### Monday at the Streetfest II

#### Posted by Guest

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’s 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.

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 $X:I\to \mathrm{Set}$, 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 $M:{I}^{\mathrm{op}}\times I\to \mathrm{Set}$ (satisfying some conditions), and a solution is one satisfying $$XisomM\otimes X$$

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

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