I have been introduced to algorithms, computability and computational complexity (as part of my minor in CS).

What are some mathematical topics that I can tackle with the new perspectives I acquired from these ideas of computer science? Could you suggest some reference books (preferably self-contained and accessible to an undergraduate)?

  • $\begingroup$ Numerical analysis seems like a very obvious choice. $\endgroup$ – Mark McClure Jan 12 '15 at 21:29

Some possible answers include

  • Machine learning: Machine learning is more or less applied statistics, using theoretical ideas from statistics to generate algorithms for analysing and grouping data. Data mining is also related to this area.
  • Computer algebra: Computer algebra studies how to devise efficient algorithms for algebraic manipulations. Many interesting algorithms from an applications point of view are still so inefficient as to often be useless in practice, and some old algorithms, like multiplication, have been improved significantly.
  • Cryptography: The basis of modern cryptography is the fact that some algorithms are computationally efficient while others are highly inefficient.
  • Numerical analysis: Many interesting problems, especially in calculus and analysis, can not be computed exactly. Therefore people have come up with clever algorithms to approximate the results for these problems instead, leading to the field of numerical analysis.

More generally, I would say that anytime you have a mathematical problem that you would like a solution for that can be used in practice, you need to consider whether or not your solution can at all be computed via an algorithm, and whether such an algorithm runs in a reasonable amount of time.

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