Tell us what you're working on anons, and discuss. I'm currently reading pic related so I can start a research project on variations of Hodge structure. Here's the gist: Just like normal manifolds we can define complex manifolds, where all the maps are holomorphic instead of smooth. This is a very restrictive condition it turns out, and we further restrict our view to Kähler manifolds, on which the usual real geometry and the complex geometry agree. It turns out that with a little work (and blackboxing a big theorem from PDEs) you can show that the cohomology of a Kähler manifold $X$ decomposes into a bigraded direct sum $$H^k(X,\mathbb{C})=\bigoplus_{p+q=k}H^{p,q}.$$ Now you can study what happens to this decomposition when you deform $X$. This is what I'm working toward.

Any questions for me? What are you guys working on? If I have to read another thread about IQ or 0.9...=/=1 I'm going to kill myself.