## July 12, 2005

### 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 $M\left(j,i\right)$ 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 $M:{I}^{\mathrm{op}}×I\to \mathrm{Set}$ (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 $M\otimes -:\left[I,\mathrm{Set}{\right]}_{\mathrm{nondeg}}\to \left[I,\mathrm{Set}{\right]}_{\mathrm{nondeg}}$. $\left[I,\mathrm{Set}{\right]}_{\mathrm{nondeg}}$ 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, $M=\left\{0,1\right\}$) 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

Posted at July 12, 2005 4:28 AM UTC

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