My favorite topic of mathematical research is an abstract area of discrete mathematics called Matroid Theory.  Hassler Whitney defined matroids in 1935 to study properties of dependence common to Linear Algebra and Graph Theory.  If you enjoyed studying linear independence in a Linear Algebra class or are intrigued by figures such as the Fano plane shown below,

 

you might like to learn about matroids.  A good place to start is the paper, "Briefly, What is a Matroid?" written by James G. Oxley.