Date: | July 19, 2018 |

Time: | 3:00PM - 3:50PM |

Location: | BLOC 220 |

Speaker: | Mikhail Ostrovskii, St. John's University |

Title: | Lipschitz free spaces on finite metric spaces |

Abstract: | Abstract: The main results of the talk: (1) For any finite metric space $M$ the Lipschitz free space on $M$ contains a large well-complemented subspace which is close to $\ell_1^n$. (2) Lipschitz free spaces on large classes of recursively defined sequences of graphs are not uniformly isomorphic to $\ell_1^n$ of the corresponding dimensions. These classes contain well-known families of diamond graphs and Laakso graphs. Interesting features of our approach are: (a) We consider averages over groups of cycle-preserving bijections of graphs which are not necessarily graph automorphisms; (b) In the case of such recursive families of graphs as Laakso graphs we use the well-known approach of Gr\"unbaum (1960) and Rudin (1962) for estimating projection constants in the case where invariant projections are not unique. Joint work with Stephen Dilworth and Denka Kutzarova |

Date: | July 23, 2018 |

Time: | 3:00PM - 3:50PM |

Location: | BLOC 220 |

Speaker: | Beata Randrianantoanina, Miami University |

Title: | Bilipschitz embeddings of Cayley graphs of lamplighter groups |

Abstract: | We prove that the sequence of Cayley graphs of finite lamplighter groups $\{\mathbb{Z}_2\wr\mathbb{Z}_n\}_{n\ge 2}$ with a standard set of generators embeds bilipschitzly with uniformly bounded distortions into any non-superreflexive Banach space, and that this sequence is a is a set of test-spaces for superreflexivity in the sense of Ostrovskii. Our proof is inspired by the well known identification of Cayley graphs of infinite lamplighter groups with the horocyclic product of trees. We cover $\mathbb{Z}_2\wr\mathbb{Z}_n$ by three sets with a structure similar to a horocyclic product of trees, which enables us to construct well-controlled embeddings. Using known results, we also observe that, for any $q$, the Cayley graph of the infinite lamplighter group $\mathbb{Z}_q\wr\mathbb{Z}$ with respect to any finite generating set is a test space for superreflexivity. Joint work with Mikhail Ostrovskii. |