Wednesday, November 26-th, 16:00, room 403, IMI-BAS
Joint meeting of the ICMS seminar and the seminar on Mathematical Foundations of Informatics
Frank Vallentin (Universität zu Köln)
Least distortion Euclidean embeddings of flat tori
Abstract: In the emerging field of bi-Lipschitz invariant theory, one studies embeddings of orbit spaces H/G (H a Hilbert space, G a subgroup of its automorphisms) into simpler Hilbert spaces via distance-preserving maps up to a constant factor.
In this talk, I present an infinite-dimensional semidefinite program that computes least-distortion embeddings of flat tori Rn/L, where L is an n-dimensional lattice, into Hilbert spaces. Using symmetry reduction techniques, this infinite-dimensional semidefinite program reduces to an infinite-dimensional linear program.
Even with this simplification, solving the program remains challenging. By combining this approach with the linear programming bound for spherical designs, combined with brute-force methods, we finally are able to determine the least-distortion embeddings of Rn/Zn, R2/A2, and R8/E8.
(Based on joint work with Arne Heimendahl, Moritz Lücke, Philippe Moustrou, and Marc Christian Zimmermann)