Lesson 5: Daniel
Daniel's lesson focused on the subject of Arithmetic Dynamics—that is, looking at iterated functions and how the groups and fields they are performed on can affect stability.
We saw the functions and algebraic structures as frameworks, out of which steady-state solutions emerged. Interesting connections to directed graph theory were possible with functions considered in fields (i.e. with modular arithmetic).
Elements of a walk could be split into three types:
Periodic (a fixed point is a periodic point with period 1)
Pre-Periodic (those that lead into a periodic loop)
Wandering (non-periodic; requires an infinite parameter space)
Natural questions arise: how long is the typical walk until a periodic point? What proportion of a domain is wandering? How do you predict if walking "backwards" in a graph results in necessary jumping outside the target field?
We sat and discussed various questions like this while Henry quickly created a walk visualizer in Mathematica.
Tomorrow is a discussion on the shapes and forms for final research proposals. Our midterm crit (with guest Prof. Brad Marston of Physics) is next week Tuesday.