2. Especially since you have a pure math background, you could work on SAGE (http://sagemath.org). The people there (mostly academics) are very friendly and open, and you should be able to find something with an AI flavour (e.g., automated empirical optimization of software, as is done in ATLAS http://math-atlas.sourceforge.net/) that has a fair chance of generating a publication, or at least generating an outcome you can point to.