Draft of dissertation:

In this paper we study properties of groupoids by looking at their $C^∗$-algebras. We introduce a notion of rapid decay for transformation groupoids and we show that this is equivalent to the underlying group having the property of rapid decay. We show that our definition is equivalent to a number of other properties which are in direct correspondence to the group case. Additionally, given two bilipschitz equivalent discrete groups we construct an isomorphism of the corresponding transformation groupoids and are able to reformulate the open problem of showing invariance of rapid decay under quasi-isometry.
We then begin to examine various notions of amenability when abstracted to measured ́etale groupoids. In the group case, the following properties are equivalent:
1) $G$ is amenable
2) $C_r^∗(G)=C^∗(G)$
3) The trivial representation decends from ^C^∗(G)$ to $C_r^∗(G)$.
In the groupoid, $G$, case we have 1) ⇒ 2) ⇒ 3), but it was shown by Rufus Willett that $C_r^∗(G) = C^∗(G)$ is not enough in general to give amenability of G. In this paper we study property 3) for groupoids, formulate some equivalent statements and show that 3) ⇒ 2) is also false in general.

Kuykendall 210

Draft report

To fulfill the specialty exam requirement, Sarah Berner will give a presentation on “p-adic analogs of the Mandelbrot set” in Keller 403. Please join us if you can.

Title: Universes of sets

Abstract: As is well-known, all mathematical objects can be coded as sets and thereby all of mathematics can be formally founded in set theory. What is perhaps less well-known is that there are many different models of set theory, each of which is powerful enough to function as a universe of sets and found (most) of mathematics, but these models can have very different properties.

This talk will aim to explore the question: what is a model of set theory? We will learn about Skolem’s paradox, that there are countable models of set theory, even though these countable models think they contain uncountable sets like the set of reals. We will be introduced to transitive models, usually considered to be the best behaved, but also meet ill-founded models, such as models which think ZFC is inconsistent. To conclude we will briefly discuss two positions in the philosophy of set theory: universism, the view that there is a unique maximal universe of sets, and multiversism, the view that there are many equally valid universes of sets.

This is an introductory talk, aimed to be understandable by those with little background in set theory. It is a prequel to my next talk, which is in turn a prequel to my talk after that.

Title: A conceptual overview of forcing

Abstract: Paul Cohen—who visited UH Mānoa in the 1990s—introduced the method of forcing to prove that the failure of the continuum hypothesis is consistent with ZFC, the standard base axioms for set theory. Since then it has become a cardinal tool within set theory, being the main method for proving independence results and even enjoys use in proving ZFC results. In this talk I will give an introduction to forcing, focusing on the big picture ideas.

This talk is a sequel to my previous talk and a prequel to my next talk.