In my research, I study Riemann surfaces, their moduli spaces, and their mapping class groups. The main tool I employ is a type of Feynman diagrams called a fatgraph which essentially gives a combinatorial representative of such surfaces. With this combinatorial perspective, one can use QFT techniques to study the cohomological aspects of moduli of Riemann surfaces and mapping class groups. More recently, I have used this combinatorial approach to study representations of mapping class groups, the classical Johnson homomorphisms, and finite type invariants of 3-manifolds. This last topic is interesting as it has a different QFT interpretations, and many of these finite type invariants also can be described in terms of similar Feynman diagrams.