Groups, rings, algebras: this I could handle. If you tell me that the circle is $\mathbb{R} / \mathbb{Z}$, I believe you. Next, we started talking about the group of loops of a topological space, and that got a little crazy. If you were to tell me that the fundamental group of the Mobius strip is $\mathbb{Z}$ since it has the homotopy-type of a circle, I will nod because I have heard this before. Nowadays, I’ve got people in my ear whispering that there’s a duality between geometric properties of a structure and algebraic properties of the ring of analytic functions on that structure. This is all getting out of hand. What’s next?
Topics often include:
- abstract algebra
- alg topo
- alg geo
- algebraic number theory (not for me)
list of papers
paper summaries/notes
“An Algebraic Approach to Chaos”
- cool and sexy paper that takes Delaney’s defn of “chaotic maps $f$” via 3 conditions and removes pointwise specification of these, allowing us to consider chaos on structures other than metric spaces, such as commutative rings
- topological transitivity - The original condition is that for two nonempty open $U, V$there exists $n$ large enough that $U \cap f^{-n}(V) \neq \emptyset$. This paper doesn’t change this condition, which already has no pointwise consideration.
- periodic points are dense - She changes this condition by introducing an algebraic formulation of periodicity via absolute periodicity on the lattice of ideals, which turns out to be equivalent sometimes. this is the bulk of the novel stuff of the paper
- sensitive dependence on initial conditions - The classic characterization of chaos as when every deviation in initial conditions eventually produces a uniformly-lower-bounded deviation in state. Turns out (1) and (2) together $\implies$ (3), so we don’t even need to consider this.