Conformal Dimension and Analysis On Fractals for Laakso-Type Spaces -

Conformal Dimension and Analysis On Fractals for Laakso-Type Spaces

Time: 2025-02-17 Published By: Xiaoni Tan

Speaker(s): Sylvester Eriksson-Bique (University of Jyväskylä)

Time: February 18 - February 22, 2025

Venue: Room 77201, Jingchunyuan 78, BICMR

Time & Location:
February 18 2025 3:00-4:00pm (Beijing Time) BICMR 77201, Peking University
February 20 2025 3:00-4:00pm (Beijing Time) BICMR 77201, Peking University

February 22 2025 3:00-4:00pm (Beijing Time) BICMR 77201, Peking University

Title: Conformal Dimension and Analysis On Fractals for Laakso-Type Spaces

Abstract:

Given a metric space X, its conformal dimension is the infimum of all Hausdorff dimensions of spaces quasisymmetric to it. This invariant is natural and arises in a number of settings, but is extremely difficult to understand well. For example, a result of Keith and Kleiner (shown by Carrasco-Piaggio) gives a natual characterization for its value, but in such a way that it is difficult to use in most explicit cases. In a different area of analysis on fractals, a similarly difficult porblem arises in constructing Dirichlet forms for self-similar fractals. The difficulty lies in the fact that certain modulus problems are very difficult to compute explicitly. This difficulty further has led to many open problems that have until now resisted efforts at their resolution. We have now been able to solve some of these.

With Riku Anttila we discovered a simple construction of a large class of fractals, Laakso-type fractals, for which everything becomes much more transparent. One can compute their conformal dimension, characterize the attainment of conformal dimension, describe Dirichlet forms explicitly and exhibit many behaviors previously only conjectured to hold. In this three lecture sequence I will give an introduction to these spaces, and use them to tell you about analysis on fractals and conformal dimension. The goal is to tell what these spaces are, and use them as simple examples to explain why Carrasco-Piaggio is true, and how one constructs Dirichlet forms on fractals. You do not need to know anything about conformal dimension or analysis on fractals as my goal will be to tell you what they are through these examples. Usually both of these are very difficult, but these spaces allow one to make the complicated objects explicit in a very illuminating, yet fairly simple, way.

Talk 1: The resolution of a conjecture of Kleiner and Laakso-type spaces

Talk 2: Conformal dimension for Laakso-type spaces,

Talk 3: Analysis on Fractals for Laakso-type spaces

Joint work with Riku Anttila, Ryosuke Shimizu and Lassi Rainio.