberaic topology, which is about 250 pages long and covers all that you do in any graduate course on topology in the first 10 pages.

I have the Benson books (Representations and Cohomology), and the May book (A Concise Course in Algebraic Topology). Since, however, I'm not in mathematics anymore, it's difficult to find time to open them.

 

[matt grime]

As someone said of Dave's book: "if I ever need to know anything about representation theory I look in there. Then, once I've found what I'm looking for, I go find some other book to explain it to me." I find them harder than Hartshorne since Hartshorne is expository and vaguely (i've not read it closely) intended to teach a course from. Where as the others are more encyclopedic with little explanation, in fact Dave's book almost stops giving exercises after the first couple of chapters.

 

[mathwonk]

If you want a tip on reading Hartshorne's book, you might be interested to know the chapters were not written in the order they are now assembled in for the book.

He first taught the chapter on curves in a course at Harvard in the 60's, and wrote up notes and handed them out, but not so complete as in the chapter in the book.

Next he taught a course on surfaces, but filled in and summarized some sheaf cohomology as needed along the way.

Then he moved to Berkeley I believe, and wrote the book in the opposite order, with all the heavy machinery up front (aside from a terse chapter on varieties at the beginning), and the applications to curves and surfaces in the back.

So the danger is a poor student tries to swallow all the sheaf theory and cohomology and duality, and general Riemann Roch theorem, derived functors, flat morphisms, semi continuity, formal functions, etc etc... in the first three chapters, but fizzles out before he reaches the part about actual curves and surfaces.

As I said Hartshorne himself did not originally teach it this way, nor write it this way. Note for example his comment on page 293: (first paragraph of chapter 4 on curves):

" Except for the [easy] proof of the Riemann Roch theorem, which uses [the statement of] Serre duality, we use very little of the fancy methods of schemes and cohomology. So if a reader is willing to accept the statement of the Riemann Roch theorem, he can read this chapter at a much earlier stage of his study of algebraic geometry. That may not be a bad idea pedagogically, because in that way he will see some applications of the general theory, and in particular will gain some respect for the significance of the general Riemann Roch theorem. In contrast the proof of the Riemann Roch theorem is not very enlightening."

One professor here starts his students essentially on chapter 4, and has them start working the exercises. That is also what Hartshorne did at Harvard. I.e. he just said here is the Riemann Roch theorem for curves, let's start using it to do some geometry.

 

[mathwonk]

There is certainly a premium on candidates for PhD degrees in US schools, especially US born, as matt said. there is real effort tor ecruit them, and that emans one cannot insist they know group cohomology and algebraic topology before showing up.


In the 1960's when i was an undergrad at harvard, the strong undergrads (not me) took 2 years of what were then grad courses, as undergrads, real (variable content) and complex analysis (Ahlfors or Cartan), and algebra (roughly Lang), and algebraic topology (say Spanier).

As freshmen or sophomores they often took an even more demanding course of Advanced calculus a la Loomis Sternberg or Dieudonne's foundations of modern analysis.

Nowadays I guarantee you that a person can be admitted to a PhD program some places (not Harvard) with barely any acquaintance with either topology or algebra. In fact I have recently felt, in some basic grad algebra courses, I was actually teaching some beginning grad students to do proofs for the first time.

In my whole 20+ year career of teaching graduate courses in algebraic curves and Riemann surfaces at a big state school, I cannot ever remember having a class that knew what either the implicit function theorem or Green's theorem said.

I also definitely do not follow the tradition from the 60's of giving automatic A's in grad courses. Here, I am reacting to my own unsuccessful attempt to first obtain a phD under that system, and not learning enough to get by without requirements.

On the other hand some students decline to take my courses for such reasons.

Basically getting a PhD involves noticing something that has not been done, in the context of learning about thigns that have been done, and beginning to inquire about it.
When I wasa young student I never asked such questions. After not succeeding in grad school, I read the wonderful little book: Psychology of Invention in the mathematical field, by Jacques Hadmard, and he told a story about a student coming to a professor in France to ask for a thesis problem.

The professor asked in return: "How long have you been taking our courses?" several years was the answer... and then he asked the student: " In all that time, have you never noticed anything that wanted further investigation?"

That blunt question hit me like a ton of bricks. I was that unimaginative unquestioning student. I set out to become a different kind, and eventually succeeded, with enormous assistance from a gifted advisor.

First I committed myself to learn the basic graduate content of books like Lang's Analysis II (now Real Analysis), Spivak's Calculus, then Calculus on Manifolds, and then Differential Geometry (I only got through volume 1), and Cartan's Analysis of one and several complex variables, by volunteering to teach them, sometimes as extra loads, or as seminars for bright undergrads. We also ran a seminar on De Rham's theorem.

Finally I was "ready" to go back to grad school. Guess what? Whjen I got there I was still overwhelmed by how hard it was. I worked night and day. They started me right off on Gunning and Rossi and Hormander's Several complex variables, and Kodaira - Morrow's Complex manifolds, and Griffiths Morgan and Friedlander's homotopy theory, spectral sequences and differential forms.

Fortunately a fine advisor offered me a chance to work on some concrete problems in Riemann surfaces, and I at length got a toe hold on one. It seemed I had a flair for geometry, and he taught me how to use it to make discoveries.

Of course after discovering something one has to prove it, and that takes a lot of technical power and tenacity. This is where all that one has learned or can find out pays off.

I got enough of a start to get my first job, but feared I would not be promoted unless I got more training, so i packed up my family and made a limited tour of good places where experts in my field could be found. I spent a couple years as visitor in some excellent places, absorbing all I could and living on peanuts, and then came home to stay.

 

[matt grime]

That raises an important topic that hasn't yet been broached in this thread. I do hope some of the original people are still reading.

Choosing an advisor is vitally important. Jon Baez has an article on it (and lots of other things well worth reading). He's at www.math.ucr.edu then follow the links (faculty and so on).


From my experience of the US system (at a very good state school) the idea that an incoming (US educated) graduate student would know what a group is would be getting far fetched fancy ideas. Never mind group cohomology. (I wrote my masters thesis equivalent (actually a part III essay, which has no direct translation to the world outside of Cambridge, shockingly) on Cohomology of groups, based upon reading Benson et al.)

That isn't to deny there are very many talented US students. Just that at some of the less prestigious places recruitment is an exercise in real politik.


I'll stay out of the Hartshorne debate - I've never read it properly (it sits on my office mate's shelf staring at me) and don't intend to for a while, though I cite some of his "more difficult stuff" in my thesis.

Actually, I must confess that I probably can't state Green's theorem (first year undergradutae material, surely) anymore. But then I quit doing applied (no offence if anyone thinks it's pure) maths in my second year of undergraduacy.


It's the same here too. If you don't take someone on for a funded place this year you may not get that funding next so it's better to have a bad student now for a better one in the future than lose all chance of having any students at all.


And if you want an analogy, bmobadillo, then, just as one can do algebraic geometry withou knowing hartshorne in detail, one can do group cohomology. If you want to understand how either are used today then you need to bite the bullet and learn about derived categories.

But i may be biased as my thesis is about general triangulated/derived categories al a Neeman, with padding from the safe shores of representation theory.

 

[JFo]

Thanks for continuing on with this thread. I have been reading it on a regular basis, but haven't replied since it has become too advanced for me to contribute anything. But that's a good thing. I've learned a lot from your interactions here on this thread. It's great to be able to listen in as a third party between the conversation of professional mathematicians. A lot of real life examples. Thanks so much.

Thanks bfd for the book recommendations and web link.

 

[bfd]

There's much more I want to add to this thread but today seems to not be the day to do it for me. I'll be sure to add more later during the coming week.

JFo: I'll be sure to add a few more links during this week as well. There's just so much information I'd like to get out there.

 

[CTS]

The information in this thread is great! It's nice for us students to hear what being/becoming a mathematician is like from those who have had the experience. Thanks for giving your advice, stories, etc.

 

[mathwonk]

If you want an opinion from a British mathematician on a suitable undergraduate book for algebraic geometry, check out Miles Reid's Undergraduate Algebraic Geometry. You wioll find it is genuinely written for undergraduates anywhere, with expertise and wit.

It is nowhere near even the level of Shafarevich, yet has some good content, and is quite readable.

Not the least of its appeal is Reid's ability and willingness to make some historical comments on the culture within algebraic geometry in the 60's when the Grothendieck school terrorized the community.

A sample of his writing style is a remark following a theorem whose proof is tedious, something like: "the reader may skip this proof without undue concern, as the author is a professional algebraic geometer of the highest moral fiber."

Reid is also the translator of a recent edition of Shafarevich's book.

 

[bombadillo]

If you want an opinion from a British mathematician on a suitable undergraduate book for algebraic geometry, check out Miles Reid's Undergraduate Algebraic Geometry. You wioll find it is genuinely written for undergraduates anywhere, with expertise and wit.

I have a copy, but I never really cared for it. I like his other book, titled, "Undergraduate Commutative Algebra", much better.

Ideally, anyone taking a course on algebraic curves should already have at least some minimal exposure to complex projective geometry, Riemann surfaces, and commutative algebra. In a well-taught algebraic curves course, these can be incorporated into the course itself. The tragedy of my class was that our instructor couldn't even answer why we were working in complex projective space -- he was using someone else's lecture notes, and didn't know the subject himself. And this was at one of the older London colleges.

There have been several good introductory books on algebraic curves published in the last several years:

1) Conics and Cubics. Bix. Springer.
2) Elementary Geometry of Algebraic Curves. Gibson. Cambridge.
3) Plane Algebraic Curves. Brieskorn and Knorrer. Birkhauser.
4) An Invitation to Algebraic Geometry. Smith,et al. Springer.
5) Complex Algebraic Curves. Kirwan. Cambridge.
6) Introduction to Commutative Algebra and Algebraic Geometry. Kunz. Birkhauser.
7) Algebraic Curves and Riemann Surfaces. Miranda. AMS
8) An Invitation to Arithmetic Geometry. Lorenzini. AMS.

I also like delving into the 2nd edition of Clemens' "A Scrapbook of Complex Curve Theory."

It's unfortunate that the subject tends to be taught exclusively at the graduate level, where a lot of abstract machinery is developed and used for application to varieties and schemes, and the geometry tends to be relegated to the background. Perhaps it's similar to Riemannian geometry -- one would like grad students to have been exposed to an undergrad course following something like O'Neill's "Elementary Differential Geometry", so that they've seen differential forms, pullbacks and pushforwards, covariant derivatives, and shape operators applied to the geometry of surfaces.

 

[bfd]

For those who are interested I found the following links:

http://mathforum.org/
(another good math forum)

http://www.math.uiowa.edu/~tomforde/GradGuide.pdf
(grad school guide for Dartmouth grad math students but as the author Mark Tomforde says "Although it was written with Dartmouth graduate students in mind, many of its suggestions apply to students in any mathematics graduate program." Includes sections on becoming a part of the math research community and debunks myths about math and graduate school)

http://www.math.uiowa.edu/~tomforde/gradstudents.html
(more helpful information from Mark Tomforde for math graduate students. Includes links for those who are looking for jobs after obtaining the degree)

http://www.maa.org/
(inlcudes a primer for graduate school)

http://www.siam.org/
(Great site for applied mathematics)

Also I found these journals helpful for many of my questions:

-Mathematics Teacher
-Mathematics Magazine

 

[mathwonk]

we had a nice undergraduate version of the course on curves successfully taught recently using the book, plane algebraic curves, by gerd fischer.

an older out of print book, that is excellent, is fulton's algebraic curves. in particular, fulton proves all the main results on curves, bezout, resolution of singularities, and riemann roch.

 

[mathwonk]

Thats an interesting list of nice books, bombadillo.

If I was unclear, I was not recommending a book to learn curves from, merely indicating that hartshorne is not at all the level thought appropriate for undergrads by at least one famous algebraic geometer in britain.

I enjoyed perusing your list as well, a few of which i own. they are very diverse in nature, and certainly include some very elementary entry level books. I add a few remarks.



1) Conics and Cubics. Bix. Springer.

apparently extremely elementary, discussing only the curves in the title, although cubics are already a sophisticated topic, at the upper end of their study, having played the lead role in wiles solution of fermats last theorem.

2) Elementary Geometry of Algebraic Curves. Gibson. Cambridge.

another quite elementary treatment, again maxing out pretty much at the addition law on cubics it seems.

3) Plane Algebraic Curves. Brieskorn and Knorrer. Birkhauser.

an impressive tome of over 700 pages, with sections on history, elementary algebraic methods (resultants), and more sophisticated topological methods, resolution of singularities and integrals of differential forms, by a master researcher (Brieskorn) on the theory and practice of analyzing singularities. far froma self contaiend boiok, but very illuminating and interesting, he discusses some prettya dvanced topics such as hodge theory and algebraic derham hypercohomology, but seems never to even prove the riemann roch theorem. his approach is often to tell you the many facts he knows about the subject, and then just pull in whatever heavy machinery he needs to deduce them.

tha advantage of his un - self - contained approach is he tells you a lot you would not hear if he were restricted to what he can fully present with all the background.

4) An Invitation to Algebraic Geometry. Smith, et al. Springer.

apparently kind of an informal overview of what goes on in algebraic geometry, not restricted to curves really, nor an introduction to them.


5) Complex Algebraic Curves. Kirwan. Cambridge.

i could not get hold of a copy, but the author is a well known algebraic geometer, a student of Atiyah I believe. this book should be a good treatment.


6) Introduction to Commutative Algebra and Algebraic Geometry. Kunz. Birkhauser.
again not an introduction to curves, but a general introduction to commutative algebra and algebraic geometry.


7) Algebraic Curves and Riemann Surfaces. Miranda. AMS

this really is a thorough going introduction to curves over the complex numbers, with complete proofs of the main results, such as riemann roch theorem (a la Weil and Serre)

8) An Invitation to Arithmetic Geometry. Lorenzini. AMS.

a lovely introduction to the arithmetic side of the subject, the algebraic concept of integral closure, due to zariski, discriminants, ideal class group, and again reproducing the proof of Weil using ideles, (did Serre add anything?) for riemann roch.

so the book to choose depends on what you want out: you can have fun playing around with elementary conics and cubics, or go for a development that includes riemann roch, and hopefully also some significant applications, to jacobian varieties and projective models of curves, such as found in p. griffiths' china lectures, introduction to algebraic curves. he proves most of the basic results (bezout, riemann roch) and applies them too.

 

[bombadillo]

we had a nice undergraduate version of the course on curves successfully taught recently using the book, plane algebraic curves, by gerd fischer.

an older out of print book, that is excellent, is fulton's algebraic curves. in particular, fulton proves all the main results on curves, bezout, resolution of singularities, and riemann roch.

I have the Fischer book, and I have the Fulton book in storage in England. Another title which I may buy sight unseen is Hulek's Elementary Algebraic Geometry, published by the AMS. I'm also looking forward to Kunz's book on plane algebraic curves, which will be puiblished by Birkhauser, and due in a month or two.

I did attend a grad course on Riemann surfaces in London, but the nincompoop of a lecturer didn't even get to the Riemann-Roch theorem.

The problem with the British university system was its rapid growth during the 60s. At that time anyone with a PhD and one or two indifferent papers could get a lecturership at the rapidly expanding universities. Needless to say, most of these lecturers were no good. Once hired, it was next to impossible to chuck them out, and they've continued to be a blight on the English academic scene for decades. Brilliant young scholars have been unable to obtain academic appointments simply because these was no room anywhere. With these mediocre time servers as lecturers, the quality of undergraduate math education has continued to slide downhill, with the possible exception of Oxbridge and Warwick. Of course, this isn't the whole story: school-leavers are arriving at universities with less technical virtuosity than of old.

 

[mathwonk]

i am not an expert on riemann roch, but do appreciate it and am a huge fan of it. I have writtena set of notes on it with an account of riemanns own proof as wella s a sketch of serre weil's proof,a nd also a sheaf theoretic version in higher dimensions and an inmtroduction to hirzebruch's version. if you like i'll send you a pdf version if you send me your email.

it has a couple typos but not too many i hope.

to make a long story short, riemann roch is an analog of the mittag leffler theorem for compact non planar surfaces.

i.e. on a compact two dimensional surface with complex structure, by the residue theorem, given a non constant meromorphic function f, then for every holomorphic differential one - form w, the residue of the meromorphic differential fw is zero. conversely a set {Pi} of local principal parts forms the polar part of a global meromorphic function f if and only if for every holomorphic differential w, the residue of the system {Piw} is zero.


This converse of the residue theorem is almost the riemann roch theorem. that theorem goes further and counts the number of meromorphic functions with given polar behavior.

in higher dimensions the theorem is weaker, and only computes by topological data, the alternating sum of a sequence of dimensions of vector spaces, one of which is the number of meromorphic functions with given polar behavior. there is however a criterion for this alternating sum to equal the desired number, due to kodaira, which holds for high degree polar behavior, but not always.

the theorem has many consequences, e.g. every complex curve homeomorphic to a projective line is also complex holomorphically isomorphic to one, given any elliptic curve (curve of genus one) and any three points on it, there is an embedding of that curve in the complex projective plane as a cubic with those three points collinear. given any curve of genus 2 there is a realization of that curve as a double cover of the projective line with 6 branch points. any curve of genus three is either a smooth plane quartic curve or a double cover of the line with 8 branch points. etc...