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Quant View on Pure Math

Apologies if my post doesn't have a direct question, I'm more interested in having a discussion.

For context, I'm a math undergraduate studying at a mid-tier university in the U.K.

I recently attended a panel talk on life in the quant industry (mostly UK quants), and whenever the question of education came up, the panel largely agreed that employers aren't much fussed about your specialist knowledge - i.e. if you're a smart Oxbridge undergraduate they don't care what you studied, as long as it demonstrates you'll be able to learn whatever's necessary on the job.

However, I did notice that those in the panel who had a math background tended to be on the applied side (quantum mechanics, fluid dynamics etc.). I personally have been going through a struggle because although I have a place secured for an MFE program in September, I am becoming somewhat depressed by the idea that my 'career' as a pure mathematician is coming to an end. I've taken courses relevant for quant work, but also courses on functional analysis and topology, which I've really been enjoying.

In the U.K, university course structures are quite rigid; for example, in the second year of my three year degree I had no elective modules, I was given a compulsory list to sit by the department, as is fairly standard. The result of this is that the knowledge of a mathematics graduate is quite varied - what I learn in three years a Cambridge undergraduate might have learned in their first 1.5-2 years.

I have an internalised battle as to whether I should defer my MFE and transfer into integrated masters program, essentially meaning in September I would study math for a fourth year as part of my degree, where I would probably take courses on algebraic topology, Galois theory etc., and then after move to focus on financial mathematics in an MSc. It seems likely to me, however, that this would make little impact career-wise, as well as cost an extra year's worth of tuition - but I can't help but worry I will regret ending my formal pursuit of pure mathematics if I don't.

There are multiple alternatives, namely
i) self-study alongside work, for enjoyment,
ii) enrol in a pure math master's program after some years in the industry.

However, both these options have their problems.

Can anyone else share similar experiences they've had? Does anyone regret not pursuing their favourite discipline? How difficult is it to resume high-level study after taking a prolonged period out of university?
 
I agree with you that it probably wouldnt advance your career anymore if u were planning on taking an MFE, other than maybe aiding masters application (irrelevant since u have an offer), but if you would regret not doing it for self fulfillment purposes then maybe you should consider it. It sounds like you might enjoy a path of pure math and research more than of quant fin?

I dont think study after a break would be difficult and could be done part time but I think it would be more of an expensive and time consuming hobby at that point and like you said you could study in your own time.

Dont know if that was any help but sometimes its just good to get another viewpoint 😅
 
I have an internalised battle as to whether I should defer my MFE and transfer into integrated masters program, essentially meaning in September I would study math for a fourth year as part of my degree, where I would probably take courses on algebraic topology, Galois theory etc., and then after move to focus on financial mathematics in an MSc. It seems likely to me, however, that this would make little impact career-wise, as well as cost an extra year's worth of tuition - but I can't help but worry I will regret ending my formal pursuit of pure mathematics if I don't.

There are multiple alternatives, namely
i) self-study alongside work, for enjoyment,
ii) enrol in a pure math master's program after some years in the industry.

At a mid-tier British university, the fourth year will be roughly the same as the third, but with slightly more advanced courses. It won't be like a Cambridge Part 3 or the Warwick M.Sc. Probably the Warwick third-year undergrad courses will be like your fourth year. The point of saying all this is that you might be disappointed in what you are exposed to and think it wasn't worth the while.
 
At a mid-tier British university, the fourth year will be roughly the same as the third, but with slightly more advanced courses. It won't be like a Cambridge Part 3 or the Warwick M.Sc. Probably the Warwick third-year undergrad courses will be like your fourth year. The point of saying all this is that you might be disappointed in what you are exposed to and think it wasn't worth the while.
This certainly seems the case, about half the courses I'm currently sitting are also being studied by graduate students.

In a dream world I'd love to work for some time and then go off to Cambridge to study Part III, but the reality is that it's hard enough coming from actively studying mathematics each day (my second year grades are good but not outstanding, certainly not good enough for admission unless I do a bit better in exams this year), never mind if you're years out of touch studying with some of the brightest young mathematicians in the country.
 
This certainly seems the case, about half the courses I'm currently sitting are also being studied by graduate students.

In a dream world I'd love to work for some time and then go off to Cambridge to study Part III, but the reality is that it's hard enough coming from actively studying mathematics each day (my second year grades are good but not outstanding, certainly not good enough for admission unless I do a bit better in exams this year), never mind if you're years out of touch studying with some of the brightest young mathematicians in the country.

I don't know about now but I remember that in my day a 2:1 from Cambridge wasn't enough to get admitted to the Part III (in mathematics). The Part III is demanding and punishing and it's designed to give a solid foundation to prospective research students (I think you need a Distinction in Part III to stay on as a Ph.D. student at Cambridge).

People keep falling by the wayside at each stage of the game: first you need to get admitted to Cambridge/Warwick/Oxford (the rest don't count). Then you need to earn a First. Then you need to get admitted to the Part III. Then you need to earn a Distinction (or at least a Merit). Then you need to be accepted by a worthwhile research supervisor (at one of the three schools above).
 

Daniel Duffy

C++ author, trainer
In very general terms, most students tend to be good at/like pure math XOR applied/numerical maths. A bit one-sided and somewhat myopic.I had the luck of doing all 3 (as well as theoretical physics) as undergrads and we hardly ever used the words pure/applied in the context that they are used here. It was all maths.
The pure maths stuff like groups, Galois, Lie, topology are not so relevant to quant. What is more relevant is what could be called 'computational' maths.' If Euler and Cauchjy were alive today they would be C++ programmers. As would John von Neumann.

And (Applied) Functional Analysis is extremely important for many reasons.

//

BTW what does Part III entail, content-wise? Is it basically 4th year advanced undergrad? I know the term but that's as far as it goes.
What I don't see is hard-code Numerical Analysis courses in general. This has major consequences for pure mathematicians in finance, ML etc. (it's > 70% numerical maths).

Just sayin'
 
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In very general terms, most students tend to be good at/like pure math XOR applied/numerical maths. A bit one-sided and somewhat myopic.I had the luck of doing all 3 (as well as theoretical physics) as undergrads and we hardly ever used the words pure/applied in the context that they are used here. It was all maths.
The pure maths stuff like groups, Galois, Lie, topology are not so relevant to quant. What is more relevant is what could be called 'computational' maths.' If Euler and Cauchjy were alive today they would be C++ programmers. As would John von Neumann.

And (Applied) Functional Analysis is extremely important for many reasons.

//

BTW what does Part III entail, content-wise? Is it basically 4th year advanced undergrad? I know the term but that's as far as it goes.
What I don't see is hard-code Numerical Analysis courses in general. This has major consequences for pure mathematicians in finance, ML etc. (it's > 70% numerical maths).

Just sayin'
I suppose my experience has been similar so far (minus the theoretical physics; unless you count some applications of DEs), prior to a departmental reshuffle, owing to COVID, my third-year degree choices would see my overall undergraduate considered 'Maths and Stats' rather than straight maths, although it's back to straight maths because of how my credits are currently weighted.

Also, what do you mean by hard-code Numerical Analysis? I've come across the Cambridge undergraduate numerical analysis course notes, and on the Part III course listing there is also a section for Applied and Computational analysis. For those with a real passion for mathematics and an interest in quant, from my (albeit inexperienced) perspective it does seem Part III is unbeatable. My interpretation is that in general sets you up for independent work much more suitably than other programmes, with the drawback being the workload and difficulty of tripos.
 
In any case, I would lean C++, ideally at QN.
C++ is great in combination with maths.
I was planning to study some C++ over summer, although the course I'm starting in September covers this content, so I wasn't sure if I'd be better off practising algorithm analysis/LeetCode style problems as interview prep.

I'm familiar with basic C++ syntax, although all the formal training I've had in numerical programming is in Fortran 95.
 
BTW what does Part III entail, content-wise? Is it basically 4th year advanced undergrad? I know the term but that's as far as it goes.

Nope, it's basically a year and a half of grad courses crammed into one year. For example take a look at this course:


Even at good US universities this will be a second-year grad course. And this builds on the previous course:


Or look at this one:


These courses are designed for the very best students, who will be going on to do Ph.D.s with the best research supervisors in mainstream mathematics.
 

Daniel Duffy

C++ author, trainer
I
Nope, it's basically a year and a half of grad courses crammed into one year. For example take a look at this course:


Even at good US universities this will be a second-year grad course. And this builds on the previous course:


Or look at this one:


These courses are designed for the very best students, who will be going on to do Ph.D.s with the best research supervisors in mainstream mathematics.
Purest of pure maths.
My son is drummer in a metal band; the base guitarist + composer is doing a PhD in algebraic topology. As soon as I mention PDE etc. he freezes. Two mathematicians separated by a common language.. The two other guys are CS, so there is some rapport :devil:

 

Daniel Duffy

C++ author, trainer
Not really a common language -- not past the idea of continuity. One step past continuity to homeomorphism and you've already lost the PDE crowd.
Are we entering a 'no true Scotsman' discussion?
In fairness, they are paid to do PDEs, not homeomorphisms.
 
Are we entering a 'no true Scotsman' discussion?
In fairness, they are paid to do PDEs, not homeomorphisms.

Um, I'm not blaming them for doing PDEs and not homeomorphisms. Someone has to do the gritty and unpleasant work of PDEs so that the more poetic and artistic types can prove theorems in etale cohomology. It's that old Morlock/Eloi divide.
 
Nope, it's basically a year and a half of grad courses crammed into one year. For example take a look at this course:


Even at good US universities this will be a second-year grad course. And this builds on the previous course:
Algebraic geometry is a monster. I've taken some unorthodox courses as an applied maths undergrad (logic, measure theory, abstract algebra, Galois theory, Coxeter groups), and it only took me 4 hours of Hartshorne to nope the hell out of that class. I have nothing but mad respect for those who can consume such topics.
 
Algebraic geometry is a monster. I've taken some unorthodox courses as an applied maths undergrad (logic, measure theory, abstract algebra, Galois theory, Coxeter groups), and it only took me 4 hours of Hartshorne to nope the hell out of that class. I have nothing but mad respect for those who can consume such topics.

It's just sheer lunacy to assign Hartshorne as first reading to grad students. There have been perfectly capable grad students who have struggled with Hartshorne for two years and then given up their Ph.D. in disgusted failure. Hartshorne should be approached after having gone through three or four gentler books on algebraic geometry/curves. That's the case with the Part III, where the students (well, at least the Cambridge ones) will have taken a Part II (i.e. 3rd year undergrad) course in algebraic geometry.

Postscript: These are the ones being recommended in Part II:

K. Hulek Elementary Algebraic Geometry
F. Kirwan Complex Algebraic Curves.
M. Reid Undergraduate Algebraic Geometry.
B. Hassett Introduction to Algebraic Geometry.
K. Ueno An Introduction to Algebraic Geometry.
R. Hartshorne Algebraic Geometry, chapters 1 and 4.

Except for the last, all are accessible to undergraduates.
 
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Not really a common language -- not past the idea of continuity. One step past continuity to homeomorphism and you've already lost the PDE crowd.
No way. Come on now, a homeomorphism is nothing but a bijective continuous map with continuous inverse. If someone says they do PDEs but don’t know what a homeomorphism is, then damn sure they have never heard of a Sobolev space, i.e. they don’t actually do PDEs.
 
It's just sheer lunacy to assign Hartshorne as first reading to grad students. There have been perfectly capable grad students who have struggled with Hartshorne for two years and then given up their Ph.D. in disgusted failure. Hartshorne should be approached after having gone through three or four gentler books on algebraic geometry/curves. That's the case with the Part III, where the students (well, at least the Cambridge ones) will have taken a Part II (i.e. 3rd year undergrad) course in algebraic geometry.

Postscript: These are the ones being recommended in Part II:

K. Hulek Elementary Algebraic Geometry
F. Kirwan Complex Algebraic Curves.
M. Reid Undergraduate Algebraic Geometry.
B. Hassett Introduction to Algebraic Geometry.
K. Ueno An Introduction to Algebraic Geometry.
R. Hartshorne Algebraic Geometry, chapters 1 and 4.

Except for the last, all are accessible to undergraduates.
Algebraic geometry was offered as a level 4 undergrad course in my university back then, albeit a rather obscure one - for good reasons. Thanks for the reading recommendations though, maybe I'll pick one or a few of them up when I feel like diving into some more PTSD-inducing maths.
 
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