### Jake Koerner, Joo Seung Lee, Oleksii Mostovyi

accepted in the Missouri Journal of Mathematical Sciences (2023)

accepted in the Missouri Journal of Mathematical Sciences (2023)

**Séminaire Lotharingien de Combinatoire (2021)**

Sém. Lothar. Combin.85B(2021), Art. 14, 12 pp.

**Proceedings of the 33rd Conference on Formal Power****Series and Algebraic Combinatorics ****Ben Drucker, Eli Garcia, Emily Gunawan, and Rose Silver**

A box-ball system is a collection of discrete time states representing a permutation,

on which there is an action called a BBS move. After a finite number of BBS moves

the system decomposes into a collection of soliton states; these are weakly

increasing and invariant under BBS moves. The students proved that when this

collection of soliton states is a Young tableau or coincides with a partition of a type

described by Robinson-Schensted (RS), then it is an RS insertion tableau. They also

studied the number of steps required to reach this state.

Samantha Forshay, Leng Mawi, Matthew Peeks.

Chris Hayes, Luke Rogers, Sasha Teplyaev.

A celebrated result in analysis and probability on fractals is the construction of a diffusion on the standard Sierpinski Carpet by Barlow and Bass. One key part of their argument is a pair of upper and lower estimates for the resistances of precarpets: if \(K_n\) denotes the level \(n\) approximation of the carpet and \(E_n\) is the minimal Dirichlet energy of a function that is identically 1 on one side of the carpet and identically 0 on the other side, then there are constants \(0<c\leq C< \infty\) so that \(c\rho^n\leq E_n\leq C\rho^n\). Estimates for \(\rho\) are known but the exact value is not.

The Sierpinski-type carpets to which the preceding estimates of resistance have been extended are all self-similar. By contrast, in the setting of post-critically finite fractals, resistance scaling has been successfully studied also in the self-affine case, initially by Fitzsimmons, Hambly and Kumagai.

The goal of this project was to investigate what aspects of the Barlow-Bass approach to resistance estimation on carpets could be extended to the self-affine case, and to make numerical computations of the behavior of resistance in this setting and its dependence on the affine scalings.