Brin Harper | Searcharxiv

arXiv2026

Dimension lower bounds in random geometry via Lipschitz functions

We prove lower bounds for the Hausdorff dimensions of various natural sets associated with the Liouville quantum gravity (LQG) metric. We prove that the set of 3-star points (i.e., starting points of three disjoint geodesics) has Hausdorff dimension at least two with respect to the LQG metric, which is conjectured to be optimal. Our proof works for a general class of planar length metrics which also includes, e.g., Kendall's Poisson roads metric. We additionally prove a dimension lower bound of one for the set of 2-star points intersected with the boundary and for the metric net intersected with the boundary, as well as a dimension lower bound of two for the intersection of two metric nets. In the particular setting of LQG, we obtain sharper lower bounds for the Hausdorff dimensions of the set of 2-star points and the LQG metric net, with respect to both the Euclidean metric and the LQG metric. Our proofs are primarily topological. The key idea is to express the sets of interest in terms of non-constancy sets of Lipschitz functions.

math.PR↗

arXiv2021

Learning to solve complex tasks by growing knowledge culturally across generations

Knowledge built culturally across generations allows humans to learn far more than an individual could glean from their own experience in a lifetime. Cultural knowledge in turn rests on language: language is the richest record of what previous generations believed, valued, and practiced, and how these evolved over time. The power and mechanisms of language as a means of cultural learning, however, are not well understood, and as a result, current AI systems do not leverage language as a means for cultural knowledge transmission. Here, we take a first step towards reverse-engineering cultural learning through language. We developed a suite of complex tasks in the form of minimalist-style video games, which we deployed in an iterated learning paradigm. Human participants were limited to only two attempts (two lives) to beat each game and were allowed to write a message to a future participant who read the message before playing. Knowledge accumulated gradually across generations, allowing later generations to advance further in the games and perform more efficient actions. Multigenerational learning followed a strikingly similar trajectory to individuals learning alone with an unlimited number of lives. Successive generations of learners were able to succeed by expressing distinct types of knowledge in natural language: the dynamics of the environment, valuable goals, dangerous risks, and strategies for success. The video game paradigm we pioneer here is thus a rich test bed for developing AI systems capable of acquiring and transmitting cultural knowledge.

cs.CL↗