[Math] Where have you used computer programming in your career as an (applied/pure) mathematician

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For background: I'm working on a book to help mathematicians learn how to program. However, I need to see some examples from people in the field that have done different kinds of things than I have.

Where have you used programming in your career as a mathematician? (If you haven't feel free to say so, though it isn't very helpful)

I've currently used programming in several math-y settings. Computational Biology, Image Processing (Fourier Transforms and other things like that), writing scripts that comply to a certain data restriction or to a library. I've looked at some computational algebraic geometry, but not much as of yet, and I'd use SAGE or sympy if I needed that.

Best Answer

Look at any of my published papers! :-)

I've

  1. Solved huge systems of quadratics in order to construct exotic subfactors.
  2. Calculated R-matrices for representations of various quantum groups, in order to systematically identify coincidences amongst small modular tensor categories.
  3. Implemented a parallelisable, caching algorithm for computing Khovanov homology, in an attempt to disprove the smooth 4-d Poincare conjecture.
  4. Enumerated bipartite graphs satisfying certain combinatorial conditions, filtered by largest eigenvalue, in order to classify subfactors up to index 5 (slides).
  5. Looked for small real cyclotomic integers which are larger than all their conjugates, in order to verify the many cases of a theorem identifying those smaller than 76/33.
  6. Proved identities involving q-binomial coefficients using the methods of A=B.

and then a whole lot of stuff that hasn't yet, or won't ever, make it into print.

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