Weakly quasisymmetric embeddings of R into C

Amenability, Tubularity, and Embeddings into R Ω

Inapproximability for Metric Embeddings into R

We consider the problem of computing the smallest possible distortion for embedding of a given n-point metric space into Rd, where d is fixed (and small). For d = 1, it was known that approximating the minimum distortion with a factor better than roughly n1/12 is NP-hard. From this result we derive inapproximability with a factor roughly n1/(22d−10) for every fixed d ≥ 2, by a conceptually very...

Amenability, Tubularity, and Embeddings into R

Suppose M is a tracial von Neumann algebra embeddable into R ω (the ultraproduct of the hyperfinite II 1-factor) and X is an n-tuple of selfadjoint generators for M. Denote by Γ(X; m, k, γ) the microstate space of X of order (m, k, γ). We say that X is tubular if for any ǫ > 0 there exist m ∈ N

Quasisymmetric embeddings, the observable diameter, and expansion properties of graphs

It is shown that the edges of any n-point vertex expander can be replaced by new edges so that the resulting graph is an edge expander, and such that any two vertices that are joined by a new edge are at distance O( √ log n) in the original graph. This result is optimal, and is shown to have various geometric consequences. In particular, it is used to obtain an alternative perspective on the re...

Embeddings of Schur functions into types B/C/D

We consider the problem of embedding the semi-ring of Schur-positive symmetric polynomials into its analogue for the classical types B/C/D. If we preserve highest weights and add the additional Lie-theoretic parity assumption that the weights in images of Schur functions lie in a single translate of the root lattice, there are exactly two solutions. These naturally extend the Kirillov–Reshetikh...