Research
Universality & Undecidability
Why is it so easy to generate complexity? One perspective is provided by universality: essentially every non-trivial system is universal, that is, capable of exploring all complexity in its domain. We understand various aspects of universality in three domains: In computer science, universal Turing machines can run any algorithm; in machine learning, simple artificial neural networks can learn any function by virtue of the universal approximation theorem; in physics, universal spin models can simulate any other spin model. These three notions of universality share many similarities (even in their proofs), but the knowledge is hitherto isolated in their respective disciplines. Our team investigates connections between these (and other) notions of universality, as well as the other side of the coin of universality, namely undecidability (see this essay).
Recent Publications
Positivity structures
Notions of positivity are central in many problems in physics, mathematics, and beyond: probabilities are nonnegative, and quantum states (which encapsulate some notion of ‘quantum probabilities’) are nonnegative in a non-commutative way. Positivity interacts in a funny way with the multiplicity of systems, and with invariance of the element. This applies to quantum many-body systems, quantum magic squares, polynomials… And quantum information theory and free semialgebraic geometry often study the same problem from different angles. Our teams investigates various aspects of the interplay between positivity structures and the multiplicity of systems.