## Departmental Ph.D. Thesis Exam – Wenbo Li

Everyone is welcome to attend the presentation. Refreshments will be served in the lounge at 3:30 p.m.!

Tuesday, April 5, 2022 at 4:00 p.m.

PhD Candidate: Wenbo Li
Supervisor: Ilia Binder
Thesis title: Quasiconformal Geometry of Metric Measure Spaces
and its Application to Stochastic Processes

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We study three topics of quasiconformal geometry in this dissertation; the quasisymmetric embeddability of metric Sierpi\’nski carpets, the quasisymmetric embeddability of weak tangents and the conformal dimension of stochastic spaces. The common tools we use to attack these three topics are different versions of Moduli and the convergence of spaces.

For the first problem, the problem of quasisymmetrically embedding spaces homeomorphic to the Sierpi\’nski carpet into the plane, we use transboundary modulus to study it. This allows us to give a complete characterization in the case of dyadic slit carpets. Every such slit carpet $X$ can be embedded into a “pillowcase sphere” $\widehat{X}$ which is a metric space homeomorphic to the sphere $\mathbb{S}^2$. We show that $X$ can be quasisymmetrically embedded into the plane if and only if $\widehat{X}$ is quasisymmetric to $\mathbb{S}^2$ if and only if $\widehat{X}$ is Ahlfors $2$-regular.

For the second problem, the problem of quasisymmetric embeddability of weak tangents of metric spaces, we first show that quasisymmetric embeddability is hereditary, i.e., if $X$ can be quasisymmetrically embedded into $Y$, then every weak tangent of $X$ can be quasisymmetrically embedded into some weak tangent of $Y$, given that $X$ is proper and doubling. However, the converse is not true in general; we will illustrate this with several counterexamples. In special situations, we are able to show that the embeddability of weak tangents implies global or local embeddability of the ambient space. Finally, we apply our results to Gromov hyperbolic groups and visual spheres of expanding Thurston maps.

For the third problem, the conformal dimension of stochastic spaces, we develop tools related to the Fuglede modulus to study it. In order to achieve this goal, we study the conformal dimension of deterministic and random Cantor sets and investigate the situation of conformal dimension $1$. We apply our techniques to construct minimal(in terms of conformal dimension) planar graph. We further develop this line of inquiry by proving that a “natural” object, the graph of one dimensional Brownian motion, is almost surely minimal.

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A draft of the thesis is available here: Wenbo Li Ph.D. Dissertation UofT Mathematics