Different definitions of conformal dimension are (essentially) equal,

Professor Sylvester Eriksson-Bique, University of Jyväskylä

Abstract:

I want to tell you about something that came out of OIST. Last May OIST hosted a work-
shop in analysis, random walks and potential theory on metric spaces, which showcased
some exciting developments in these areas. I was fortunate to participate, and the topics
of the conference prompted discussions with another participant Mathav Murugan, who
told about an open problem regarding the conformal dimension of the Sierpinski carpet.
The question asks, if different definitions of this notion are equal. Through discussions at
the workshop, I solved this question. I will explain the problem, the notions of conformal
dimension, and the crucial tool: a new notion of discrete modulus. There are roughly two
approaches classically to define discrete modulus, and this new approach lies roughly in
between the two — in such a way, that it can benefit from good estimates in both of the
worlds. The talk will be fairly mathematical, but I will try to give definitions of the main
concepts and some motivation.

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