C2

Product systems represent powerful contemporary tools in the study of mathematical structures. A major success in the theory came from Katsura (2007), who provided a complete description of the gauge-invariant ideals of many important C*-algebras arising from product systems over Z+. This result recaptures existing results from the literature, illustrating the versatility of product system theory. The question now becomes whether or not Katsura's result can be bolstered to product systems over semigroups other than Z+ and, if so, what applications do we obtain? An answer has been elusive, owing to the more pathological nature of product systems over general semigroups. However, recent strides by Dor-On and Kakariadis (2018) supply a more tractable subclass of product systems that still includes the important cases of C*-dynamics, row-finite higher-rank graphs, and regular product systems. 

In this talk we will build a parametrisation of the gauge-invariant ideals, starting from first principles and gradually increasing in complexity. We will pay particular attention to the higher-rank subtleties that are not witnessed in Katsura's theorem, and comment on the applications.
 

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Transportation cost spaces are of high theoretical interest,  and they also are fundamental in applications in many areas of applied mathematics, engineering, physics, computer science, finance, and social sciences. 

Obtaining low distortion embeddings of transportation cost spaces into L_1 became important in the problem of finding nearest points, an important research subject in theoretical computer science. After introducing

these spaces we will present some results on upper  and lower estimates of the distortion of embeddings of Transportation Cost Spaces into L_1

The notion of biexactness for groups was introduced by Ozawa in 2004 and has since become a major tool used for studying solidity of von Neumann algebras. We introduce the notion of biexactness for von Neumann algebras, which allows us to place many previous solidity results in a more systematic context, and naturally leads to extensions of these results. We will also discuss examples of solid factors that are not biexact. This is a joint work with Jesse Peterson.

In the world of von Neumann algebras, the factors that do not have a trace, the so-called type III factors, are the most difficult to study. Some of their key structural properties are still not well-understood. In this talk, I will give a gentle introduction to Connes's Bicentralizer Problem, which is the most important open problem in the theory of type III factors. I will then present some recent progress on this problem and its applications.

I will present novel freeness results in ultraproducts of tracial von Neumann algebras. As a particular case, I will show that if a and b are the generators of the free group F_2, then the relative commutants of a and b in the ultraproduct of the free group factor are free with respect to the ultraproduct trace. The proof is based on a surprising application of Lp-boundedness results of Fourier multipliers in free group factors for p > 2. I will describe applications of these results to absorption and model theory of II_1 factors. This is joint work with Adrian Ioana.