# Mathematical Sciences Research Institute

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# Seminar

AGRS Research Seminar Series: Conformal Walk Dimension February 16, 2022 (02:00 PM PST - 03:00 PM PST)
Parent Program: The Analysis and Geometry of Random Spaces MSRI: Simons Auditorium, Online/Virtual
Speaker(s) Mathav Murugan (University of British Columbia)
Description No Description
Video

#### AGRS Research Seminar Series: Conformal Walk Dimension

Abstract/Media

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In the field of diffusion processes on fractals, the transition density of the diffusion typically satisfies analogs of Gaussian bounds which involve a space-time scaling exponent \beta greater than two and thereby are called sub-Gaussian bounds. The exponent \beta, called the walk dimension of the diffusion, could be considered as representing how close the geometry of the fractal is to being smooth''. Kigami (2008) observed that, in the case of the standard two-dimensional Sierpi\'{n}ski gasket, one can decrease this exponent to two (so that Gaussian bounds hold) by suitable changes of the metric and the measure while keeping the associated Dirichlet form (the quadratic energy functional) the same. Then it is natural to ask how general this phenomenon is for diffusions. This leads to the notion of conformal walk dimension. I will explain its definition and its universal value. Our work suggests some new methods to understand the attainment of Ahlfors regular conformal dimension on fractals. I will also present some conjectures in this direction.

This talk is based on joint works with Naotaka Kajino (Kyoto University).

 Conformal Walk Dimension 3.86 MB application/pdf