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.