# Wollongong OANCG Seminar

A weekly research seminar hosted at the University of Wollongong on the topics of operator algebras, noncommutative geometry, and related fields. The seminar has been running continuously since 2011.

## Key Info

• Regular time: Thursday 3:30 pm Wollongong time (this may vary occasionally and will be noted on the talks below)
• Regular location: 39C.174 - Campus map
• Zoom: Link - Meeting ID: 861 2230 0349 (password provided on request)
• Organisers: Abraham Ng & Alexander Mundey

Talks typically run for around 55 minutes. We welcome speakers ranging from seasoned researchers to graduate students. We hold talks on campus and host remote speakers via Zoom.

## Talk Schedule - Autumn 2023

Click on "Abstract" to reveal the abstract.

### Upcoming talks

• Mar 23: Enrique Pardo (Universidad de Cádiz)
Modeling groupoid algebras using left cancellative small categories
Abstract A decade ago, Spielberg described a new method of defining C*-algebras from oriented combinatorial data, generalizing the construction of algebras from directed graphs, higher-rank graphs, and (quasi-)ordered groups. To this end, he introduced left cancellative small categories, and endowed any such category with a C*-algebra encoding categorical information; he showed that this algebra is the groupoid algebra of a Deaconu-Renault étale groupoid.
In this talk, we will try to explain why such algebras are relevant. Also, we will show that they are also Exel's groupoid C*-algebras associated to a suitable inverse semigroup $$\mathcal{S}_\Lambda$$. Subsequently, we study groupoid actions on left cancellative small categories and their associated Zappa-Szép products using the same strategy. We show that certain left cancellative small categories with nice length functions can be seen as Zappa-Szép products. Then, we can characterize properties of them, like being Hausdorff, effective and minimal, and thus simplicity for these algebras. Also, we determine amenability of the tight groupoid under mild, reasonable hypotheses.
The contents of this talk are a joint work with Eduard Ortega (NTNU Trondheim, Norway)
• Mar 30: Rodrigo Frausino (University of Wollongong)
TBA
Abstract TBA
• Apr 6: Aidan Sims (University of Wollongong)
TBA
Abstract TBA
• Apr 13: Malcolm Jones (Victoria University Wellington) - Zoom broadcast
TBA
Abstract TBA
• Apr 20: Aidan Sims (University of Wollongong)
TBA
Abstract TBA
• Apr 27: Alistair Miller (University of Southern Denmark) - Zoom broadcast
TBA
Abstract TBA
• All following dates are all TBA!

### Recent talks

• Mar 16: Becky Armstrong (University of Münster)
Conjugacy of local homeomorphisms via groupoids and C*-algebras
Abstract A Deaconu–Renault system consists of a partially defined local homeomorphism on a locally compact Hausdorff space, and for each such system there is an associated amenable Hausdorff étale groupoid. Deaconu–Renault systems give rise to a large class of (groupoid) C*-algebras that, in particular, includes graph C*-algebras, crossed products by actions of the integers, and all Kirchberg algebras satisfying the UCT. In this talk I will introduce a notion of (topological) conjugacy of Deaconu–Renault systems, and I will show how to recover the conjugacy class of a Deaconu–Renault system from its associated groupoid or groupoid C*-algebra. (This is joint work with Kevin Aguyar Brix, Toke Meier Carlsen, and Søren Eilers.)
• Mar 9: Eva-Maria Hekkelman (University of New South Wales)
A Dixmier trace formula for the density of states and Roe's index theorem
Abstract The density of states (DOS) is a measure associated to an operator, which is an important object in solid state physics. In this talk, it will be shown this measure can be recovered via a Dixmier trace in various settings. This Dixmier trace formula for the DOS resembles Connes' integration formula for the Lebesgue measure. As an application, we will give a scandalously brief overview of Roe's index theorem for non-compact manifolds (which is one way of extending Atiyah–Singer's index theorem) and show that in certain cases everything can be assembled to a Dixmier trace formula for Roe's index.
• Mar 2: Jonathan Taylor (Georg-August-Universität Göttingen)
Essential Cartan pairs of C*-algebras and their twisted groupoid models (Australian man lost in Europe for three years; comes home speaking non-Hausdorff)
Abstract Classifying all C*-algebras is hard, so we don't do that. Instead, one approach is to look at C*-algebras with a particularly 'nice' subalgebra, such as one that is maximal abelian. This has been studied by Kumjian and Renault, showing that (with a few extra assumptions) one may show that these C*-algebras are twisted groupoid C*-algebras. Many people have worked to remove some of the underlying assumptions and aquire similar results, such as removing separability (Kwaśniewski-Meyer, Raad) or commutativity (Exel, Kwaśniewski-Meyer), or maximal abelian-ness (Exel-Pitts, Exel-Pardo). One other assumption was the existence of a faithful conditional expectation from the larger algebra down to the subalgebra. In my doctoral thesis, I show that this can be relaxed to a 'pseudo-expectation'. On the groupoid side of the coin, this translates to the underlying groupoid model being non-Hausdorff, as certain functions you would expect to be continuous just aren't. In this talk, I'll briefly describe how Renault's proof works, where it doesn't work in the non-Hausdorff setting, and what we do to fix it.
• Feb 23: Two talks!
• Galina Levitina (The Australian National University) at 2:30 pm AEDT
Spectral shift function for massless Dirac operator in dimension two and higher
Abstract In this talk we show that spectral shift function can be expressed via (regularised) determinant of Birman-Schwinger operator in the setting suitable for higher order differential operators. We then use this expression to show that the spectral shift function for massless Dirac operator is continuous everywhere except possibly at zero. Behaviour of the spectral shift function at zero is influenced by the presence of zero eigenvalue and/or resonance of the perturbed Dirac operator.
• Serge Richard (Nagoya University) at 3:30 pm AEDT
Topological Levinson's theorem applied to group theory: a starter
Abstract In this talk, we present the spectral and scattering theory of the Casimir operator acting on the radial part of SL(2,R). After a suitable decomposition, the initial problem consists in studying a family of differential operators acting on the half-line. For these operators, explicit expressions can be found for the resolvent, the spectral density, and the Moeller wave operators, in terms of hypergeometric functions. Finally, an index theorem is introduced and discussed. It corresponds to a topological version of Levinson's theorem. This presentation is based on a joint work with H. Inoue.
• Feb 16: Astrid an Huef (Victoria University Wellington)
Twisted groupoid C*-algebras and finite nuclear dimension
Abstract Let $$E$$ be a twist over a principal étale groupoid $$G$$. I will talk about the main ideas of joint work with Kristin Courtney, Anna Duwenig, Magdalena Georgescu and Maria Grazia Viola, where we proved that the nuclear dimension of the reduced twisted groupoid $$C^*$$-algebra is bounded by a number depending on the dynamic asymptotic dimension of $$G$$ and the topological covering dimension of its unit space. This generalises an analogous theorem by Guentner, Willet and Yu for the $$C^*$$-algebra of $$G$$. Our proof uses a reduction to the unital case where $$G$$ has compact unit space, via a construction of groupoid unitisations'' $$\widetilde{G}$$ and $$\widetilde{E}$$ of $$G$$ and $$E$$ such that $$\widetilde{E}$$ is a twist over $$\widetilde{G}$$.
• Feb 9: Jonathan Mui (University of Sydney) - Zoom broadcast
Eventual positivity and compactness of operator semigroups
Abstract One-parameter semigroups of linear operators provide an abstract framework in which to study linear evolution equations on Banach spaces. Moreover, in many applications, we are interested in modelling a quantity which is naturally positive, such as in the context of diffusion equations and population density models. For this reason, we are lead to study positive operators acting on ordered function spaces, in particular Banach lattices (where the order structure is compatible with the norm). The study of semigroups of positive operators on Banach lattices is by now a classic part of the abstract theory of evolution equations. However, much more recently, and motivated by PDE examples, a systematic theory of eventually positive operator semigroups was initiated by Daners, Glück and Kennedy in 2015. The purpose of this talk is to give an accessible survey of recent progress in this topic. Loosely speaking, an eventually positive semigroup yields the solution to a linear evolution equation with the property that if the initial state is positive, then the solution eventually becomes and stays positive after a sufficiently large time $$t>0$$. It turns out that compactness plays an essential role in many results about such semigroups. One of my main goals is to highlight the intriguing interplay between the order structure of the underlying space, compactness, convergence, and spectral properties of operator semigroups. Time permitting, I will also mention how some of my PhD work fits into the story, and outline some open problems.