# Does it matter where I do my PhD?

I did object to the exams though as they seemed to expect obscene amounts of scholarship over problem solving; while that's necessary and useful "in-the-wild", I still maintain that it's more important for a mathematician to be able to present a creative solution to a tough problem than have an encyclopedic memory.

There used to be a time -- though they've stopped doing it -- when they would sneak in an unsolved research problem into the exam papers. And occasionally these problems would be solved -- by one of your freakishly clever people -- under exam constraints.

#### nickaelwen

Hey debaters, after reading all your comments. I am really jealous of you guys. You already got the chance to experience the coolest course, why do you so care about how other people see about it? But I recognize one thing, you all try to compare math on a single line, like cambridge is 100 and other places are 90 so on. To me, it just not true. My A-level sucks, my undergrad was in BA finance. But i got first class. My master also sucks because it was very hard for me,but I got distinction in financial mathematics. The thing is, haven't done good A-level math (actually I did not even take A-level math exams). I can understand calculus, actually only one idea makes me understand the whole calculus, that is the beauty of "limits". During masters, I engaged differential equations, which enlightens my eyes and the beauty of modeling differentials can give you the whole picture of the underlying variables, and then from ordinary to partial. I master them because I can see the beauty and powerfulness of it, actually all the works I done best had these kind of moments. Unlike you, you study it because it is super hard and very few can master it.

Why don't you tell me in simple words what these theory can do? Lie group, finite group, other different groups, it looks to me they all groups? Ok, so what group theory can do?

Secondly, I tried to study Cambridge's mathematical tripos according to the list of part 1,2 and 3. But no luck, by looking at recommended texts and try to study on my own just are not working. I never find those very important intuitions I described above, so I quit. Maybe you have some insights and can help me? Just give me some central idea of those theories and some blue prints of what it can do?

#### Andy R

What is your argument? 50% of people doing part III come from universities other than cantab. The fact that cantab admits them and thinks they'll be able to cope is testament to the fact that Cambridge's standard of education is not head and shoulders above decent courses at other decent UK universities, and that those who do well in said courses will be able to cope just fine with part III or a cantab phd.
Isn't this really saying that intelligent people exist at other Universities? I don't really think that anyone doubts this (even though I'm likely to argue that they show up in higher densities at Oxbridge as a consequence of admissions tests, STEP and interviews). There will be self-selection for Part III (being the traditional preparation for a PhD in the UK), so you're always going to get the strongest external applicants applying: although I did well, I found the Oxford MFoCS to be very tough even though I got a first in my BA. If I'd scraped it, I think I'd have really struggled and I can't imagine that Part III would be any easier, Cambridge let their own lot continue automatically with a 65. While I'm not saying that all Cambridge students are more intelligent than all Leeds students, say, and while I do believe that most UK Russell group BA degrees are broadly comparable, it does seem to be the case that Cambridge are offering a very more rigorous and intensive curriculum, even for Part I and II and this higher standard of education ends up being far better preparation for Part III. Evidence that this is the case for Oxford Physics, at least, can be found here: http://www.physics.ox.ac.uk/teach/Exam_Reports/2010-2011/ExternalExaminersReports_Complete.pdf ... I'll also base this on the experience of my room-mate of two years who failed his first year at Cambridge, moved to Bristol and got a first. He's highly intelligent and pretty well motivated but just couldn't cope with the pace (I'm not really convinced that quickness are intelligence are so closely related).

And as I mentioned, what courses they offer is frankly superfluous, and is often at the whim of the faculty. One course some of my friends took last year repeated the part II course for 50% of the material, and the other 50% was the lecturers personal way of solving a particular differential equation encountered in medical imaging. That's a far cry from the standard of the part III black holes course, for example. There is no standardisation. You cannot add apples to oranges. The courses are not universally good, or challenging.
That's true and it annoys the hell out of me. I often regret not taking the easy courses (mostly to be found in the form of a few suspect stats courses and two or three softer applied courses) and walking out with a stellar transcript (I knew a guy who learned numerical linear algebra in a week because he didn't understand Algebraic Geometry, he's now doing a DPhil...). Great pity that there isn't a way to compare between subjects and modules with any degree of fairness.
And further to my original point, if you offer a course on string theory but the only questions you get asked are bookwork, what is the point in that? Sure, you've memorise some equations and derivations, but have you learnt how to be a professional mathematician or physicist, more so than at any other institution? No. And that ultimately is the point of a masters course.
There's the part III essay (most PhD applicants will do it but it's usually expository instead of research) and the courses are very tough: not even Oxford do a course on Hodge Theory. It's pretty telling when you see the lecturer struggling: just to understand some of the stuff is hard and develops some pretty sophisticated thinking and, tbh, *most* people can't memorize by rote without developing the understanding. I completely agree with the points about excessive amount of bookwork though.
I can hands down say I'd be much more impressed with someone who has a 2:1 at a russell group uni but has done a decent undergraduate/masters project than someone who got a distinction in part III, where you do no original research or project.

Distinction at Part III doesn't always make good mathematicians either. Miles Reid (FRS) barely passed Part III and there are countless senior wranglers who have never amounted to anything. Although most eminent UK mathematicians seem to be of COW lineage (with emphasis on the CO bit) there are always exceptions and even then, it's quite likely that this is a consequence of having big departments and for historical reasons. Caucher Birkar is one truly dazzling exception (by many accounts). It's very hard to predict who'll make a good mathematician; the best way is usually to see if they've done anything within 5 years of the PhD, if not they'll probably never do anything.

#### Barny

There used to be a time -- though they've stopped doing it -- when they would sneak in an unsolved research problem into the exam papers. And occasionally these problems would be solved -- by one of your freakishly clever people -- under exam constraints.
Isn't this really saying that intelligent people exist at other Universities? I don't really think that anyone doubts this (even though I'm likely to argue that they show up in higher densities at Oxbridge as a consequence of admissions tests, STEP and interviews). There will be self-selection for Part III (being the traditional preparation for a PhD in the UK), so you're always going to get the strongest external applicants applying: although I did well, I found the Oxford MFoCS to be very tough even though I got a first in my BA. If I'd scraped it, I think I'd have really struggled and I can't imagine that Part III would be any easier, Cambridge let their own lot continue automatically with a 65. While I'm not saying that all Cambridge students are more intelligent than all Leeds students, say, and while I do believe that most UK Russell group BA degrees are broadly comparable, it does seem to be the case that Cambridge are offering a very more rigorous and intensive curriculum, even for Part I and II and this higher standard of education ends up being far better preparation for Part III. Evidence that this is the case for Oxford Physics, at least, can be found here: http://www.physics.ox.ac.uk/teach/Exam_Reports/2010-2011/ExternalExaminersReports_Complete.pdf ... I'll also base this on the experience of my room-mate of two years who failed his first year at Cambridge, moved to Bristol and got a first. He's highly intelligent and pretty well motivated but just couldn't cope with the pace (I'm not really convinced that quickness are intelligence are so closely related).

That's true and it annoys the hell out of me. I often regret not taking the easy courses (mostly to be found in the form of a few suspect stats courses and two or three softer applied courses) and walking out with a stellar transcript (I knew a guy who learned numerical linear algebra in a week because he didn't understand Algebraic Geometry, he's now doing a DPhil...). Great pity that there isn't a way to compare between subjects and modules with any degree of fairness.

There's the part III essay (most PhD applicants will do it but it's usually expository instead of research) and the courses are very tough: not even Oxford do a course on Hodge Theory. It's pretty telling when you see the lecturer struggling: just to understand some of the stuff is hard and develops some pretty sophisticated thinking and, tbh, *most* people can't memorize by rote without developing the understanding. I completely agree with the points about excessive amount of bookwork though.

Distinction at Part III doesn't always make good mathematicians either. Miles Reid (FRS) barely passed Part III and there are countless senior wranglers who have never amounted to anything. Although most eminent UK mathematicians seem to be of COW lineage (with emphasis on the CO bit) there are always exceptions and even then, it's quite likely that this is a consequence of having big departments and for historical reasons. Caucher Birkar is one truly dazzling exception (by many accounts). It's very hard to predict who'll make a good mathematician; the best way is usually to see if they've done anything within 5 years of the PhD, if not they'll probably never do anything.

Well exactly. My point is simply that you can't determine who is or isn't good by which university they went to, and I believe that very, very strongly. The average at Cantab might be higher, but it's meaningless, because the average students are exactly that - average, and you wouldn't them to be anything other than accountants or similar. The competition is between those who are coming top of their year at their respective institutions, and those coming top of their year at Bristol can stand up to anyone at Cambridge in terms of intellect. Sure, the BA (Cantab) may have memorised more mathematics, but that's meaningless when it comes to research, where skill is measured in terms of originality and depth of understanding.

And btw the part III essay is a joke. It takes a couple of weeks, nothing comparable to an in-depth research project like you do in e.g. your diploma in Germany.

#### Andy R

Why don't you tell me in simple words what these theory can do? Lie group, finite group, other different groups, it looks to me they all groups? Ok, so what group theory can do?
It's Pure Maths, just done for it's own sake. Group Theory is essentially the study of symmetry but it's of very, very limited application in the real world (no use to any quant, for example). If you're a Theoretical Physicist you might be interested in certain Lie Groups (but heed my point about of being of limited application to the real world). Chemists have a slight interest in Character Theory, which is a branch of Mathematics useful in understanding Group Theory. It's really only done for the intellectual thrill though. ISBN 0387966757 is a pretty nice introduction, if you're interested. I'm not sure what's happening in research any more though (I think it went out of fashion in the 80s).

#### DominiConnor

I'm going to be careful about separating reality and perception, but if I fail to make it clear, as a headhunter I observe that the "branding" of a school/course and the value it adds are correlated but far from the same thing.

People who manage the recruitment of grads are spectacularly biased and frankly often stupid. One very large firm that does a lot of recruitment had (note the past tense) a head of grad recruitment who was asked why the firm now only seemed to get newbies from Oxford and Cambridge and the top man in London asked why at at meeting listing some other universites, including Southampton. HR goes off on a small rant how Southampton grads could not possibly make the grade for this firm.
She actually have a short laugh about Southampton
You know where this is going don't you ?
Yep, top man was a Southampton grad, as were two other people in the meeting....

Some managers hiring perceive that your undergrad degree is the most reliable measure of your ability, rationalising that the signal is cleaner since undergrad degrees aren't so specialised, so you go to Cambridge (for instance) without (usually) wanting to be taught be a specific professor in a specific subject. PhDs are quite different, if the main man in your target is at Cambridge or QMW (where I went) or Princeton or even Fordham or Reading, that's were you would choose to go. Also some qualifications like MFE have a large cost element in the decision which is not a measure of abilty.
The PIII/MMast at Cambridge is of course quite cheap, so that noise term is largely absent and there is an irony that one of the relatively few universities that doesn't have an MFE or equivalent course actually provides one of the best masters level Quant programmes that has almost no explicit financial teaching.

As for the "people from place X are better than others" argiument, I have contempt for this and generally regard people who assert it as morons who I will never ever choose to represent for a job since they are likely to say such dumb shit at interveiws and thus embarrass me for recommending them.
This is a quant forum right ?
So we understand stochastic dominance ?
Univcrsity A may dominate B in some subjects, that's fine.

What we have here is the dichotomy between search and evaluation.
If I wanted to hire a very tall person, I'd look amongst Dutch or New Zealand males because their average height is the greatest amongst humans(if you're an evangelical, note that this is not quite stochastic dominance)

A Nepalese woman is really quite unlikely to be very tall, they are utterly SD'ed in height by Dutch and also men SD women.
But...
A 2 metre tall Nepalese is exactly the same height as a 2 metre Kiwi, Dutchman or Klingon and taller than me.

Given that search has costs that increase with the number of candidates, it is perceived by many campus recruiters / entry level HR that it is not worth looking outside the likely places. This is because their utility function doesn't reflect very well the cost and value of the people they recruit. To put this in money terms... Finding a candidate that is 1% higher up the distribution of ability is worth many thousands to the bank per candidate hired, conversely hiring someone 1% further down has costs and risks.
However the HR's bonus is parlty driven by her percieved productivity, but more by some random exogenous factors like the banks overall performance and what some senior exec thinks aobut HR.
So the incentive is to focus on "obvious"" places, especially since if they deliver a mediocre candidate from an "obvious" places there is no real downside and there is no upside for locating a source of good people. If a trader (or headhunter) found a source of untapped value he would become richer and his status would be increases, a HR who discovered that a non-obvious university appened to prooduce exactly what they wanted to hire would most likely not get a bigger bonus and might get hassle for spending time on a university that no one had heard of. Imagine some Chinese university tapped into a vast pool of smart PRC citizens and delivered a great course, it would take (in my estimation) a decade at least to become a "target school" and it might never happen at all. A US or UK school would face the same problems, albeit it might take a little less time.
Note I use words like "likely" and "obvious", too much of these discussions are driven by branding rather that the value that a course adds, which given this is a finance forum is depressing.

#### nickaelwen

It's Pure Maths, just done for it's own sake. Group Theory is essentially the study of symmetry but it's of very, very limited application in the real world (no use to any quant, for example). If you're a Theoretical Physicist you might be interested in certain Lie Groups (but heed my point about of being of limited application to the real world). Chemists have a slight interest in Character Theory, which is a branch of Mathematics useful in understanding Group Theory. It's really only done for the intellectual thrill though. ISBN 0387966757 is a pretty nice introduction, if you're interested. I'm not sure what's happening in research any more though (I think it went out of fashion in the 80s).

I looked over some terms on wiki, and I intersected at "abstract algebra". Many references points all of them to abstract algebra, and after a brief reading, it makes some sense to me, it actually trys to give a structure and property of why we can do "A + B = C" by generalize the algebra we know to some sorts of condition that for some qualified set and operations on sets such that we can create a new algebra system, like {some set} + {some other set} = {other set} if this set and operation satisfy the 4 required conditions. Does this mean that we can actually create new algebra to deal with objects other than numbers? It blows up my mind, if it is, it is applicable everywhere. Pls, anyone let me know if I am wrong?

#### Andy R

I looked over some terms on wiki, and I intersected at "abstract algebra". Many references points all of them to abstract algebra, and after a brief reading, it makes some sense to me, it actually trys to give a structure and property of why we can do "A + B = C" by generalize the algebra we know to some sorts of condition that for some qualified set and operations on sets such that we can create a new algebra system, like {some set} + {some other set} = {other set} if this set and operation satisfy the 4 required conditions. Does this mean that we can actually create new algebra to deal with objects other than numbers? It blows up my mind, if it is, it is applicable everywhere. Pls, anyone let me know if I am wrong?

Algebra is the study of mathematical structure. That's a very vague statement, but it's difficult to be more specific as algebra is very broad and well developed as a discipline. TBH, this probably isn't the right place to go into detail. If you're interested, here's a standard undergraduate text from which I learned a great deal http://www.amazon.com/Topics-Algebra-I-N-Herstein/dp/0471010901 . It's quite gentle, maybe the first year and a half of a UK undergraduate degree. But it's worth reading. Again, Algebra is Pure Mathematics: it's not written with the intention of solving a specific problem (such as understanding the aerodynamics of an aeroplane wing) and done more for the intellectual satisfaction: rightly or wrongly, it has a reputation for being quite difficult (compared to undergrad statistics, for example, this is certainly true). On the other hand, you never know which parts of Pure Maths might eventually become useful in other areas: number theory, which was once thought to be completely useless but full of many beautiful theorems and arguments, now forms the basis by which all secure communications (such as sending credit card details over the internet) are made and many other ideas from Pure have since moved over to Applied (which is, by definition, useful). For this reason, it's worth keeping a few Pure Mathematicians employed (after all, they're very cheap: academic salaries are notoriously low).

And yes, there are other algebras besides the ordinary algebra of everyday arithmetic.

Anyway, slightly off topic...

#### wlfgngpck

I looked over some terms on wiki, and I intersected at "abstract algebra". Many references points all of them to abstract algebra, and after a brief reading, it makes some sense to me, it actually trys to give a structure and property of why we can do "A + B = C" by generalize the algebra we know to some sorts of condition that for some qualified set and operations on sets such that we can create a new algebra system, like {some set} + {some other set} = {other set} if this set and operation satisfy the 4 required conditions. Does this mean that we can actually create new algebra to deal with objects other than numbers? It blows up my mind, if it is, it is applicable everywhere. Pls, anyone let me know if I am wrong?

Algebra is the study of mathematical structure. That's a very vague statement, but it's difficult to be more specific as algebra is very broad and well developed as a discipline. TBH, this probably isn't the right place to go into detail. If you're interested, here's a standard undergraduate text from which I learned a great deal http://www.amazon.com/Topics-Algebra-I-N-Herstein/dp/0471010901 . It's quite gentle, maybe the first year and a half of a UK undergraduate degree. But it's worth reading. Again, Algebra is Pure Mathematics: it's not written with the intention of solving a specific problem (such as understanding the aerodynamics of an aeroplane wing) and done more for the intellectual satisfaction: rightly or wrongly, it has a reputation for being quite difficult (compared to undergrad statistics, for example, this is certainly true). On the other hand, you never know which parts of Pure Maths might eventually become useful in other areas: number theory, which was once thought to be completely useless but full of many beautiful theorems and arguments, now forms the basis by which all secure communications (such as sending credit card details over the internet) are made and many other ideas from Pure have since moved over to Applied (which is, by definition, useful). For this reason, it's worth keeping a few Pure Mathematicians employed (after all, they're very cheap: academic salaries are notoriously low).

And yes, there are other algebras besides the ordinary algebra of everyday arithmetic.

Anyway, slightly off topic...

I want to add a few comments to this because while it is off topic, it was still my area of research for my dissertation. :P

The best way to think about algebra is to think of it more as the mathematics between structures, and not just of the structures themselves. I always like to think of algebra as being similar to a study of translating languages. Say you are Chinese, and want to tell us something. But unfortunately we don't speak Chinese so to get us to understand you need to translate into English, and you hope that your translation remembers enough of what you wanted to say that we understand. That's what algebra does. Group theory is mostly used this way in today's mathematics. For example, showing that the sphere and the torus are different topological structures is difficult to do using only techniques of topology (there may actually be a way, but I don't know it). But translating the situation to group theory (using the homology of the spaces, if you care to look that up) makes this problem a very simple group theory problem.

To answer another question, pure group theory is still researched but research in that area is very slow because most of the problems that are still open are unbelievably difficult (and maybe even impossible). One of the very early developments and uses of group theory was in the study of algebraic equations (which we now call Galois Theory). That problem has pretty much been resolved. So group theory does have applications....it's just that those applications remain in other areas of mathematics.

To add an additional reference, the standard book on Abstract Algebra is the book by Dummit and Foote; the book literally begins assuming you know nothing about algebra at all, and ends with introductions to the beginnings of areas where research takes place (such as algebraic geometry). It's a great subject, but if your aim is to work as a quant, then it is pretty much useless to you. Still, if you have interest, check it out.

Leading UK groups in Pure Mathematics RAE 2008

RankInstitutionFTE Category A staff submitted4*3*2*1*
1 Imperial College London 21.8 40 45 15 0
2 University of Warwick 32 35 45 20 0
3 University of Oxford 55.16 35 40 25 0
4 University of Cambridge 55 30 45 25 0
5= University of Bristol 34.53 30 40 25 5
5= Heriot-Watt University 10 25 45 30 0
5= University of Edinburgh 31 25 45 30 0

#### Barny

University rankings are a meaningless piece of crap, martingaletrader. Use with extreme caution. Also, no employer cares about what the subject rankings are, they only care/know about universities overall reputations. That's why you're far better off doing to Oxbridge/Imperial/LSE/UCL for any subject than Bristol/Edinburgh/Durham for a specific subject.

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