Is PhD in pure math worth it?

[Daniel Duffy]

I strongly disagreed.

Pure maths is the backbone for FM application.

The word 'pure' is out of place and not applicable here. IMO,

 

[muttiah]

I'm taking classes with CS PhDs right now and yes, there is definitely no hand-holding. The classes are painful (algorithms) and stressful but so far manageable. Is the math PhD much harder?

My career goal is to work as a trader or quant. I trade with my IRA portfolio and have gotten good returns over the past 10 years (~25%) and would love to do it professionally. Is it possible to become a trader with a MSCS or will I get pigeonholed as an IT person forever?

I do know that a MSCS will not give me a shot at any "quant" type jobs.

PhD level algorithms (or complexity theory) course will be similar to level of a PhD level pure math courses. However I believe getting through the qualifying exams (general and comprehensive exams) are more difficult in pure math (Fwiw I have master's in pure math). You will need to know about a wide range of basic topics

Real analysis
Measure theory (Lebesgue and abstract)
Complex analysis
Algebra (Basic algebra, Galois theory, maybe basic algebraic and analytic number theory)
Topology (metric space, general topology, and some algebraic topology)
Geometry (some differential geometry, some algebraic geometry)
Miscellaneous topics (depends on the department. Sometimes logic, probability theory)

All these courses will be theoretical. Unlike undergraduate courses where it's enough to know the results and the basic proofs, in grad level you are required to really internalize the theorems and apply them in different ways, come up with your own examples and counterexamples, come up with your own conjectures and try to prove/disprove them.

 

[muttiah]

I was looking at NYU's math PhD program and noticed that Marco Avellaneda is a well-respected financial math researcher there. Wouldn't you qualify for the best quant roles in the industry if a respected financial math researcher were your dissertation advisor? And NYU doesn't seem to offer a PhD in applied math either. Just "math".

NYU is generally considered the top applied math department in the US. That's because they are very strong in the traditional applied fields like PDEs, finance, probability theory etc. However the program/courses are highly theoretical. Some schools have just a dept of math and others have split departments for pure and applied.

 

[Daniel Duffy]

PhD level algorithms (or complexity theory) course will be similar to level of a PhD level pure math courses. However I believe getting through the qualifying exams (general and comprehensive exams) are more difficult in pure math (Fwiw I have master's in pure math). You will need to know about a wide range of basic topics

Real analysis
Measure theory (Lebesgue and abstract)
Complex analysis
Algebra (Basic algebra, Galois theory, maybe basic algebraic and analytic number theory)
Topology (metric space, general topology, and some algebraic topology)
Geometry (some differential geometry, some algebraic geometry)
Miscellaneous topics (depends on the department. Sometimes logic, probability theory)

All these courses will be theoretical. Unlike undergraduate courses where it's enough to know the results and the basic proofs, in grad level you are required to really internalize the theorems and apply them in different ways, come up with your own examples and counterexamples, come up with your own conjectures and try to prove/disprove them.

IMO these mathematical 'pure' topics (btw I did them in undergrad) are not so useful in computational finance.

Of course, they don't have to be.

 

[muttiah]

IMO these mathematical 'pure' topics (btw I did them in undergrad) are not so useful in computational finance.

Of course, they don't have to be.

Oh I agree. I was simply stating the typical curriculum for PhD math students. Since OP has his eye on finance it makes sense to take/audit applied courses for electives.

 

[bigbadwolf]

Real analysis
Measure theory (Lebesgue and abstract)
Complex analysis
Algebra (Basic algebra, Galois theory, maybe basic algebraic and analytic number theory)
Topology (metric space, general topology, and some algebraic topology)
Geometry (some differential geometry, some algebraic geometry)
Miscellaneous topics (depends on the department. Sometimes logic, probability theory)

All these courses will be theoretical. Unlike undergraduate courses where it's enough to know the results and the basic proofs, in grad level you are required to really internalize the theorems and apply them in different ways, come up with your own examples and counterexamples, come up with your own conjectures and try to prove/disprove them.

And each of these -- real, complex, algebra, topology, Riemannian geometry -- will typically be year-long first-year graduate courses. And the grad students will usually have taken lighter versions of these as undergrads (e.g., Churchill for complex, Herstein for algebra). In the strong departments, there will be advanced courses for second- and third-year PhD students that will take them close to the research frontier. A good PhD from a strong department is not easy.

 

[tfors]

These pure math topic seem very much like overkill to me for most applications in computational finance. But I am just a student still.

 

[tfors]

(and btw. pure math, so far as I have taken knowledge of it, is not fun at all compared to applied, and that's a more common opinion it seems)

 

[Daniel Duffy]

(and btw. pure math, so far as I have taken knowledge of it, is not fun at all compared to applied, and that's a more common opinion it seems)

I suppose this is similar

"In pure you prove existence/unqueness of objects, in applied you find/construct them.'

 

[bigbadwolf]

I suppose this is similar

"In pure you prove existence/unqueness of objects, in applied you find/construct them.'

The existence/uniqueness part was an aberration of 20th century math. Most of that math (aka "pure") probably won't survive. As Courant put it sixty or seventy years ago, the only math that will survive will be the math connected to physics, biology, etc. -- applied/applicable math. The pure stuff was an indulgence in an economic "golden era," when real growth was to be had, times were easy, and universities were expanding.

 

[JesseLevermore]

I've heard/thought about the Courant quote also and I feel that while true, it may not be a change in regards to math research. I feel you will see that as time goes on, people will use more and more mathematical tools that were previously thought to be abstract.

Look at where economics research is right now. There are microeconomists using group theory and macroeconomics has required high level analysis for decades now.

I also don't know that "pure" mathematicians don't construct either, just their constructions have no immediate use. Computational group theory is such an example.

 

[bigbadwolf]

The math that's needed might be developed on an ad hoc basis. Something analogous to the Dirac delta function, which was initially created by physicists. Later on mathematicians developed the structure and machinery of the theory of distributions which legitimises/makes sense of such an ad hoc construction. In the future there may not be such mathematicians -- they'll be seen as an indulgence in an age of decline and austerity. And these things have happened in the past -- e.g., Hamiltonian and Lagrangian mechanics was employed because it worked, not because it was properly understood (it needed the 20th century framework of jet and/or contact bundles for that).

 

[JesseLevermore]

But is it not the "indulgence" of proof that produces the new mathematics necessary to advance scientific thought? How would we have ended up with analysis if not for some "pure" rigor on calculus leading us to higher results.

 

[bigbadwolf]

But is it not the "indulgence" of proof that produces the new mathematics necessary to advance scientific thought? How would we have ended up with analysis if not for some "pure" rigor on calculus leading us to higher results.

There seem to be differing points of view. One is that of Hadamard, who thinks that intuitions should be translated into definitions, and then theorems will come out of them. On the other, people like (maybe) Ulam and Feynman, who were more at home with an intuitive and ad hoc style of thinking. I'm inclined towards the latter. The problem with the former is that second-rate minds like myself end up trapped in an axiomatic straitjacket created by more powerful minds. And I can't see the world any other way than from the perspective provided to me by that axiomatic system. The creative playfulness is lost.

 

[JesseLevermore]

I like the fact that you realize that some men get trapped into their schools of thought but the more people I converse with the more I'm inclined to believe that this is more due to lack of diverse experiences (or creativity as we know it) more than an innate brilliance.

Mathematics has this odd duality I think, and the best analogy to describe it is with a mathematics construction! Much like the discrete metric space where subsets are both open and closed, the system of math takes minds and both opens and closes them. I think it is a matter of perspective (or what property you need to finish your proof). Now the ones without a diverse enough range of experience bind themselves to their school because it is all they know, it's where they are comfortable and they see no connection to anything outside.

It makes for a very much dangerous academic setting I think. One where future professors are very claustrophobic and research is pure only in the sense of naivety.

But to me that implies we need really a new renaissance. Men should again strive for diversity in experiences and not pigeonhole themselves immediately into one tiny research area. Intuition is nice but I have a sincere question: how will we build on the intuition? Will we not place rigor onto the intuition? And if not applied by the intuitionist (why is this not a word in Quantnet?) wouldn't there almost assuredly be a loss of his original framework?

And I don't think you have a second rate mind either! I've read your posts and you seem quite brilliant; probably just another case of wrong place, wrong time, wrong adviser =).

 

[Yike Lu]

As I perceive it, the role of pure math is to distill and abstract.

Reduce proofs to require as weak assumptions as possible, so that commonalities can be seen, then create an overarching framework around that.

So the inevitable progression of pure math is to be more and more abstract.

But there are lots of useful and interesting special cases, some of which simply do not fit into an already existing framework, others which do but the framework hasn't been applied (or the framework is too arcane for people to see the connection).

These are the situations where application leaps ahead of purity, the situations where intuition gets rewarded. As BBW alludes to, these are Feynman-esque situations where you need to invent the tool, do some basic sanity checks and plunge forward and let the mathematicians do the clean up work later.

 

[bigbadwolf]

So the inevitable progression of pure math is to be more and more abstract.

At some stage an area of math has a tendency to become a glass bead game. The structure of post-war academia has encouraged this abstract and self-serving navel-gazing. This era is coming to an end. Courant foresaw this. Terence Tao has some insightful things to say on what makes good math. My hunch is that in an era of civilisational decline (collapse?), not much of the math of the last two centuries is going to survive.

 

[Yike Lu]

At some stage an area of math has a tendency to become a glass bead game. The structure of post-war academia has encouraged this abstract and self-serving navel-gazing. This era is coming to an end. Courant foresaw this. Terence Tao has some insightful things to say on what makes good math. My hunch is that in an era of civilisational decline (collapse?), not much of the math of the last two centuries is going to survive.

Yes, there is likely going to be a culling. There are so many results that many of them just feel like noise.

In a broader sense, the structure of academia itself needs to change.

Oh, and in case OP hasn't realized it: no a pure math PhD is not worth it.

 

[Daniel Duffy]

Oh, and in case OP hasn't realized it: no a pure math PhD is not worth it.

I don't agree, necessarily; if you really want to do pure maths, do it because you might become the most famous mathematician of all time.

If you want to make money best way is to start your own company.

 

[bigbadwolf]

I don't agree, necessarily; if you really want to do pure maths, do it because you might become the most famous mathematician of all time.

I don't think any active research mathematician does what he does for fame, and the odds of gaining such fame are staggeringly unfavorable. Other than the fame part, the level of ability among the best mathematicians is such that no merely ambitious mortal can hope to compete with them in solving the difficult problems or getting the prize jobs.