Activities This Week

One way of viewing coarse geometry is that it is the perfect tool for treating metric spaces like groups and doing representation theoretic/harmonic analytical techniques in a much larger setting. R. Willett and G. Yu define a property (T) for coarse spaces using the uniform Roe algebra. In this talk, I plan to define coarse spaces and show that the uniform Roe algebra is nice tool that acts like the group C* algebra. I will define geometric property (T) and discuss some of its properties.

The geometry of graphs is usually studied in the locally finite setup: each vertex has finitely many neighbors. By compactness arguments, one proves some useful and classical regularity theorems for such graphs. Such theorems are easily disproved for locally infinite graphs, but finding homogeneous counter-examples (transitive or Cayley) leads to interesting constructions. I will explain why the geometry of locally infinite graphs is worth studying, present my results, and state some questions I currently cannot answer.

We observe that several elementary definitions in point-set topology can be reformulated in terms of finite topological spaces and elementary category theory. This includes compactness of Hausdorff spaces, being connected, discrete, the separation axioms.

Though elementary, these observations raise a few open questions. For example, I was not able to prove that this reformulation of compactness gives the correct answer for non-Hausdorff spaces, or whether implications between various topological properties can also be proved entirely in terms of finite topological spaces, without any additional axioms.

We will give an overview of questions one might ask about the first-order theory of free groups and related groups: how much information can first-order formulas convey about these groups or their elements, what algebraic interpretation can be given for model theoretic notions such as forking independence, etc. It turns out that techniques from geometric group theory are very useful to tackle such problems. Some of these questions have been answered, others are still open - our aim is to give a feel for the techniques and directions of this field. We will assume no special knowledge of model theory.

An operator space is a complex vector space V together with a sequence of (complete) norms on square matrices of all sizes over V satisfying certain compatibility conditions. These conditions are due to Ruan who showed that they are necessary and sufficient for the sequence of matrix norms to be induced from a linear embedding of V as a closed subspace into the space of bounded linear operators on a Hilbert space. There are notions of completely bounded maps and complete isometries between operator spaces that correspond to bounded maps and isometries between Banach spaces. There is also a notion of the dual operator space.

There are (infinitely) many ways to extend the given norm on a Banach space to a sequence of matrix norms to obtain an operator space. In particular, there is a variety of natural operator space structures on a Hilbert space H, none of which turns out to be self dual. Pisier showed that there is a unique operator space structure on H that is self dual, i.e., such that the canonical isometry from H to the conjugate Hilbert space is a complete isometry; he called this operator space the (corresponding) operator Hilbert space, or OH.

There are two constructions of OH, a rather abstract one using complex interpolation for operator spaces and a more concrete one, using a noncommutative version of the Cauchy–Schwartz inequality that is due to Haagerup. In this talk, I will review some operator space basics, and then present a variation of the second construction that is motivated by the recent theory of completely positive noncommutative kernels (see Ball–Marx–Vinnikov, arXiv:1602.00760).