In the last four to five decades, classical geometric function theory has evolved to encompass dynamical, metric, and probabilistic aspects. One of the notable advancements in these fields is Schramm's introduction of Schramm-Loewner Evolutions (SLE’s), which have been influential in understanding the relationship between conformally invariant and probabilistic structures. The elucidation of the intricate structure of the Mandelbrot set has lead to further insights into small scale structures in holomorphic dynamics. Thurston’s groundbreaking geometrization program has been another fundamental contribution, revolutionizing low dimensional topology.

Concurrently, there has been a growing recognition of the synergy between dynamical and geometric concepts. Sullivan’s influential dictionary between the dynamics of rational maps and Kleinian groups, provides a powerful framework to translate problems between dynamics and geometry. For instance, Cannon’s conjecture, which has spurred extensive research in fractal geometry and hyperbolic groups, can be related under Sullivan’s dictionary to Thurston’s topological characterization of rational maps. While the central questions of each of these topics are distinct, all of these areas have benefitted from developments in the others and this conference aims to further this cross-pollination of ideas.

While some of these developments, such as Thurston's Geometrization Conjecture, have been successfully realized, others, like whether the Mandelbrot set is locally connected (MLC) or Cannon's conjecture, remain open questions. The network of connections between holomorphic dynamics, geometry, and random processes, akin to Sullivan's dictionary, continues to expand.

Many notable advancements in these topics have come from collaboration between several of the proposed speakers and organizers. For instance, Bishop and Rempe recently showed that all non-compact Riemann surfaces may be obtained by gluing equilateral triangles. Random triangulations are well studied in the probabilistic setting, for instance in the context of Liouville quantum gravity. Bonk, Lyubich, and Merenkov used tools from the metric setting, specifically tools from quasiconformal geometry, to study the quasisymmetries of Sierpinski carpet Julia sets. Dudko, Lyubich, and Selinger have made progress on local connectivity of the Mandelbrot set, and hence density of hyperbolicity, a central topic in dynamics, by introducing new powerful renormalization tools.

This workshop aims to unite researchers working in these areas and related fields, fostering collaboration and exchange of ideas for the mutual benefit of all. Equally important is the goal of nurturing the next generation of researchers. In addition to inviting emerging scholars as speakers, we will actively promote the workshop to attract participation from graduate students and post-doctoral fellows, with a focus on enhancing diversity among participants.