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 3:30pm 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: Alexander Mundey, Alex Paviour, and Ada Masters

If you are interested in either giving a talk or being 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.

Talks from previous years

Talk Schedule - 2025

Click on "Abstract" to reveal the abstract.

Upcoming talks

• May 29: Ali Raad (American University in Bulgaria) - Zoom Talk
  Constructing (non)-unique inductive limit C*-diagonals in AH-algebras via generalized Bratteli diagrams
  Abstract

  Cartan subalgebras are distinguished MASAs of C*-algebras. They are important for various reasons, one being that certain rigidity questions regarding C*-algebras arising from dynamical systems or geometric group theory are equivalent to questions about uniqueness of Cartan subalgebras. They also feature in the classification program for C*-algebras as every classifiable C*-algebra contains a Cartan subalgebra. The proof of this relies on being able to construct Cartan subalgebras in inductive limit C*-algebras as a limit of Cartan subalgebras of the building blocks. In this talk I will highlight how certain inductive limit C*-algebras containing canonical C*-diagonals yield a generalized Bratteli diagram which can be used to understand the spectrum of the C*-diagonals, which can further be exploited in order to construct unique and non-unique C*-diagonals in various classes of inductive limit C*-algebras.
• June 5: Jens Kaad (University of Southern Denmark) - Zoom Talk
  Noncommutative metric geometry of quantum spheres.
  Abstract

  In this talk we investigate the noncommutative metric geometry of the higher Vaksman-Soibelman quantum spheres. More precisely, we shall see how to endow a given quantum sphere with the structure of a compact quantum metric space by means of a seminorm arising from noncommutative differential geometric data. We view our quantum sphere as a noncommutative circle bundle over the corresponding quantum projective space. Using techniques from unbounded KK-theory this point of view allows us to construct vertical and horizontal differential geometric data on the quantum sphere in question. The vertical data comes from the generator of the circle action and the horizontal data comes from the unital spectral triple on quantum projective space introduced by D’Andrea and Dabrowski. An interesting feature of our setting is that the horizontal geometric data yields a twisted derivation on the coordinate algebra whereas the vertical geometric data produces a derivation in the usual sense. Nonetheless we can assemble these two (twisted) derivations into a single seminorm on our quantum sphere and show that the corresponding metric on the state space metrizes the weak*-topology.
• June 12: Bartosz Kwaśniewski (University of Białystok) - Zoom Talk
  TBA
  Abstract

  TBA
• June 19: TBA
  TBA
  Abstract

  TBA
• June 26: Gongfest: So Long, and Thanks for All the Fish

Recent talks

• May 22: Magnus Fries (Lund University) - Zoom Talk
  Spectral triples without locally compact resolvent and a characterization of Fredholmness
  Abstract

  The Fredholm index in Kasparov's \(KK\)-theory appears as the functorial map corresponding to the unit of the algebra. In particular, this means that it is most convenient in index theory for our algebra to be unital. Using the usual notion of unbounded Kasparov modules this would require compact resolvent of the operator which is stronger than being Fredholm and is rather restrictive for differential operators on non-compact manifolds. In this talk we look at the property of being Fredholm and give a neat geometric characterization. The characterization appeared as an assumption in Bunke's work on the relative index theorem in \(KK\)-theory, and we can combine this with a different flavor of unbounded Kasparov modules due to Wahl in order to obtain a handy tool for index theory.
• May 15: Alex Paviour (University of Wollongong)
  Some gauge theory
  Abstract

  I will discuss (classical) gauge theories, which are the foundation of much of modern physics, and their mathematical description in terms of non-metric affine geometries on vector bundles carrying representations of particular Lie groups. In particular, I will explain the somewhat ad-hoc Higgs mechanism and describe how some particles acquire mass through spontaneous symmetry breaking. I will then describe the Connes-Lott reformulation of the standard model in terms of noncommutative geometry, focusing on the reinterpretation of the Higgs field as one major advantage of this approach.
• May 8: Dimitris Gerontogiannis (IMPAN) - Zoom Talk
  Ideal quantum metrics from fractional Laplacians
  Abstract

  This talk presents a new framework for constructing computable Monge–Kantorovich metrics using Schatten ideals and commutators of fractional Laplacians on Ahlfors regular spaces. These “ideal” metrics admit explicit spectral formulas and naturally respect underlying dynamics. Our methods introduce new tools in noncommutative geometry, including a fractional Weyl law and Schatten-class commutator estimates. As an application, we extend the construction to expansive \(Z^m\)-actions and their associated \(C^*\)-algebras, illustrating the reach of fractional analysis across dynamics, fractal geometry, and quantum metric spaces. This is joint work with Bram Mesland.
• May 1: Dani Czapski (University of New South Wales)
  Freeing Kruglov’s operator
  Abstract

  The Kruglov operator in classical probability theory was first defined by S. Astashkin and F. Sukochev in the 1990s as part of their study into norm bounds on sums of independent, integrable random variables in symmetric Banach function spaces. This culminated in a generalisation of the Johnson–Schectmann inequalities and a powerful statement regarding precisely which symmetric spaces such inequalities exist in. If we replace \(L^\infty\) with a general semifinite von Neumann algebra \(\mathcal{M}\), the expectation with a faithful, normal, tracial state \(\tau\) and equip this with the free independence of D. Voiculescu, we obtain a "non-commutative probability space" \((\mathcal{M},\tau)\). We seek to define a free analogue of the Kruglov operator on \(L^1(\mathcal{M})\) and attempt to answer a question posed by R. Speicher on such on operator on spaces where the trace is no longer finite. I will provide some background on the Kruglov operator and free probability, explain our current plan of attack and discuss what progress has been made.
  
  This is joint work with Dmitriy Zanin.
• April 24: Anna Duwenig (University of New South Wales)
  Cartan subalgebras and C*-diagonals in the twisted C*-algebras of non-principal groupoids
  Abstract

  The reduced twisted C*-algebra A of an étale groupoid G has a canonical abelian subalgebra D: functions on G's unit space. When G has no non-trivial abelian subgroupoids (i.e., G is principal), then D is in fact maximal abelian. Remarkable work by Kumjian shows that the tuple (A,D) allows us to reconstruct the underlying groupoid G and its twist uniquely; this uses that D is not only masa but even what is called a C*-diagonal. In this talk, I show that twisted C*-algebras of non-principal groupoids can also have such C*-diagonal subalgebras, arising from non-trivial abelian subgroupoids, and I will discuss the reconstructed principal twisted groupoid of Kumjian for such pairs of algebras.
• April 17: Adam Rennie (University of Wollongong/Sydney)
  Cayley transform, spectral flow, and scattering theory
  Abstract

  I will describe the Cayley transform picture of odd K-theory, and how to take Kasparov products in this picture. This gives an approach to spectral flow, and a fascinating example arises in scattering theory to exemplify the issues.
  
  Work in progress with A. Alexander, C. Bourne, A. Carey, G. Levitina, A. Masters.
• April 10: Lynnel Naingue (Mindanao State University - Iligan Institute of Technology)
  On Graded Algebraic Pairs and Graded Twisted Steinberg Algebras
  Abstract

  In this talk, we show how to construct a graded discrete twist over an ample Hausdorff groupoid \(G\) from what we call a graded algebraic quasi-Cartan pair \((A,C)\) and show that its twisted Steinberg algebra is graded isomorphic to the graded algebra \(A\) we started with. Next, if we start from a suitable graded discrete twist \(\Sigma\) over an ample Hausdorff groupoid \(G\), we show that the pair \((A_R^{\mathrm{gr}}(G;\Sigma), A_R(G^{(0)}, q^{-1}(G^{(0)})))\) is a graded algebraic quasi-Cartan pair. Then we show that following our construction recovers \(\Sigma\). Our work generalises results on reconstruction of (ungraded) twisted Steinberg algebras, and we use the homogeneous normalisers in place of more general normalisers. With weaker hypotheses, we cover larger classes of algebras. This is a joint work with Prof. Lisa Orloff Clark and Prof. Jocelyn P. Vilela.
• April 3: Jamie Bell (University of Münster)
  Stable rank one for crossed product \(C^*\)-algebras
  Abstract

  Stable rank was introduced by Marc Rieffel in 1983 and has since played a crucial role in the study of operator \(K\)-theory and its generalisations. In particular, \(C^*\)-algebras of stable rank one—the lowest possible stable rank—have been extensively studied and are now recognised as a large and important class. In this talk, we survey key results on stable rank one in the context of crossed product \(C^*\)-algebras and discuss ongoing work to generalise these results. This is based partly on joint ongoing work with S. Geffen and D. Kerr.
• March 27: Ada Masters (University of Wollongong)
  Parabolic noncommutative geometry
  Abstract

  I will discuss a new framework, generalising spectral triples, capable of handling noncommutative geometries with anisotropic behaviour. This framework, of 'tangled spectral triples', involves replacing the single Dirac operator with a collection, required to satisfy certain compatibility conditions informed by tropical combinatorics. I will discuss three motivating examples: the Rumin complex on a contact manifold, crossed product C*-algebras of parabolic dynamical systems, and C*-algebras of nilpotent Lie groups. I will also briefly discuss the limitations of tangled spectral triples and ongoing work to incorporate Bernstein–Gelfand–Gelfand complexes into spectral noncommutative geometry. This is joint work with Magnus Fries and Magnus Goffeng and appears in a recent preprint.
• March 20: Shay Tobin (Macquarie University)
  Characterising the category of Hilbert spaces
  Abstract

  In 2022, the category of real or complex Hilbert spaces and bounded linear maps was characterised in purely categorical terms by Chris Heunen and Andre Kornell. The tensor product of Hilbert spaces played a key role in this characterisation. In this talk I will present a new but related characterisation which does not involve the tensor product. As a bonus, we are also able to characterise the category of quaternionic Hilbert spaces.
• March 13: Dilshan Wijesena (University of New South Wales)
  A tensor product for representations of the Cuntz algebra
  Abstract

  In this talk we introduce a tensor product \(\boxtimes\) for a large class of representations of the Cuntz algebra \(O_2\), providing the first explicit notion of fusion for such representations. We do this by using our previous work on the classification of \(Rep(O_2)\) via Jones' technology and Pythagorean representations. We demonstrate many interesting applications of \(\boxtimes\). For example, we are able to produce many interesting tensor categories consisting of representations of \(O_2\) with computable fusion rules which is rather rare in the literature. Moreover, using \(\boxtimes\) we deduce Lie group actions that define smooth deformations of representations of the Thompson groups \(F,T,V\) and \(O_2\), which recover many previously known representations. This is a joint work with Arnaud Brothier.
• March 6: Adam Rennie (University of Wollongong)
  Conformal transformations and equivariance in unbounded KK-theory
  Abstract

  I will give (at least) the third talk on this material, the two previous being by Ada, as is the bulk of this work. In this talk I will focus on the novel structures unveiled by probing conformality and equivariance. These include the essential appearance of ternary rings of operators, and the more general spectral-triple-like objects that result.
  
  Joint work with Ada Masters, arXiv:2412.17220.
• February 27: Alexander Mundey (University of Wollongong)
  Fibrewise compactifications and generalised limits (non)commutative topology
  Abstract

  The category of compact spaces and continuous maps is "nice". The category of locally compact spaces and continuous maps is less "nice"—infinite products and projective limits of locally compact spaces are not necessarily locally compact. To address this, I will introduce fibrewise compactifications, a family of constructions that extend continuous maps between locally compact spaces to proper maps. These provide a systematic way to construct limits of locally compact spaces, with examples including the boundary path space of a directed graph. I will also discuss how these constructions translate into the C*-algebraic setting via extensions, providing a new entry in the noncommutative topology dictionary.
• February 20: Chris Bourne (Nagoya University)
  A C*-module framework for interfaces of discrete quantum systems
  Abstract

  Loosely speaking, an interface describes a spatial mixing of several systems described by a C*-algebra of observables such that these mixed dynamics are not felt 'at infinity'. I will introduce work-in-progress to adapt work by Măntoiu to formalise a mathematical description of (discrete) interfaces using C*-modules and (discrete) crossed products. Several examples will be given as well as spectral and index-theoretic properties.
• February 13: Astrid an Huef (Victoria University of Wellington)
  Nuclear dimension of \(C^*\)-algebras of groupoids with large isotropy subgroups, and applications to \(C^*\)-algebras of graphs
  Abstract

  Let \(G\) be a locally compact, Hausdorff groupoid. Guentner, Willet and Yu defined a notion of dynamic asymptotic dimension (dad) for étale groupoids, and used it to find a bound on the nuclear dimension of \(C^*\)-algebras of principal groupoids with finite dad. To have finite dad, a groupoid must have locally finite isotropy subgroups which rules out, for example, the graph groupoids and twists of étale groupoids by trivial circle bundles. I will discuss how the techniques of Guentner, Willett and Yu can be adjusted to include some groupoids with large isotropy subgroups, including an applications to \(C^*\)-algebras of directed graphs that are AF-embeddable. This is joint work with Dana Williams.
• Talks from previous years