Applied Math Seminar

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Walks in Rotation Spaces Return Home When Doubled and Scaled

Speaker: Tsvi Tlusty, Ulsan National Institute of Science & Technology

Location: Warren Weaver Hall 1302

Date: Friday, May 2, 2025, 2:30 p.m.

Synopsis:

The dynamics of numerous physical systems, such as spins and qubits, are series of rotations, that is, walks in the manifold of the rotation group. This provokes a natural question: How likely and under what conditions will such walks return to the origin, such that the physical system will return to its initial state?     The surprising result is that almost every walk in SU(2) or SO(3) – even a highly convoluted one – will preferentially return to the origin simply by traversing the walk twice in a row and uniformly scaling all rotation angles. We will see how this question arose from the problem of rolling bodies along arbitrary paths on the plane [1,2]. Then, we will explain how its “two-period” solution followed from the distribution of random rotations and Minkowski’s theorem, and why traversing the walk only once almost never suffices to return [3]. An ideal conclusion would be the audience proposing how this property of SU(2) can be used.

[1] Sobolev et al. Solid-body trajectoids shaped to roll along desired pathways. Nature, 2023.
[2] Eckmann et al. Tumbling Downhill Along a Given Curve. AMS Notices, 2024.
[3] Eckmann et al. Walks in Rotation Spaces Return Home When Doubled and Scaled. arXiv:2502.14367.