# Seminars & Events for Minerva Lectures

##### Minerva Lecture I: An introduction to the Ribe program

A classical theorem of Martin Ribe asserts that finite dimensional linear properties of normed spaces are preserved under uniformly continuous homeomorphisms. Thus, normed spaces exhibit a strong rigidity property: their structure as metric spaces determines the linear properties of their finite dimensional subspaces. This clearly says a lot about the geometry of normed spaces, but it turns out that one can also use it to understand the structure of metric spaces that have nothing to do with linear spaces, such as graphs, manifolds or groups. PLEASE CLICK ON LECTURE TITLE FOR COMPLETE ABSTRACT.

##### Minerva Lecture II: Dichotomies and universality in metric embeddings.

It is a basic fact that the space of continuous functions on the interval [0,1] contains an isometric copy of every separable metric space, and in particular of every finite metric space. However, it is a subtle question to

decide whether or not a given metric space is universal in the sense that it contains a copy of every finite metric space with O(1) bi-Lipschitz distortion. PLEASE CLICK ON LECTURE TITLE FOR COMPLETE ABSTRACT.

##### Minerva Lecture III: Super-expanders and nonlinear spectral calculus

While the topic of this talk initially arose in the context of the Ribe program, it will take us further afield. A bounded degree n-vertex graph $G = (V, E)$ is an expander if and only if for every choice of $n$ vectors ${x_v}_{v\in V}$ in $R^k$ the average of the Euclidean distance between $x_u$ and $x_v$ is within a constant factor of the average of the same terms over those pairs ${u, v}$ that form an edge in $E$. PLEASE CLICK ON LECTURE TITLE FOR COMPLETE ABSTRACT.