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 R^n / 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 R^n / Z^n, R^2 / A_2, and R^8 / E_8.

(Based on joint work with Arne Heimendahl, Moritz Lücke, Philippe Moustrou, and Marc Christian Zimmermann)