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Programs Best Places to Recieve a Degree (Maths) From?

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Okay lets all chill out here - I was just interested in what other people thought as I am not an academic - and so I was jsut throwing it out there - I have my opinion like everyone else but i was looking for ideas for other places - so thank you to those that have given me food for thought!!

-NewScientist
 
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From two seperate sources apparently the Top Ten Universities 2004 were:

1. Harvard University
2. Stanford University
3. University of Cambridge
4. University of California - Berkley
5. Massachusetts Institute of Technology (MIT)
6. California Institute of Technology (Caltech)
7. Princeton University
8. Oxford University
9. Columbia University
10. University of Chicago

And from The Times 2004, they were:

1. Harvard University
2. University of California at Berkeley
3. Massachusetts Institute of Technology (MIT)
4. California Institute of Technology (Caltech)
5. University of Oxford
6. University of Cambridge
7. Stanford University
8. Yale University
9. Princeton University
10. ETH Zurich

The Top Ten Universities vary so much from source to source, generally the Top 50 Universities will give you a damn good Degree.
 

cronxeh

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I dont get you guys. What is the point of any of those top 10 ratings if the faculty at any, and many, state and city colleges and universities is alumni from those top10 schools? Is it the flaw of logic or just inept inability to understand what 'knowledge' means?`
 

mathwonk

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well i am a faculty member at a state school, so I am a student of a graduate of one of those schools, and many of my colleagues are graduates of those schools. still there is a difference between the strength of the faculty at those schools and the strength of their graduates.

so I am sort of a top "10 college twice removed" faculty member, and my colleagues are "once removed", and the faculty at those places are the best.

lets take a bilbical example, jesus was the master teacher, and his 12 disciples were his personal students students. then they went out and taught further. being a student of say james or peter is not the same as hearing it from the masters lips directly.
 
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Since math can be done (more or less) anywhere, I never understood how college choice mattered...
I suggest tearing through as much math as you can, and hopefully, you'll know what you want to focus on by the time you graduate. Then decide where you want to go based on specific fields/professors, not subjective ratings.
 
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College choice matters a great deal. The teaching and extra-studial word (such as applications of math into physics, computing etc) is different at different institutions.

Saying college choice doesn't matter is like saying that going to the north sea is just the same as the carribean because they both have water!

-NS
 
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In my opinion a University with family accomodation which is nice and quiet is good enough for me, like York or Durham. As long as the University lies within the top 20 for Physics in the UK I am not particularly bothered, the core content is generally the same.

The location and facilities are most important to me.
 

brewnog

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Peter.E said:
In my opinion a University with family accomodation which is nice and quiet is good enough for me, like York or Durham. As long as the University lies within the top 20 for Physics in the UK I am not particularly bothered, the core content is generally the same.

The location and facilities are most important to me.

This seems pretty sensible. In terms of getting a degree, I would have been happy with any of the top 20 UK universities, - getting a Cambridge degree would have had no advantage over the one I've got, for example. So I chose based on the city environment, student life, location, and cost of living.
 
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Hey, I sent a link of this to a friend and he responded thus:

Getting a degree from Oxbrdige has several advantages in my view:
1) Any networking you achieve is with some of the best and richest in their fields (netwroking is getting a group of contacts - people you know etc).
2) The Kudos of oxbridge is massive and a degree from there gives your CV a huge boost - if you have a degree from Salford, and somebody else has the same degree from Cambridge, the guy from cambridge will be chosen
3) The lecturers and professors at Oxbridge tend to be near the top of the best professors around nad are great educators
4) The level of degree achieved there is greater than at most other universities. This is why you used to be able to pay £50 to get a BA turned into an Ma.

-NS
 

brewnog

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NewScientist said:
Getting a degree from Oxbrdige has several advantages in my view:


1) Any networking you achieve is with some of the best and richest in their fields (netwroking is getting a group of contacts - people you know etc).
Many people can get on perfectly well without an old-boys network, thank you very much!

2) The Kudos of oxbridge is massive and a degree from there gives your CV a huge boost - if you have a degree from Salford, and somebody else has the same degree from Cambridge, the guy from cambridge will be chosen
Depending on what field you're studying (and yes, I know your thread was just about maths), this just doesn't hold water any more. Using Salford is a poor example, because in academic terms it does not compete with Oxbridge. Statistics show that an engineering degree from my university is worth more, in the eyes of employers, than an equivalent Oxbridge degree. (For arts-based subjects, however, the reputation of the Oxbridge universities is still as powerful as ever.)

Your example is poor: A University of Nottingham graduate is more likely to be selected over a Nottingham Trent graduate, for a technical job. So what?

3) The lecturers and professors at Oxbridge tend to be near the top of the best professors around nad are great educators
Yes, of course they are. But other world-class universities also have lecturers and professors at the top. Several of my lecturers served as lecturers at Oxbridge. So what?

4) The level of degree achieved there is greater than at most other universities.
What do you mean?
 
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I'm calling him and typing now! Here goes.....

Point 1 : Yes, many people do but I came from a poor family and I desperately wanted to work in London but could not afford it, however, as I had been at Cambridge, I had a number of friends who were very keen to let me use their houses in London. Without 'the network' I wouldn't hav ebeen able to.

Point 2 : Engineering is a poor example too for a number of reasons. Engineering is on the decline, and also oxbridge's approach is far too technical and theory based than a degree from elsewhere which is more practical based - and therefore more useful - however we are both using poor example so hey! My point was that the kudos of Cambridge is an important factor.

Point 3 : I was making the point that the level of teaching at Oxbridge is higher than at a great deal (not all by a long way) of other universities.

Point 4 : Well, the subject content included at Oxbridge (can) be more demanding than at other Unis. It is based on this why a Oxon or Cams degree is percieved as better.

Anyway, I (NewScientist) am going for a drink so that is why I won't reply for a while!!

-NS
 

brewnog

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NewScientist said:
Point 1 : Yes, many people do but I came from a poor family and I desperately wanted to work in London but could not afford it, however, as I had been at Cambridge, I had a number of friends who were very keen to let me use their houses in London. Without 'the network' I wouldn't hav ebeen able to.
What's Cambridge got to do with that? I've made plenty of friends at uni with whom I could stay with in London! But yes, the old boys network is a quirk of Oxbridge

Point 2 : Engineering is a poor example too for a number of reasons. Engineering is on the decline, and also oxbridge's approach is far too technical and theory based than a degree from elsewhere which is more practical based - and therefore more useful - however we are both using poor example so hey! My point was that the kudos of Cambridge is an important factor.
Engineering is on the decline? I'm not even going to try and address that one, especially since you don't have a source.

Oxbridge does indeed have kudos, but a lot of other universities do too. I know for a fact that my course contained almost exactly the same modules as its Cambridge counterpart, and the quality of teaching is of a similar standard. The one thing that Cambridge has is it's name. This does not mean that it is necessarily better for teaching or research than other top universities. Just take a look at the Times Good University Guide. Yes, Cambridge and Oxford appear in the top 20 for many subjects, but you'll see a lot (perhaps 5-10) of other universities consistently making the top 20 too.

Just glancing through, the courses I see Cambridge and Oxford appearing in the top 10 for, I consistently see Bath, Imperial, Sheffield, Nottingham, Warwick, Queen's (amongst others) appearing up there too, often above both Oxbridge universities.


Point 3 : I was making the point that the level of teaching at Oxbridge is higher than at a great deal (not all by a long way) of other universities.
Yes, and I was making the point that the teaching at, say, Durham, Birmingham, Warwick, Manchester and Imperial is also higher than at a great deal of other universities. Cambridge and Oxford are NOT unique in this respect.

Point 4 : Well, the subject content included at Oxbridge (can) be more demanding than at other Unis. It is based on this why a Oxon or Cams degree is percieved as better.
It can be, but again, this is not a unique feature of Oxbridge, and you'll find that subject content at other universities (especially in technical disciplines) is identical.

Oxford and Cambridge are NOT the only good universities in the UK, and they are NOT always the best for individual fields. The one thing that sets Oxbridge apart is it's reputation and history, and NOT any particular academic advantage over other universities.

For example (and here, I'm afraid I'm using the 2001 version!), the Times Good University Guide rates Cambridge as being #1 overall for some courses (such as Architecture), yet other courses, such as Business, don't even make it into the top 20.

While Cambridge and Oxford tend to appear frequently in the top twenty overall for arts and humanities (history, English, music, geography, French etc), they are often overshadowed by other universities in technical subjects.
 

mathwonk

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it is not impossible to be well educated anywhere but there are many reasons "better" schools are really better places to learn.

1) the other students are both smarter and harder working, so you learn more from them than at a weaker school.

2) with better students the faculty are more motivated to do a good job in the classroom of actually teaching good material instead of focusing on explaining basics to the dull ones.

3) the faculty are able to use better books for the courses and generally raise the level of the courses when they can assume everyone in the class is "on board".

4) the students are also more motivated by the higher level of standards at the better school, to do their best. (after a year or more of struggling to keep up or catch up with people at a top school, i actually found myself getting "smarter", i.e. quicker at noticing things than before.

5) the top places are like magnets attracting the latest information, before it is available elsewhere. i have been handed an unpublished paper at a top school, that a professor had received from a colleague and asked to present it in a seminar, before it became current anywhere else. the ideas in that paper later played a role in some of my best work.

6) the professors are so strong at top places they know a lot that is not in books or papers at all, and cannot be learned elsewhere. a prof at a top place once responded to a challenge from me by producing an argument no one else knew anywhere, since he never made it public, and this idea too played a role in research done by me and by friends of mine with whom i shared it.

some "negatives":

I once proved something that did not seem to impress the profs at a top place so i did not publish it. 2 years later I heard someone speak on this same result at an international conference, and it became his chance to publish it, and not mine.

I was so content at the approval of my advisors at a top place that I neglected to publish even things that they did like, thinking their approval was enough. when i left and went elsewhere, people who did not know enough to question me, evaluated me more on my publications than on my knowledge.
 
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mathwonk said:
the ideas in that paper later played a role in some of my best work.
which is ???

marlon
 

mathwonk

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it was a simple idea that in order to show a particular algebraic variety is irreducible, i.e. has only one piece, first proceed by finding a point which must lie on every possible piece. Then you have reduced the global irreducibility problem to a local one, namely local irreducibility at that one point. If you are lucky then that can be proven by then showing the variety has an irreducible tangent cone at that point.

We were not actually able to do that, and besides we were adapting the idea to showing a certain variety had exactly two components instead of only one. so we developed a tool for estimating the multiplicity of certain discriminant loci (the type of variety we were considering), and showed that as you approached our special point along one of the components, then multiplicity went up by exactly one. hence the moving point could have come to lie on only one more component. Since we had already shown it did lie on all possible components, we were done!

does this answer your question, in some way?

if this ids not your area, notice that showing a variety is irreducible, or counting its coimponents, is analogous to showing a number is prime, or an ideal is prime, and it is technically often very dificult. in fact we won a little award for this work, and the general idea was developed by others into a general theory of non isolated "milnor numbers".

we never did get a good criterion for non emptiness of higher dimensional milnor loci though, analogous to milnor's work for isolated singularities. i am still curious about that but it has been almost 20 years since i thought about it.
 
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matt grime

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From the perspective of gaining an undergraduate degree in 3 years Cambridge* is the best, in my opinioon, in the US and UK**. what does best mean in this context though? indeed, if we were to believe the Guardian (newspaper) then St Andrews, in their circa 2000 course evaluation, came top for maths. Cambridge rarely appears in the top 5. my reasons for placing Cambridge in a different league from other UK universities are vastly different from the criteria they were using, and are roughly as follows (remember i am not including russia, japan, or degrees one obtains after the age of 21/22)

1. the course content is the most demanding, and it heavily emphasizes intellectual rigour. deep mathematical understanding is preferred over simple plug and chug courses as i taught in the US. (for instance if you attempt two questions it is better to get one fully correct and one wrong than to get both half correct. the questions are long here, not simple one liners. your final mark comes down to four 3 hour exams, one of which is marked upon your best 3 questions; you are allowed to attempt 4. the other exams vary but in 3 hours doing4 short and 3 long questions was, if i recall correctly, moer than enough to get you a good 2.1 or better)

2. the teaching you recieve is (almost) without compare: tutorial groups with 2 students and 1 teacher (only oxford can match it)

3. the exams are tough but fair eveni if the resulting mark scheme is only understandable by those obtaining a 2.1 or higher.

4. the facilities are better than (almost) anywhere in the UK, and rival their US counterparts. however, as cambridge turns out 250 graduates in maths a year to harvard's 10 the fact that cambridge has more books per student should be a relative "win" for cambridge

5. the college system makes for an incredibly good place to work. it is cheaper to be a student at cambridge than almost anywhere else. the sense of community creates an ideal way to cope with the stress of the extra expectations placed upon you.


but this isn't to say it is without its faults. there used to be (probably still is) a computing assignment that was heavily biased in favour of those who could program before they arrived. you weren't, for instance, taught how to do so by the university. the claim was that the quality of the progam was immaterial. of course that may be true if you can make it work in the first place, which is a quality issue. the marks for this were sufficient that it was pssoible ot have obtained a 2.2 before you even entered the exam hall. the marks were not transferrable, ie there was not an extra exam paper you could sit to make up for it.

other comments: looking at other good universities in the UK, cambridge has the edge because it tends to do in 3 what they do in 4, and this includes oxford *for maths*. cambridge has higher selection standards, though they aren't foolproof. germany's degrees are arguably a higher standard in parts but they take much longer to obtain. when doing part three i noticed how much better prepared the german students were for the course, but also how much older they were. i know little about russian universities hence i excluded them. american *undergraduate* courses in mathematics, even the ivy league ones, are several yards off the pace of their euiopean counter parts. a good student at cambridge would after 3 years have been taught (if they chose to) complex analysis up to and including the construction of riemann surfaces, representation theory of finite groups, and some lie groups/algebras, differential manifolds, functional analysis (say the stone weierstrass theorem, spectral theory of self adjoint operators), algebraic curves (riemann roch), number fields, dynamical systems, measure theory, markov chains, linear programming, algebraic topology and geometry. if you chose to do a 4th year you could be taking courses like "infinite descent and ellpitic galois cohomology", but i was excluding part 3 from the discussion. in any case cambridge offers far more courses than you can ever take, so many that they have to deliberatley clash lectures hoping that the two clashes are so far removed no one wants to do both courses. it is also entirely possible to obtain a fantasitcally good first without having done any of those courses at all and instead done QM, SR, GR, fluid dynamics, partial differential equations etc. compare that, within the uk, to Bristol where I now work and by the 4th year it is possible to have done a small fraction of these courses, and from what i have seen of the syllabuses to nowhere near as much depth.

if we are to extend to higher degrees then it becomes much more interesting. in short, as an undergraduate inst. i could name 10-20 places in the UK that would be ahead of any in the US, and, although i know little about mainland euopean schools, i could probably extend that to include 50-60 places in europe that are ahead of the US, and add a few more if i include australia.

if we pass to graduate programs then it is alomst the exact opposite as is implied by the incredible nuimber of euopean, indian, korean, and australasian students in the US grad schools. they are there for a reason and it isn't (directly) financial or because they like Wendy's.



* there is no need to make a distinction between the colleges really: you are all lectured by the same people and predominantly supervised by the same people. indeed i was supervised by some trinity fellows as an undergrad and the cleverest student i taught there was not from trinity

** other euopean countries have a different attitude towards education, as does Australia. France for instance has universal education free to all, but with a drop out rate of 50% after 1 year. geographically australian students are predominantly restrcited to going to their local uni. german degrees, as i have mentioned, are long and rigorous, but perhaps too long to be directly compared. japan, russia, and the powerhouse of Hungary are alien to me.
 
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mathwonk

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Matt, that sounds really good, and true. I have a young friend now at Cambridge as a Gates fellow and he loves it there. And I believe Harvard is an attempt to imitate Cambridge in the US. Is it feasible for a (strong) American high school student to aspire to admission to Cambridge? If so, how would they go about it? Just "apply"?
 

mathwonk

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Another remark on local reducibilitya nd irreducibility of varieties for the interested. Neither local nor global reducibility imply each other, but a connected variety which is globally reduciblwe is also locally so at any point common to two or more components.


This is used to prove that such a variety is singular (not a manifold) at such points as follows:

for affine varieties, and all varieties are locally affine, irreducibilioty corresponds precisely to the ring of functions, or local ring of functions being a domain, i.e. to the ideal of functions defining the bariety being prime.

then there is a big theorem that at all smooth (non singular, manifold) points, the local is a regular local ring, and also that all such rings are domains, u.f.d.'s in fact (after Auslander and Buchsbaum in general).

so every non singular variety is everywhere locally irreducible. Hence the union of two varieties is locally reducible at any intersection point, hence also singular.

this interplay between zero divisors and components is just one aspect of the beautiful relationship between algebra and geometry revealed in modern algebraic geometry.
 
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mathwonk said:
Is it feasible for a (strong) American high school student to aspire to admission to Cambridge? If so, how would they go about it? Just "apply"?
I don't see why not. They would have to apply through UCAS, just like we do. There is a deadline for Oxbridge applications (15th October, IIRC. It's quite early) and you're only allowed to apply to either Oxford or Cambridge as an undergraduate.
 
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Nylex said:
) and you're only allowed to apply to either Oxford or Cambridge as an undergraduate.
No, you can apply to both, but whichever is second in your choices will instantly ignore you and bin your application!

-NS
 
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NewScientist said:
No, you can apply to both, but whichever is second in your choices will instantly ignore you and bin your application!

-NS
On the form, you don't list choices in order of preference, they're listed alphabetically. UCAS would probably send the form back to you, if you put both down.
 

matt grime

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mathwonk said:
Matt, that sounds really good, and true. I have a young friend now at Cambridge as a Gates fellow and he loves it there. And I believe Harvard is an attempt to imitate Cambridge in the US. Is it feasible for a (strong) American high school student to aspire to admission to Cambridge? If so, how would they go about it? Just "apply"?
some background.

in the UK at A-level there are two maths qualifications: maths and futher maths. cambridge students are expected to have taken both of these. i think technically you can apply and get in with single maths but it is just that almost every applicant has both. of course there are small schools unable to offer both so it is flexible. thus the modern view is that it is not assumed they have both qualifications, and given the relatively small core overlap from different A-level exam grades, the key material is "retaguht" however it is taught in such a way that the 2 years of material of these A-levels (which comprise at least half of the material you learn between the ages of 16 and 18) is given, collectively, about 6 hours. you will be given no worked examples, for instance you will be given the statement of de Moivre's theorem (the proof will be left as an exercise) and that's it, next topic.

it would thus be beneficial for the incoming student to know

complex numbers, 2 and 3d real vectors, matrices, determinants of 2x2 and 3x3 matrices, dot and cross product, 2nd order differential equations, all their trig identities, integrals via substition etc, hyperbolic trig

but if they don't they will get a crash course in it.

there is also a crash course in physics for those who didn't take a-level physics. the fact it is there means we need say no more about it.

there is another aspect though, entrance exams. cambridge sets and administers STEP, sixth term examination papers, in mathematics that are almost always required for entry (for home students). these are obtainable over the internet and give a good indication of the level required.

http://www.maths.cam.ac.uk/undergrad/admissionsinfo/admissionsguide/text/node6.html [Broken]

here is what the university itself says (none maths specific)

http://www.cam.ac.uk/admissions/undergraduate/international/

but back to what i know of it as a student there.
there are lots of exchange schemes with the US so that students may experience cambridge for a term or so. there is certainly one with MIT in engineering. even with these students the difference in the system is dramatic. some were suprised to find a mark of 0 on their work as they'd just written down all of the information they thought relevant to the question which would be 'positively' graded in their own classes but was ignored. we (cambridge) do not have mid term exams, nor multiple choice finals, there is no cram and forget, no pulling an all nighter the day before a test: that would just be unfeasible given that you have to reproduce potentially anything from 100 to 150 lectures of material (that may not sound much but the 30 lecture course i taught at penn state had sufficient material in it for perhaps 1 lecture in the style i was used to, less as we weren't proving the results).


if the prospective student is prepared for an entirely different culture then it is feasible since knowledge isn't what cambridge look for in a student, it is ability. if they have the ability we can teach them the knowledge. (one thing that cambridge fosters is a sense of pride: i still refer to it as if i am there. this attitude is common in the US, from my experience, but rare in the UK and only the college systems of cambridge oxford and durham seem to have that effect).


the fees would worry me, but then i was paid to be an undergraduate and i think all higher ed should be free and the fact that students are expected to take out loans is unfair. perhaps to a US student they would be perfectly reasonable.
 
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Ooh, I forgot about STEP :eek:.
 

matt grime

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NewScientist said:
No, you can apply to both, but whichever is second in your choices will instantly ignore you and bin your application!

-NS
No, you may not apply to both, as Nylex says. (I don't think this has changed recently at any rate)
 
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Nylex said:
On the form, you don't list choices in order of preference, they're listed alphabetically. UCAS would probably send the form back to you, if you put both down.
Damn it, how times have changed since I was that age!
 

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