The String Coffee Table

David here - I thought I’d mention some of the stuff that went on in the afternoon - a talk by Tom Leinster and one from Boris Chorny.

Leinster talk on self similarity (ref: math.DS/0411343) from a general point of view - a self similar object looks like copies of another object glued to each other, each of which look like copies of another object etc, such that we have a finite pool of objects to draw on to glue together. The last part of that sentence is of course ill-defined, but one gets the idea. Really we can define the above “glued together” relation as a sort of eigenvalue system.

He gave the example of a particular Julia set of an endomorphism of the Riemann sphere - and broke it down into sets $X_0, X_1, X_2, X_3$ with $X_0=*$ and the others looking like copies of each other glued by $*$. There were certain recursive equations for each $X_i$ and we can define $M(j,i)$ to be the number of copies of $X_j$ in the equation for $X_i$.

Really the solution $X$ is a functor $X:I\to Set$, and we can make the concept of self similar system hard and fast by defining it to be a small category $I$ and a functor $M:I^{op}\times I \to Set$ (satisfying some conditions), and a solution is one satisfying $$X\isom M\otimes X$$

There is a concept of a universal solution, defined to be a coalgebra for the endofunctor $M\otimes - : [I,Set]_{nondeg} \to [I,Set]_{nondeg}$. $[I,Set]_{nondeg}$ is a full subcategory of the cat of homotopy classes of maps $I\to Set$

Examples of course include the the interval, the Cantor set C($I$=one object cat, $M=\{0,1\}$) and $\Delta$.

Someone pointed out the interesting example of a Cantor-Mobius band - the nontrivial C-bundle over $S^1$ - a Mobius strip with the interval replaced by C

Chorny’s talk was on the homotopy theory of small functors over large categories, and contained a lot I didn’t understand with model categories and something about the calculus of functors of Goodwillie.

The machines Marni and I are working on are Win XP, with IE as our browser so mathematical content is largely uncheckable when previewing. Blame Macquarie uni’s IT dept.

More later.

David