Look at any of my published papers! :-)
- Solved huge systems of quadratics in order to construct exotic subfactors.
- Calculated R-matrices for representations of various quantum groups, in order to systematically identify coincidences amongst small modular tensor categories.
- Implemented a parallelisable, caching algorithm for computing Khovanov homology, in an attempt to disprove the smooth 4-d Poincare conjecture.
- Enumerated bipartite graphs satisfying certain combinatorial conditions, filtered by largest eigenvalue, in order to classify subfactors up to index 5 (slides).
- 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.
- 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.