The Geometry of Thought: Perspectives on Neural and Artificial Representations

Understanding how complex systems represent, transform, and act on information is a shared challenge across neuroscience, machine learning, and cognitive science. In both biological and artificial systems, internal representations exhibit structure that goes beyond individual variables and can often be fruitfully described using mathematical concepts such as geometry, dynamics, and optimization.

In this lecture, I will present a selection of ideas from our recent work that explore how mathematical structures emerge in neural and artificial representations, and how they can be used to compare, interpret, and constrain models of cognition and learning. The examples will be drawn from empirical neural data, deep learning models, and learning agents interacting with complex environments.

Rather than focusing on a single method or result, the talk will emphasize conceptual connections. The goal is to offer an intuitive entry point for mathematicians into current questions at the interface of data-driven neuroscience and (Machine) learning theory, highlight fruitful applications of mathematical thinking to real systems, and outline open problems where further mathematical insight could play a decisive role.