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

If you are interested in giving a talk or to be added to the mailing list, please contact the organisers at wollongong.oancg[at]gmail.com.

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.

Previous years

Talk Schedule - Autumn 2023

Click on "Abstract" to reveal the abstract.

Upcoming talks

• June 8: Richard Garner (Macquarie University)
  [B|M]-sets for operator algebra
  Abstract

  In the last talk we met matched pairs of algebras [B|M] as a way of presenting ample topological groupoids, or, more generally, ample topological categories, and to any matched pair [B|M] we associated a category of [B|M]-sets. This category has various nice properties, such that any category with these same properties can be exhibited as a category of [B|M]-sets for an explicitly calculable [B|M]. In this talk, we apply this "reconstruction" result to recapture a range of well-known ample topological groupoids arising in operator algebra.
• June 15: Abraham Ng (University of Wollongong)
  TBA
  Abstract
• June 22: Thomas Futcher (University of Wollongong)
  TBA
  Abstract

  TBA
• June 29: Alexander Mundey (University of Wollongong)
  TBA
  Abstract

  TBA
• July 6: Alan Stoneham (University of New South Wales)
  TBA
  Abstract

  TBA
• July 13: TBA
  TBA
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  TBA
• July 20: TBA
  TBA
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  TBA
• All following dates are all TBA!

Recent talks

• June 1: Richard Garner (Macquarie University)
  B-sets, M-sets and matched pairs
  Abstract

  The goal of this talk is to describe a "delinearization" of the Steinberg algebra of an etale topological groupoid. This delinearlization will involve a Boolean algebra B (of clopen sets), a monoid M (of partial sections), and actions of B on M and of M on B. The nice thing about all of this is that it faithfully captures the groupoid but is purely algebraic. So, for example, we can start translating useful properties of a groupoid like minimality into properties of the associated pair (B,M). The goal this time is to concentrate on the theory; next time we will look at familiar examples of groupoids from this perspective.
• May 25: David Robertson (University of New England) - Zoom broadcast
  Self-similar quantum groups part 2
  Abstract

  In a recent project with Nathan Brownlowe, we asked ourselves the following question: is there a sensible notion of self-similarity for compact quantum groups? We believe the answer is yes! Over two talks we will describe our ideas and results. Part 1 will be an introduction to the basics of compact quantum groups, and in part 2 we will recall the definition of a self-similar group, and will describe our notion of self-similarity for compact quantum groups.
• May 18: Nathan Brownlowe (University of Sydney)
  Self-similar quantum groups part 1
  Abstract

  In a recent project with Dave Robertson, we asked ourselves the following question: is there a sensible notion of self-similarity for compact quantum groups? We believe the answer is yes! Over two talks we will describe our ideas and results. Part 1 will be an introduction to the basics of compact quantum groups, and in part 2 we will recall the definition of a self-similar group, and will describe our notion of self-similarity for compact quantum groups.
• May 11: Adam Rennie (University of Wollongong)
  I was framed!
  Abstract

  I will remind everyone about frames in Hilbert modules, their properties, applications and favourite foods. Mostly not new, but a useful tool and I will point at applications in previous work with numerous people.
• May 4: Alexander Mundey (University of Wollongong)
  Splittings for topological graphs and C*-correspondences
  Abstract

  In the 1970s, Williams showed that two subshifts of finite type (SFTs) are conjugate if and only if their associated adjacency matrices are strong shift equivalent. Moreover, he showed that any conjugacy of SFTs can be realised as a finite composition of elementary conjugacies coming from in-splits and out-splits.
  
  In 2008, Muhly, Pask, and Tomforde introduced a notion of strong shift equivalence for noncommutative dynamics arising from C*-correspondences. In this talk I will extend upon this work by introducing the notion of in-splits and out-splits for C*-correspondences, and specialise this definition to topological graphs. I will show that in-splits induce gauge equivariant isomorphisms of Cuntz-Pimsner algebras, and that out-splits induce gauge equivariant Morita equivalences.
  
  This is based on joint work with Kevin Brix and Adam Rennie.
• Apr 27: Alistair Miller (University of Southern Denmark) - Zoom broadcast
  Groupoid correspondences, homology and K-theory
  Abstract

  Correspondences of étale groupoids are a groupoid analogue of C*-correspondences. I will give examples of a broad range of morphisms of groupoids that may be viewed as groupoid correspondences, including Morita equivalences and étale homomorphisms. Correspondences enjoy a fruitful relationship with groupoid homology and C*-algebra K-theory. I will exploit this to compute the K-theory of inverse semigroup C*-algebras and left regular C*-algebras of left cancellative small categories.
• Apr 20: Aidan Sims (University of Wollongong)
  The ideal structure of C*-algebras of actions of free abelian monoids (instalment II)
  Abstract

  The ideal structure of a C*-algebra is maybe its most fundamental and low-tech invariant. However, it’s bloody hard to calculate. In particular, given an irreversible dynamical system, like an action of a free abelian monoid by local homeomorphisms, the ideal structure of its C*-algebras ought to have something to do with the space being acted upon, the extent to which the action mixes it around, and (via Pontryagin duality) the ranks of the stabiliser submonoids. But figuring out exactly what is very hard work. I’ll describe a way of figuring out the answer, using groupoids and a smidgeon of Fourier analysis, for systems whose groupoids admit nice enough families of bisections. I’ll say a bit about how to see that there are plenty of systems where this does happen, and illustrate what the results say with some concrete examples. There will be a two-week hiatus separating the previous two sentences.
  This is joint work with Kevin Brix and Toke Carlsen
• Apr 13: Malcolm Jones (Victoria University Wellington) - Zoom broadcast
  Ample groupoid of a (finitely aligned) P-graph where P embeds in a group
  Abstract

  Beginning in the 80s with the work of Enomoto, Fujii and Watatani, directed graphs and their generalisations (such as the P-graphs of Brownlowe, Sims and Vittadello) have been used to model C*-algebras. Associating an auxiliary groupoid to the graph has been a fruitful technique. This was demonstrated for directed graphs by Kumjian, Pask, Raeburn and Renault in the 90s, and demonstrated for finitely aligned higher-rank graphs by Kumjian, Pask, Farthing, Muhly and Yeend in the 2000s. In the present decade, Spielberg has given us a groupoid model for the non-finitely aligned setting. In fact, Spielberg's techniques apply to all left cancellative small categories, making the techniques as complex as the class of left cancellative small categories is broad. Moreover, the length functions that moderate the behaviour of the graph are omitted. We are interested in finding a more accessible groupoid for the non-finitely aligned setting. We reintroduce length functions to make this feasible.
  
  In this talk, we associate a topological groupoid to any P-graph, where P is only assumed to be a submonoid of a group. We describe a sufficient condition for the groupoid to be ample. When the P-graph is finitely aligned, the sufficient condition holds, so the groupoid is ample. We discuss how to modify our construction to include non-finitely aligned P-graphs. Lastly, we highlight that our construction applies to discrete groups, in which case the groupoid is isomorphic to the group itself.
  
  This work is being undertaken with my supervisors Lisa Orloff Clark and Astrid an Huef.
• Apr 6: Aidan Sims (University of Wollongong)
  The ideal structure of C*-algebras of actions of free abelian monoids (instalment I)
  Abstract

  The ideal structure of a C*-algebra is maybe its most fundamental and low-tech invariant. However, it’s bloody hard to calculate. In particular, given an irreversible dynamical system, like an action of a free abelian monoid by local homeomorphisms, the ideal structure of its C*-algebras ought to have something to do with the space being acted upon, the extent to which the action mixes it around, and (via Pontryagin duality) the ranks of the stabiliser submonoids. But figuring out exactly what is very hard work. I’ll describe a way of figuring out the answer, using groupoids and a smidgeon of Fourier analysis, for systems whose groupoids admit nice enough families of bisections. I’ll say a bit about how to see that there are plenty of systems where this does happen, and illustrate what the results say with some concrete examples. There will be a two-week hiatus separating the previous two sentences.
  This is joint work with Kevin Brix and Toke Carlsen
• Mar 30: Rodrigo Frausino (University of Wollongong)
  Reconstruction of topological graphs and their Hilbert bimodules
  Abstract

  In this talk, I would like to share with you part of my ongoing PhD project, which is a joint work with Aidan Sims and Abraham Ng. Our work focuses on the C*-algebraic triple consisting of the Toeplitz algebra of a compact topological graph, its gauge action, and the commutative subalgebra of functions on the vertex space of the graph, and how we can recover the Hilbert bimodule (the graph correspondence) associated with a compact topological graph from this triple.
  More than that, we introduce a notion of local isomorphism of topological graphs and show that a compact topological graph can be recovered up to local isomorphism from its Hilbert bimodule. This is an exciting result because it provides a way to reconstruct a graph (up to this notion of local isomophism) from the C*-algebraic data mentioned above. However, we also show that it is not possible to recover the graph up to isomorphism, and we provide a counter-example for this.
  If time permits, I will also briefly discuss a joint project with Jonathan Taylor that aims to extend our results for compact topological graphs to the non-compact settings or even to the more general setting of correspondences over commutative C*-algebras.
  Looking forward to any questions or comments you may have!
• 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 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.