In this post we review the proof of the Bott periodicity theorem for C*-algebras, generalizing the original Bott periodicity theorem from homotopy theory (and topological K-theory).
Continue reading “Theorem of the month – Bott Periodicity for operator K-theory”The rigid world of symmetric spaces of noncompact type
The current post is based on a talk I gave at the university of Leiden which unfortunately I completely butchered due, in part, to the sudden realization that I had not chance to get through this material in the given time. Nothing here is new or proved, though some sketches are added when I find it relevant.
The (unattained) goal of the presentation was to show how much of the geometry of a symmetric space of noncompact type can be determined from it boundary sphere alone (in rank 1 cases) and a simplicial complex called the boundary at infinity (in higher rank cases) by following the evolution of the Mostow rigidity theorem from its origins (closed quotients of real hyperbolic spaces) to higher ranks.
Continue reading “The rigid world of symmetric spaces of noncompact type”Actions of certain groups on the Gromov boundary of their Cayley graphs
The object of study of this very short post are groups of the form 
where 
. We will stick to the standard choice of generators 
and look at their associated (undirected) Cayley graph 
and boundary space 
. Recall that the Cayley graph of 
is the graph given the
Equivalent ways to define topology
In every course in basic topology one learns the standard way to define a topology on a set 
, how to induce it from a basis, ambient space or a collection of functions into or out of the set. There are however other ways to associate a topology to a given set which may come up in practice. Here is a list of some of the once I have come across, let me know if you think there should be more elements to the list, as I am not an expert on this stuff.
Existence of a maximal G-compactifications
Having had a hard time finding this material in English I decided to make a post on it myself. Here is a proof of the existence of a universal compactification of -space of various types.