Mathematical Conversations

Some people think that the brain is something like a (conscious) computer. But if a brain can compute, why can't a rock, or a river stream? This basic question has been considered by philosophers, physicists, and mathematicians.

It is not entirely obvious how one can pose questions about the computational complexity of continuous dynamical systems, but the existing approaches largely fit into a common framework, and lead to meaningful mathematical questions. I will give some historical overview, explain a (precise) variant of this framework, and describe some interesting results and open problems in the subject and in related areas. If I have time, I will state a result about the shape of real algebraic sets.