The ACO Seminar (2025–2026)

When can a low--regularity object, such as a metric space or a graph, be embedded into a high regularity object, such as a normed space? This is a fundamental question in metric geometry, specifically the Ribe program, with wide-ranging applications in algorithm design, geometric group theory, and functional analysis. Traditional approaches to such problems rely on heavy machinery from analysis and geometry. We will introduce a recent program---joint with P. Dodos, K. Tikhomirov, and K. Tyros---towards developing direct combinatorial and probabilistic methods for studying (random) graph embeddings. Some highlight results include the resolution of a long--standing question on the asymptotics of Bourgain's metric dimension reduction modulus, as well as a solution to an outstanding problem of Jon Kleinberg.

No prior knowledge of metric geometry will be assumed; the first portion of the talk will aim to give a high--level overview of some of the key definitions, techniques, and questions in the field.