Abstracts of Blumenthal Lectures 2015

I. THE PROJECTIVE HEAT MAP

I'll describe a polygon iteration which starts with

a polygon and does a straight line construction to produce

a new one with the same number of sides. The construction

is especially natural when it acts on the 2 dimensional

space of pentagons modulo projective equivalence. I'll

explain my theorem that the map converges almost every

(class of) pentagon to the projectively regular class, except for

a connected "Julia set" of planar measure 0.

The highlight of my result is something like

a coarse topological model for the Julia set. The

talk will have a lot of computer demos in it, and

it will start from scratch.

II. THE PLAID MODEL

I will introduce a construction which produces

embedded lattice polygons in the plane. The model

is highly structured and has both a combinatorial

and number-theoretic feel to it. I call the model

the plaid model because one of its descriptions

is in terms of grids of parallel lines. The plaid

model exhibits a hierarchical multi-scale kind of

structure. It is closely related to outer billiards

on kites, and also seems similar in spirit to Pat

Hooper's Truchet tile system. Mainly I will show

off the model with computer demos and explain what

theory about it I have worked out so far.