Research on AI seems to be getting wider these days (2016). First, "obvious" few departments (no order):
- Computer Science (e.g. computation theory, algorithms): AI researchers there assume that intelligence is a kind of computation, under various forms (e.g. a neural network, a logic system).
- Software Engineering: Assuming we find a good model for AI, how do you make it? This is what the engineer will want to figure out. And it can be hard to map mathematical models to an engineered piece.
- Statistics and Probabilities (more specific than just Mathematics, which is also close to Computer Science): This is about Data Science, notably as a foundation to Machine Learning, the most active branch in AI---which "just" covers the learning part.
- Physics: This is particularly relevant now for hardware (see below).
- Neuro Science: Understand how the brain works, as an inspiration to create an artificial one, is the home for Connectionists. Recently, Hassabis and his team at Google Deepmind made several breakthroughs related to reinforcement learning, memory, attention, etc.
Recently Electric Engineering is getting a lot of light, together with the related branches of Physics. Several public and private laboratories focus on "brain chips". To name a few: IBM (who's working on that for some time already), Nvidia, and Facebook. Circa 2010, it became clear that techniques like deep learning require horsepower, thus an increasing focus on creating more powerful, smaller, more energy efficient chips. And on top of that, there is all the work in Quantum Computing.
But the thing is, there seems to be many more fields that are getting involved in AI research. We should mention Chemistry and Biology, as both inspiration and tools to make new models or hardware (e.g. chips that do not use silicon, so they can get smaller).
As for 2016, the above fields are the most active, and promise to remain very active for quite some time. Pick your own depending on your interest, skills, or mere intuition!
To finish, we may be surprised in a few years when we look back at where AI has come from. I believe that if we manage to build an AGI, it will leverage all these fields anyway. I guess the thrill is to be part of the story.