Seminarium KMMF "Teoria Dwoistości"

The signature of a path X in R^d is a noncommutative formal power series σ(X) that uniquely characterizes X up to a mild equivalence. Signature tensors have been introduced by Kuo-Tsai Chen in 1954 and, after decades of oblivion, have recently risen again in the context of topological data analysis. In this talk, based on a recent preprint written in collaboration with F. Galuppi (MiMUW) and P. Santarsiero (University of Bologna), I will show that a path X is the solution of a given (system of) ODE(s) if and only if its signature σ(X) fulfills a certain system of algebraic equations. As a "pretext" for introducing signature tensors, I will briefly recall how ChatGPT predicts the next token for a given prompt and show that the so-called "attention" mechanism is nothing but a vector field on the space where signature tensors live.