Best courses to take for PhD application in Applied Math/Operation Research

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ghjk
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Joined: Sat Apr 21, 2012 8:42 pm

Best courses to take for PhD application in Applied Math/Operation Research

Postby ghjk » Fri Dec 18, 2015 3:44 pm

Hi everyone, I'm a 2nd-year MS student in Applied Mathematics at an average, unknown state school. As I'm planning for which classes to take the next semester, I would love to hear your feedback about my selection. The reason is because I'm almost completing the M.S program (this one is probably my second-to-last semester) before I apply to PhD programs in Operation Research/Financial Math/Applied Math. My main interest lays in derivative pricing (aka, options/basket of options, etc), asset allocation, portfolio optimization and optimal routing in logistics, optimal pricing scheme for products to maximize business's profitability. I think it is quite tough for me to display serious interest in OR or Mathematical Finance, as I have only taken 2 OR graduate courses in Linear and Nonlinear Programming (got A and A+ in them), while I kinda "overemphasized" with math courses.

Right now, I can only take at most 8 more grad courses, and 2 of them I have decided to be on Measure Theory and Stochastic Differential Equations (book by M. Steele) , I still have 2 more grad courses to choose for next semester, and here's my potential list (ranked from most to least important):

Numerical Solutions of Differential Equations (purely Finite Difference method) - A first course in Numerical Analysis of DEs (Cambridge Series Textbook)

Dynammic Programming - Models + Application book by Eric Denardo

Complex Analysis I - no book decided yet

Regression and Time Series - Econometrics Model + Economics Forecast book by Pindyck

Integer Programming - Applied Integer Programming - Modeling and Solution by Chen

My current temporary pick is the first two on the list, but that might be wrong since many strong/top-ranked applied math programs require Complex Analysis as one of their core courses. Any thought would really be appreciated.

Thanks everyone for your time! Look forward to hearing from some of you.
Last edited by ghjk on Sat Dec 19, 2015 5:15 am, edited 1 time in total.

ghjk
Posts: 81
Joined: Sat Apr 21, 2012 8:42 pm

Re: Best courses to take for PhD application in Applied Math/Operation Research

Postby ghjk » Sat Dec 19, 2015 5:15 am

Nobody minds giving me some advice on which course to take? It's quite a hard choice to choose between Complex Analysis, Regression and Time series, and Integer programming. Anyone has experienced this situation, can you kindly give some help?

Enigmatic
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Joined: Sat Jul 11, 2015 11:04 am

Re: Best courses to take for PhD application in Applied Math/Operation Research

Postby Enigmatic » Sat Dec 19, 2015 1:34 pm

Are you planning a PhD in applied math or OR? Most of your stated interests are clearly OR, and most schools have separate applied math and OR programs (e.g. Columbia, Cornell, Princeton etc.).

Applied math is a blanket term that can mean a thousand different things. However, applied math PhD programs tend to focus mostly on (a) computational fluid dynamics; (b) stochastic simulations at a molecular level; (c) mathematical modeling of physical, biological, or chemical processes. None of these are directly tied to your interests, so I would encourage you to apply to OR programs.

Naturally the requirements for AM and OR programs are different since the research focus is different. No OR program I am aware of requires a differential equations "grad" course. Complex analysis is never used in OR, unless you work on esoteric topics at the interface of signal processing or control theory.

Looking at your course list, dynamic programming is a no-brainer. I would also pick time series as the second option, with integer programming a close third. Differential equations and complex analysis are least useful for OR. On the other hand, for applied math integer programming is likely useless and differential equations is a must.

ghjk
Posts: 81
Joined: Sat Apr 21, 2012 8:42 pm

Re: Best courses to take for PhD application in Applied Math/Operation Research

Postby ghjk » Sat Dec 19, 2015 4:29 pm

Enigmatic wrote:Are you planning a PhD in applied math or OR? Most of your stated interests are clearly OR, and most schools have separate applied math and OR programs (e.g. Columbia, Cornell, Princeton etc.).


Thank you very much for giving out your thought. It's quite hard to clearly state whether my passion for quantitative finance (in particular, derivatives pricing, asset allocation, portfolio optimization) and structure of capital markets would be well-defined under the OR or Applied Math or Finance department. The reason for this is quite simple: some schools (schools like UT-Austin, UCSB, UMD, UPenn, etc. with their good IROM, Probability and Statistics, AMSC, AMCS programs, respectively, include some math/finance professors who specifically work in Mathematical Finance). The difficulty when choosing to decide to specialize in OR is that the faculty is way too broad, and they tend to come from different fields who has different levels in mathematics, so it might lead to undervalue the difficulty/significance of the math courses that I took and did quite well, while weighting more on the number of OR grad courses and how well I took.Obviously I didn't have time to take a lot of courses in OR, except the two core courses in Mathematical Programming (Linear and Nonlinear), but the math in OR is far far simpler (we gotta admit this truth). Columbia is definitely an awesome school for Quantitative Finance, but again, the highest level of math appearing in their faculties' research is only up to stochastic differential equations and Monte-Carlo simulation (these things are not easy at the highest level of research, but it's still a no-brainer to compare them with PDEs, Algebra, or Functional Analysis). There is no such thing as PDE, Functional Analysis, or even showing convergence rate within H^1 or L^2-space, and that wasted three of my math courses. But again, their stuffs are clearly useful, and not as dry as the pure math.


Enigmatic wrote: Applied math is a blanket term that can mean a thousand different things. However, applied math PhD programs tend to focus mostly on (a) computational fluid dynamics; (b) stochastic simulations at a molecular level; (c) mathematical modeling of physical, biological, or chemical processes. None of these are directly tied to your interests, so I would encourage you to apply to OR programs.

Naturally the requirements for AM and OR programs are different since the research focus is different. No OR program I am aware of requires a differential equations "grad" course. Complex analysis is never used in OR, unless you work on esoteric topics at the interface of signal processing or control theory.

Looking at your course list, dynamic programming is a no-brainer. I would also pick time series as the second option, with integer programming a close third. Differential equations and complex analysis are least useful for OR. On the other hand, for applied math integer programming is likely useless and differential equations is a must.


I really appreciate your summary in the focus of AM programs, and you're definitely right that none of (a), (b) or (c) really interests me (although I'm a fan of PDEs!!). It's weird that some of the professors in AM (at Cornell, UT-Austin, UNC, UCSB, etc) actually listed their research interest in Financial Engineering/Mathematical Finance, so I don't really know whether I should try applying for both? If that's the case, should I take Complex Analysis I over time series?

Why you said dynamic programming is a "no-brainer" btw? It's one of the very advanced and important techniques in OR? Integer programming is listed as a core course at places like Cornell, but as far as relative difficulty and spectrum of applications, I'm not so sure which one is harder and more useful. Some people said to me that Complex Analysis I is useful because of stuffs like Fourier Transform and Inverse Fourier Transform, which are usually used by OR's professors whose work is in option pricing. But if I skip this Complex Analysis I, will it put me at a severe disadvantage place if I decide to apply to AM programs (say, UPenn's AMCS program), given that complex analysis is one of the 4 topics that is on their prelim exams?

Finally, as I summarize, your advice would be to take Numerical Methods + Time Series? What if I can also choose to take Statistical Inference or Mathematical Statistics II, should I choose either of them over these two courses? Really thankful to your insight, Enigmatic!

Enigmatic
Posts: 18
Joined: Sat Jul 11, 2015 11:04 am

Re: Best courses to take for PhD application in Applied Math/Operation Research

Postby Enigmatic » Sun Dec 20, 2015 2:12 am

There seems to be multiple aspects to your question: (a) admission chances; (b) fit with department and interests; (c) your love for certain areas of math. I think it is worthwhile to take a step back and analyze things without biases.

Firstly (a) should not be your primary concern. How your courses play into admissions is indeed important, but it would be a sad scenario if you end up in a department whose focus has nothing to do with any of your interests. Pick the field first, and pick the universities next depending on where you see yourself in the admission pecking order.

(b) is the most important criteria, and IMO your stated interests are very clearly OR. Now understand that certain areas of math are required for certain application focus. For most OR applications, including your own interests, complex analysis and many other topics you listed are simply not required. Further, I wouldn't comment on the quality of mathematics in OR, since it is an applied discipline, but is certainly the most math intensive applied discipline (more so than Physics). For example, combinatorial optimization, complexity theory, graph theory, game theory etc. are big in OR, but rarely stressed in pure mathematics. These may not be as pure as algebraic geometry, but is certainly no less interesting, and infinitely more useful for practical applications.

I think (c) is clouding your judgement a bit. You have clearly stated 4-5 interests. Look up the mathematical chops and tools required for those areas and you will get a clearer picture. This is not to say that your earlier math courses become pointless. Quite the contrary, many OR programs would love to have very mathematically minded PhD students joining them, since it is fairly easy to pick up the applied disciplines during your first year. You may have to work a lot of the computational aspects though, since OR is vaguely 50% math, 25% stats, and 25% CS.

With regards to your other specific questions, dynamic programming is the go-to tool for most OR applications. I think you should definitely take this because it will likely help reduce some deficiency towards the algorithmics end. Maybe you should read a bit about it and then decide? Integer programming (IP) is again an extremely algorithmic subject. Posing an IP is fairly straightforward, but it is NP-hard. In some sense the course will likely teach you when it is worth posing a problem as an IP, and when it is a lazy idea (since posing it is easy). It might also teach you algorithms to computationally solve such IPs with provable guarantees - you can never get exact solutions, but can prove something like the output is at least 85% of the maximum possible value in a maximization problem. These require advanced tools from convex analysis, semidefinite optimization etc. A famous example is the expander flows and graph partitioning work of Sanjeev Arora.

Complex analysis, as I said will be useful in the context of areas bordering control theory and signal processing. Stuff like stochastic control (uses dynamic programming a LOT), wavelet analysis, Fourier transform etc. roughly fall into this category. Personally, these are my research areas, and I would say are the most useful tools for portfolio optimization, asset allocation, and to a lesser extent pricing (can come under something like H-infinity control). However, I wouldn't recommend a *grad* level complex analysis course for this because that level of complex analysis is rarely used! An undergrad level class in the same will more than suffice and if anything more is required, you can learn on the fly.

Statistical inference is also quite useful, but probably not mathematical statistics. If I have to pick 2, I would still recommend the same: dynamic programming and time series. Statistical inference and integer programming come 3rd and 4th. Best wishes!

ghjk
Posts: 81
Joined: Sat Apr 21, 2012 8:42 pm

Re: Best courses to take for PhD application in Applied Math/Operation Research

Postby ghjk » Sun Dec 20, 2015 4:17 am

Enigmatic wrote:There seems to be multiple aspects to your question: (a) admission chances; (b) fit with department and interests; (c) your love for certain areas of math. I think it is worthwhile to take a step back and analyze things without biases.

Firstly (a) should not be your primary concern. How your courses play into admissions is indeed important, but it would be a sad scenario if you end up in a department whose focus has nothing to do with any of your interests. Pick the field first, and pick the universities next depending on where you see yourself in the admission pecking order.

(b) is the most important criteria, and IMO your stated interests are very clearly OR. Now understand that certain areas of math are required for certain application focus. For most OR applications, including your own interests, complex analysis and many other topics you listed are simply not required. Further, I wouldn't comment on the quality of mathematics in OR, since it is an applied discipline, but is certainly the most math intensive applied discipline (more so than Physics). For example, combinatorial optimization, complexity theory, graph theory, game theory etc. are big in OR, but rarely stressed in pure mathematics. These may not be as pure as algebraic geometry, but is certainly no less interesting, and infinitely more useful for practical applications.

I think (c) is clouding your judgement a bit. You have clearly stated 4-5 interests. Look up the mathematical chops and tools required for those areas and you will get a clearer picture. This is not to say that your earlier math courses become pointless. Quite the contrary, many OR programs would love to have very mathematically minded PhD students joining them, since it is fairly easy to pick up the applied disciplines during your first year. You may have to work a lot of the computational aspects though, since OR is vaguely 50% math, 25% stats, and 25% CS.

With regards to your other specific questions, dynamic programming is the go-to tool for most OR applications. I think you should definitely take this because it will likely help reduce some deficiency towards the algorithmics end. Maybe you should read a bit about it and then decide? Integer programming (IP) is again an extremely algorithmic subject. Posing an IP is fairly straightforward, but it is NP-hard. In some sense the course will likely teach you when it is worth posing a problem as an IP, and when it is a lazy idea (since posing it is easy). It might also teach you algorithms to computationally solve such IPs with provable guarantees - you can never get exact solutions, but can prove something like the output is at least 85% of the maximum possible value in a maximization problem. These require advanced tools from convex analysis, semidefinite optimization etc. A famous example is the expander flows and graph partitioning work of Sanjeev Arora.

Complex analysis, as I said will be useful in the context of areas bordering control theory and signal processing. Stuff like stochastic control (uses dynamic programming a LOT), wavelet analysis, Fourier transform etc. roughly fall into this category. Personally, these are my research areas, and I would say are the most useful tools for portfolio optimization, asset allocation, and to a lesser extent pricing (can come under something like H-infinity control). However, I wouldn't recommend a *grad* level complex analysis course for this because that level of complex analysis is rarely used! An undergrad level class in the same will more than suffice and if anything more is required, you can learn on the fly.

Statistical inference is also quite useful, but probably not mathematical statistics. If I have to pick 2, I would still recommend the same: dynamic programming and time series. Statistical inference and integer programming come 3rd and 4th. Best wishes!



What an awesome answer!! I love the maturity and an unique perspective of your answer, given your background. I can see how much time you invested in thinking of ways to help me, and I think you pretty blew my expectation off! Now, regards to your mention about graph theory/combinatorial optimization/game theory, should I take any of these courses for the sake of showing strong fit with the department? As for option pricing, I almost never see a paper which uses any of these things. Most of the times, I see them use things like Monte-Carlo simulation, stochastic DEs , integrations, Markov chain, some numerical methods like Finite Difference/FEM, Optimization (a very hardcore researcher in Math Finance, like Michael Steele/Steven Shreeve or Yves Achdou/Olivier Pironneau/Robert Kohn, would either look at a problem from a pure probability + SDEs point of view (those are strengths of M.Steele/S.Shreeve) or numerical + functional analysis (Y.Achdou/O.Pironneau) through the use of Black-Scholes' equation, and some modifications to fix some of its flawed assumptions. As you might imagine, the 2nd one is the one who actually need to prove convergence rate of H^1, L^2 or H^{infinity} norm on some bounded domains)

However, I realize that being way too narrow-minded by focusing purely on FE when applying might just completely kill off my fit with top OR programs, as an OR professor who uses OR to price options might say: "I like this guy's background in Analysis + PDE, but I don't know much about it. How am I supposed to advise him on such topic?" My current list includes Columbia's IEOR, UT-Austin's IROM, Northwestern's IEMS, UCSB's Probability and Stats, UPenn's AMCS, Maryland's AMSC, Cornell's ORFE, among other programs. As you probably see from the list, the major reason behind my decision to apply to some applied/computational math programs (like UPenn's AMCS, Maryland's AMSC, UCSB's Prob and Stats) is that these programs include Mathematical Finance as one of their research fields, and their professors usually are Math/Finance professors, not so much as OR professor. And without doubt, those professors would love to see students take Numerical Methods and Complex Analysis (grad level) more than Time Series and Dynamic Programming. I also find it surprising that the CMU's MSCF program requires Numerical Method as one of their core courses, besides Time Series and Financial Optimization (see here: http://tepper.cmu.edu/prospective-stude ... curriculum). Darn it, I wish the Numerical Methods for DEs will be offered again in the Fall, but it isn't:(

Thank you so much for your wish, and for your kindness in helping me figure out which path is the best for my preparation. I hope we can continue this conversation further, or even better, I'd love to learn more about your research areas that you mentioned (PM me if you can share that). Best of luck with your research as well.

Enigmatic
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Joined: Sat Jul 11, 2015 11:04 am

Re: Best courses to take for PhD application in Applied Math/Operation Research

Postby Enigmatic » Sun Dec 20, 2015 5:21 am

ghjk wrote:Now, regards to your mention about graph theory/combinatorial optimization/game theory, should I take any of these courses for the sake of showing strong fit with the department?

I suspect taking these courses will be useful.
ghjk wrote:Most of the times, I see them use things like Monte-Carlo simulation, stochastic DEs , integrations, Markov chain, some numerical methods like Finite Difference/FEM, Optimization (a very hardcore researcher in Math Finance, like Michael Steele/Steven Shreeve or Yves Achdou/Olivier Pironneau/Robert Kohn, would either look at a problem from a pure probability + SDEs point of view (those are strengths of M.Steele/S.Shreeve) or numerical + functional analysis (Y.Achdou/O.Pironneau) through the use of Black-Scholes' equation, and some modifications to fix some of its flawed assumptions. As you might imagine, the 2nd one is the one who actually need to prove convergence rate of H^1, L^2 or H^{infinity} norm on some bounded domains)

My understanding is that all this is close to stochastic optimal control theory (https://en.wikipedia.org/wiki/Stochastic_control). Google its connections with optimal pricing, Hamilton-Jacobi-Bellman equations etc. Essentially any multi-stage decision making or optimization problem is a control problem. You can be robust (H-inf control) - useful if you know uncertainty set, but not a distribution; stochastic (L-2 control) if you have an explicit probability distribution over uncertainty set; or do reinforcement learning - learn the probability distribution by interacting with system.

ghjk wrote: Darn it, I wish the Numerical Methods for DEs will be offered again in the Fall, but it isn't:(

I had assumed that you would have done at least a few numerical methods courses. If not, you should probably do that in lieu of DP or TSA. OR and FE are mostly computational disciplines.

ghjk
Posts: 81
Joined: Sat Apr 21, 2012 8:42 pm

Re: Best courses to take for PhD application in Applied Math/Operation Research

Postby ghjk » Sun Dec 20, 2015 4:59 pm

Enigmatic wrote:My understanding is that all this is close to stochastic optimal control theory (https://en.wikipedia.org/wiki/Stochastic_control). Google its connections with optimal pricing, Hamilton-Jacobi-Bellman equations etc. Essentially any multi-stage decision making or optimization problem is a control problem. You can be robust (H-inf control) - useful if you know uncertainty set, but not a distribution; stochastic (L-2 control) if you have an explicit probability distribution over uncertainty set; or do reinforcement learning - learn the probability distribution by interacting with system.


Thank you so much for pointing out that very cool stuff on optimal control theory, and your detailed elaboration (very useful for me to know!). It indeed links to optimization and the Hamilton-Jacobi-Bellman equation, but I haven't been able to find any references that shows its specific connection with optimal pricing. I also realize one thing: it uses DP extensively!! May you please recommend me a book about that special relationship (for example, this article is quite interesting to me: http://pubsonline.informs.org.mutex.gmu ... .2013.1240)?

Enigmatic wrote:
I had assumed that you would have done at least a few numerical methods courses. If not, you should probably do that in lieu of DP or TSA. OR and FE are mostly computational disciplines.


I only took a Numerical Analysis grad course, but not any specialized courses dedicated to specific technique like the Numerical Method for DEs (this course would only focus on Finite Difference and its use to solve BVP and PDEs. Not sure if that's a bad or good thing?). But if I take this course, I must choose between DP and Time Series. In your opinion, Which one would be more useful for competitive OR and/or some applied math PhD programs with mathematical finance as one of their research areas?

ghjk
Posts: 81
Joined: Sat Apr 21, 2012 8:42 pm

Re: Best courses to take for PhD application in Applied Math/Operation Research

Postby ghjk » Sun Dec 20, 2015 5:10 pm

Would love to hear more thoughts from other people as well, so if anyone can, please jump in to help/discuss:)




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