# From phase space to Krylov space: complexity and the geometry of chaos

- Institution: CERN
- Speaker: Julian Sonner
- Year: 2026
- Language: English
- Subjects: Mathematics, Astronomy and space
- Source: [CERN](https://videos.cern.ch/record/3025662)
- Licence: [CERN](https://copyright.web.cern)
- Who can watch: Tweens profiles and older (sources give no ratings; placed by Invisible Network from the source's own words)

We develop a classical counterpart of the Krylov complexity framework by running the Lanczos algorithm directly on the algebra of observables of a Hamiltonian system on a compact symplectic manifold. This construction arises naturally as the semiclassical limit of the quantum Lanczos algorithm. We demonstrate the usefulness of classical Krylov complexity in characterisation early-time dynamics in chaotic quantum systems with a semiclassical limit, and derive a Krylov-Ehrenfest theorem, which captures the eventual divergence of semiclassical and exact Krylov dynamics.

00:00:00 Slide 1

00:00:36 Slide 2

00:02:58 Slide 3

00:06:04 Slide 4

00:07:57 Slide 5

00:13:45 Slide 6

00:18:50 Slide 7

00:19:11 Slide 8

00:19:23 Slide 9

00:19:53 Slide 10

00:22:03 Slide 11

00:26:02 Slide 12

00:28:06 Slide 13

00:30:52 Slide 14

00:31:43 Slide 15

00:32:07 Slide 16

00:33:37 Slide 17

00:35:31 Slide 18

00:38:05 Slide 19

00:39:17 Slide 20

00:39:41 Slide 21

00:40:55 Slide 22

00:41:25 Slide 23

00:41:47 Slide 24

00:43:06 Slide 25

00:45:01 Slide 26

00:49:56 Slide 27

00:50:14 Slide 28

00:53:38 Slide 29

00:55:27 Slide 30

00:56:16 Slide 31

00:57:08 Slide 32

01:00:44 Slide 33

01:02:26 Slide 34

## Lectures

1. [From phase space to Krylov space: complexity and the geometry of chaos](https://invisiblenetwork.ai/learn/cern/3025662/1a881c8d4e8a), 1 h 6 min (CERN's player)