If you are a PhD student at UI CS and are exploring the possibility of working with me, you should first glance at some of my publications (courtesy DBLP). These days, I am mostly into approximation algorithms for geometric optimization problems.
It usually takes a certain preparedness to really be in a position to dive into such research. For starters, it helps if you took our graduate algorithms course (22C:231) and enjoyed the content. You should also check out our Algorithms Reading Group, where we meet once a week and discuss Theory/algorithms papers. In addition:
I will now list a couple of general references. This is for the sake of information only, so don't feel overwhelmed by thinking of them as required background. For an introduction to approximation algorithms (not necessarily on geometric problems), a good book is the one by Vazirani titled "Approximation Algorithms". For background on discrete and computational geometry, great sources are "Lectures on Discrete Geometry" by Matousek and "Computational Geometry: Algorithms and Applications" by de Berg, Cheong, van Kreveld, and Overmars.
Finally, I like to attend, when possible, CS conferences including SoCG, SODA, STOC, and FOCS.