What would constitute the very most important mathematics used in AI and Robotics? Not just game AI, all AI, I realize the obvious answer here is "everything" but I am looking for something more specific. Like, I can see probability being used in machine learning, linear algebra is a given. Anything else?

Nobody knows enough statistics.

Short post:

Convex optimization.

In reality, there are about four kinds of problems in the world that you can solve, and everything consists of reducing other problems to them:

1.) Linear system of equations

2.) Eigenvector/eigenvalue problems

3.) Convex optimization problems (which really includes 1, and mostly also includes 2)

4.) Low-dimensional dynamic programming

I exaggerate slightly, but not by much.

Long post:

To follow up on alvaro's comment about control theory:

There are a couple of core ideas that a lot of different people study, from slightly different angles, and with slightly different tools. These people include,

- control theoreticians

- roboticists

- classical AI (planning) people,

- machine learning researchers, and

- operations researchers.

The differences between these groups are as much social, cultural, and historical, as they are technical.

In my own view, there are a small number of core problems that they all aim to solve. The two that really stand out are,

1.) "Markov decision problems" -- a term that I use broadly to include both discrete time and state spaces (what people normally mean when they say "Markov Decision Problem" (MDP)), and with continuous time and state spaces (which people normally call "optimal control"). Additionally, problems here may or may not contain randomness, and each of these variants have very slightly different corresponding theories, but the underlying ideas are essentially the same. Do you want to find the shortest path through a maze? Or how to steer a fighter jet to perform a barrel roll? Or how to move a robot arm around an obstacle? These are MDP/optimal-control problems.

2.) State estimation problems ("filtering," "smoothing," and trajectory estimation). Whereas in my last bullet point the goal was to get a system to a particular state, here the goal is to figure out what state a system is in. Depending again on whether you are interested in discrete- or continuous- time- or state-spaces, depending on what noise model you assume, and depending on which of these problems you want to solve (filtering, smoothing, trajectory estimation) you end up with e.g. the "forward algorithm" for hidden Markov models, the Kalman filter for linear systems, or the Viterbi algorithm, among other closely-related algorithms.

There are generalizations and combinations of these ideas in all sorts of directions.

E.g., an operations researcher might say that Markov decision problems
are "just" optimization problems with a particular structure and go off
to look at Lagrange multipliers and duality gaps. He might also point
out all the problems that don't most naturally have this structure. Likewise, some
machine learning people might say that the idea of different "states"
at different "times," again, "just" describes a particular structure of
Bayes net. You can also combine #1 and #2 to arrive at POMDPs, which
are the most general (single-player) discrete--time-and-state problem. But despite all this, I think the two problems I listed above capture the essence of most things.

As for "statistics:" I do not think that there is a single unified intellectual edifice with this name. The only thing that makes any sense to me philosophically is personalist Bayesian statistics, and even that I'm still figuring out.

.......Why has nobody included basic calculus...?

Selenaut

.......Why has nobody included basic calculus...?

There is something called Control Theory that deals with many of the problems involved, especially in robotics. It does involve a lot of statistics, for sure, but some calculus is also required.