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I am math major junior considering math grad school. I need to decide whether to take complex analysis or cryptology this semester. Complex analysis seems to be a recommended course for people considering graduate school, but cryptology is strongly recommended by my advisor because 1. I will be exposed to many different fields of math in crypto so it is better for my overall mathematical maturity and 2. my current research interest is in probability and I may want to study probability in grad school. Also, I will see lots of applications of probability in cryptology and possibly find a new research idea.

Besides these two classes, by the time I graduate I will have taken calc, linear, modeling, diffeq, probability, math stats, sets and proofs, topology, graph theory, abstract algebra, real analysis, stochastic process, abstract algebra II, combinatorics, and numerical analysis/dynamical systems.

Any suggestions on whether I should take complex analysis or cryptology?

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    $\begingroup$ I'll let others decide, but this is probably not the right site for your question; I'd consider Mathematics StackExchange or Mathematics Educators (also part of the SE network) instead. I don't think I agree with your advisor regarding cryptology as being better for "mathematical maturity". Complex analysis is a core topic, and exposure to it before graduate school might be more useful in preparing for prelims, depending on the graduate school (e.g. if prelims test on broad exposure to core topics, rather than something more niche like cryptology). $\endgroup$
    – Todd Trimble
    Commented Jan 17, 2016 at 15:50
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    $\begingroup$ To avoid confusion given the close reason, the question is off-topic on Mathematics too. $\endgroup$
    – quid
    Commented Jan 17, 2016 at 17:19
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    $\begingroup$ @quid That makes sense; thanks. $\endgroup$
    – Todd Trimble
    Commented Jan 17, 2016 at 17:38
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    $\begingroup$ Complex Analysis is a rich theory with broad applications, and will especially strengthen your understanding of partial differential equations and the integral, not to mention open you up to some big picture "eureka" moments. An graduate prep course map is incomplete without it. Disclaimer: listen to your advisor. $\endgroup$
    – Andrew
    Commented Jan 18, 2016 at 0:19
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    $\begingroup$ I think this depends heavily on how the complex analysis course and how the cryptology course is taught at your university. Either could be taught as a soft elective without much in the way of proofs or almost as a first-year graduate course. $\endgroup$
    – Alexander Woo
    Commented Jan 18, 2016 at 8:27

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I will vote for complex analysis. The main theme of complex analysis is that differentiable complex functions are incredibly rigid, which is completely different from the behaviour of differentiable real functions. The main results of complex analysis are truly astonishing, when you think about them properly. In my opinion, complex analysis is the first subject in the usual undergraduate sequence where the main results have this character.

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An argument can be made in favor of Cryptography, not because it is more fundamental than Complex Analysis, or more beautiful, but because:

(1) You have not yet taken Number Theory, and you will learn quite a bit of Number Theory in a course on Cryptography. Here's a typical course: "Introduction to Cryptography and Computer Security", Brown Univ., 2014.

(2) It looks like you have learned a lot of math but not seen it applied directly to practical problems in your courses. The latest research moves from fundamental number theory to practical questions.

Ultimately, there is no wrong choice here: Either course would be both interesting and beneficial.

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    $\begingroup$ Cryptography without number theory would just be a unending torture of incomprehensible methods. $\endgroup$
    – vonbrand
    Commented Jan 17, 2016 at 21:39
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    $\begingroup$ @vonbrand: Most cryptography courses (in north America) introduce the number theory needed as part of the course. Rarely (at the undergraduate level) do they require number theory as a prerequisite. $\endgroup$
    – Joseph O'Rourke
    Commented Jan 17, 2016 at 22:17
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    $\begingroup$ I did a module in cryptography without Number Theory and found exactly that; they gave us enough Number Theory to get through. Incidentally, that Number Theory we did was enough to spark my interest and made me do more in my own time! Agree entirely with this answer, though I would say it depends a lot on what the OP wants to do afterwards.. Cryptography would open up a lot more options (both academic and not), but the absence of Complex Analysis would be considered a major disadvantage by many in the Academic Community. $\endgroup$
    – Ieuan Stanley
    Commented Jan 18, 2016 at 9:55
  • $\begingroup$ Thank you for your answer! My main concern is if I take cryptology rather than complex analysis, would grad school admission commitee view the absence of complex as a significant disadvantage? $\endgroup$
    – angie123
    Commented Jan 18, 2016 at 13:57
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    $\begingroup$ @angie123: I would hope not, but I should let those with experience on such admission committees answer. $\endgroup$
    – Joseph O'Rourke
    Commented Jan 18, 2016 at 17:32
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As you seem to have already studied essentially (expected to be) applied math, I suggest you keep going that way and maybe learn complex analysis aside by yourself for your own culture. I would have recommended complex analysis if you had been into physics or (analytic) number theory, where complex analysis is an essential tool.

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  1. Your advisor may be recommending one instructor over another; if so, trust that advice.

  2. Complex analysis will prepare you better than cryptology for classes in math grad school.

  3. For research topics in probability, an undergraduate math class is a poor source of ideas. You'd get more ideas from a class in econometrics or public health. If you want research ideas about probability in cryptology specifically, read up on Bitcoin debates.

I usually recommend applications, but the cryptography course doesn't seem to match your goals.

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My advice is to take complex analysis, but a more applied version (weak on proofs, strong on contour integration). Just based on your question and what you said about yourself, I think you will enjoy this more. And you can take more theoretical analysis versions later (if you choose). But at least you will have some exposure to a fundamental field.

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