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: Angus Alexander & Alexander Mundey

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 - 2024

Click on "Abstract" to reveal the abstract.

Upcoming talks

• October 31: Adam Rennie (University of Wollongong)
  Bivariant Chern characters
  Abstract

  When the unbounded Kasparov product breaks, maybe we can walk around it, at least for index computations. I will describe the problem and a partial solution, and hopes for the future.
  
  Joint work with: Uli Kraehmer, Jens Kaad, David Kyed, Bram Mesland and anyone else who shows up before we are done.
• November 7: TBA
  TBA
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  TBA
• November 14: TBA
  TBA
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  TBA
• November 21: Thomas Sheckter (University of New South Wales)
  TBA
  Abstract

  TBA
• November 28: TBA
  TBA
  Abstract

  TBA
• December 5: Malcolm Jones (Victoria University of Wellington)
  TBA
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  TBA
• December 12: TBA
  TBA
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  TBA

Recent talks

• October 24: Tyler Schulz (University of Victoria)
  KMS states in relation to number theory - Part 2
  Abstract

  Continuing where we left off, I will talk about the Bost-Connes system, in particular, its spontaneous symmetry-breaking and connections to class field theory. I will then describe the Toeplitz algebras of the affine semigroup over the natural numbers and some of the parallels to the BC system. In work with Marcelo Laca, we showed that the right Toeplitz system exhibits its own spontaneous symmetry-breaking with an unusual twist. Our main result is a parametrization of the high-temperature states and their behaviour under endomorphisms.
• October 17: Christian Voigt (University of Glasgow)
  Self-similar quantum symmetries
  Abstract

  In this talk I’ll explain the notion of quantum symmetry of a graph, focussing on the case of regular trees. There is a natural notion of self-similarity arising in this context, and I’ll discuss a concrete example reminiscent of Grigorchuk’s first group.
  
  (joint with N. Brownlowe, D. Robertson, M. Whittaker)
• October 10: Tyler Schulz (University of Victoria)
  KMS states in relation to number theory
  Abstract

  KMS states were introduced in the 60's as a notion of equilibrium states for statistical mechanics of quantum systems with infinitely many degrees of freedom. Since then, they have been extensively studied as a tool for relating between C*-algebras and number theory, in a broad sense. The most important example is the Bost-Connes system, which carries such interesting features as a non-trivial phase transition and spontaneous symmetry-breaking with respect to the maximal abelian Galois group of Q. In this talk, I will describe this system along with two related examples, the Toeplitz algebras of the affine semigroups over the natural numbers, with an emphasis on their simplices of KMS states and symmetries.
• October 3: Kevin Brix (University of Southern Denmark)
  Ideal structure of reduced group C*-algebras
  Abstract

  I will discuss the (difficult) problem of understanding the ideal structure of reduced group C*-algebras. We now have a good understanding of C*-simple groups due to work of Breuillard, Kalantar, Kennedy, Ozawa (in various constellations), and with Chris Bruce, Kang Li, and Eduardo Scarparo, we managed to make some progress for maximal ideals. A key tool is the Furstenberg boundary of a discrete group.
• September 26: Shiqi Liu (University of New South Wales)
  Weyl asymptotic for hypoelliptic operators
  Abstract

  Weyl-type asymptotics have a long history dating back to H. Weyl. For decades, spectral asymptotics for negative order pseudodifferential operators on noncompact manifolds have been a central theme in spectral theory. In this talk, I will present a spectral asymptotic formula for negative powers of hypoelliptic operators on stratified Lie groups. One of the highlights of our proof is a uniform elliptic estimate for hypoelliptic operator. This reveals that hypoelliptic operator can process similar analytic properties as elliptic ones.
  This is a joint work with Edward McDonald, Fedor Sukochev, and Dmitriy Zanin.
• September 19: Astrid an Heuf (Victoria University of Wellington)
  Groupoid \(C^*\)-algebras that are subhomogeneous
  Abstract

  Let \(G\) be a second countable, locally compact, Hausdorff groupoid (sometimes it will be étale, but not always). I will discuss a characterisation of groupoids whose \(C^*\)-algebras are subhomogeneous, and properties of their \(C^*\)-algebras including a composition series and bounds on their nuclear dimension.
• September 12: Jacob Bradd (Penn State University)
  A Paley-Wiener approach to the Connes-Kasparov isomorphism
  Abstract

  I will talk about my thesis work, which proves a refinement of the Connes-Kasparov isomorphism by studying (the Fourier theory of) the "Casselman algebra" of rapidly decreasing functions on a real reductive group. In fact, I show that this Casselman algebra, which encodes nonunitary representation theory, and the reduced group \(C^*\)-algebra, which encodes tempered unitary representation theory, are built in very similar ways from similar elementary components. This uses Delorme's proof of the Paley-Wiener theorem for real reductive groups, which describes the Fourier transform of compactly supported smooth functions. We will journey into the treacherous world of representation theory and Fourier theory for such groups, just dipping our toes in enough to see what I mean by "elementary components".
• September 5: Edward McDonald (Penn State University)
  Dave-Haller's Weyl law and the tangent groupoid
  Abstract

  Recently there has been significant progress in the theory hypoelliptic operators, based in part on the construction of the tangent groupoid adapted to a filtration. One advance has been Dave and Haller's proof of a generalisation of Weyl's asymptotic formula to H-elliptic differential operators on an arbitrary filtered closed manifold. I will give some background on filtered manifolds and van Erp-Yuncken's pseudodifferential calculus and then briefly explain how it can be used to prove Dave and Haller's formula.
• August 29: Jonathan Taylor (University of Potsdam)
  Two approaches to inductive limits of Cartan-like pairs of C*-algebras
  Abstract

  In 2018, Xin Li proved that the colimit of an inductive system of C*-algebras with Cartan subalgebras also has a Cartan subalgebras (assuming the connecting maps preserve the Cartan structure). This was achieved by explicitly constructing a groupoid model for the limit C*-algebra from the groupoids of Cartan building blocks. This theorem forms a pivotal connection between the worlds of combinatorial C*-algebras and the C*-algebra classification program, as it proves that every classifiable C*-algebra arises as a twisted groupoid C*-algebra.
  
  There are at least two natural questions that one may ask about the setup of Li's theorem, each going in a different direction. Firstly, can one use C*-algebraic methods to show directly that the inductive limits of Cartan pairs are again Cartain pairs, without passing to the underlying groupoid construction?
  
  Second, can Li's techniques be used for inductive systems of more general groupoid C*-algebras to construct inductive limit groupoids without requiring the induced C*-algebra pairs to be Cartan pairs?
  
  I aim to discuss and answer these two questions in this talk. This is joint work with Ali Raad and Ralf Meyer.
• August 22: Two talks!
  
  • Ulrich Krähmer (Technische Universität Dresden) at 2:00pm
    The ring of differential operators on a monomial curve is a Hopf algebroid
    Abstract

    The ring of differential operators on a cuspidal curve whose coordinate ring is a numerical semigroup algebra is shown to be a cocommutative and cocomplete left Hopf algebroid, which essentially means that the category of D-modules is closed monoidal. If the semigroup is symmetric so that the curve is Gorenstein, it is a full Hopf algebroid (admits an antipode), which means that the subcategory of D-modules that are finite rank vector bundles over the curve is rigid. Based on joint work with Myriam Mahaman.
  • Robert Yuncken (Institut Élie Cartan de Lorraine) at 3:30pm
    Crystallised functions on compact Lie groups as higher-rank graph algebras
    Abstract

    When Woronowicz introduced the \(C^*\)-algebra of continuous functions on the compact quantum group \(SU_q(2)\), he observed that it is isomorphic to a graph \(C^*\)-algebra by considering its limit as \(q\) goes to \(0\). Hong-Szymanksi and Giselsson generalized this result to quantum projective spaces and \(SU_q(3)\) respectively. We will show how these \(q=0\) limits can be obtained from Kashiwara and Lusztig's crystal basis theory, and hence obtain the crystal limit of any function algebra of a compact semisimple Lie group as a higher-rank graph algebra.
• August 15: Nigel Higson (Pennsylvania State University)
  A new look at the quantization commutes with reduction problem
  Abstract

  Symplectic reduction is a type of quotient operation in the theory of Hamiltonian group actions on symplectic manifolds. It plays many useful roles in symplectic geometry, but it also has an index-theoretic aspect. This was noted first by Guillemin and Sternberg, who formulated a conjecture related to this index-theoretic aspect. The conjecture was subsequently proved by Meinrenken—using techniques in symplectic surgery—and by Tian and Zhang—using techniques in geometric analysis. I shall explain both the conjecture, including all the terms above, and the Tian-Zhang approach to its proof, including some modest simplifications of the argument that were obtained in joint work with Qiaochu Ma and Yiannis Loizides.
• August 8: Torstein Ulsnaes (Leiden University/SISSA) - Zoom Talk
  Crossed product \(C^*\)-algebras of lattice actions on boundaries of symmetric spaces
  Abstract

  The classical Mostow rigidity theorem tells us that a large family of Riemannian locally symmetric spaces \(X\) are classified by their fundamental group. A crucial ingredient of the proof is the dynamical system consisting of the (geodesic) boundary at infinity of the universal cover of \(X\) with the action of the fundamental group of \(X\).
  In this talk I will define and study the crossed product \(C^*\)-algebra arising from such dynamical systems, and show that they fail to define a complete set of invariants for the corresponding locally symmetric spaces.
• August 1: Aidan Sims (University of Wollongong)
  "Spectral permanence" for abelian open isotropy in etale groupoids
  Abstract

  If \(G\) is an etale groupoid and the interior \(I\) of its isotropy is abelian, then there is an inclusion of \(C^*\)-algebras \(C^*(I) \subseteq C^*(G)\) that extends the obvious inclusion of compactly supported functions. Since elements of \(C^*(G)\) are honest \(C_0\) functions on \(G\), there is also an abelian subalgebra \(A\) of \(C^*(G)\) consisting of those elements that vanish on the complement of \(I\). It seems obvious that these two \(C^*\)-subalgebras coincide. I thought so anyway. It turns out that it is true but, as far as we can see, very not-obvious. I'll explain the question and where it came from, and then outline the ingredients of the (surprisingly nontrivial) proof. This is joint work with Carlsen, Duwenig, Ruiz and Tomforde.
• July 25: Francesca Arici (Leiden University) - Zoom Talk
  Spheres, Euler classes and the \(K\)-theory of \(C^*\)-algebras of subproduct systems
  Abstract

  In this talk, we shall consider equivariant subproduct system of Hilbert spaces and their Toeplitz and Cuntz-Pimsner algebras. We will provide results about their topological invariants through \(K(K)\)-theory. More specifically, we will show that the Toeplitz algebra of the subproduct system of an \(SU(2)\)-representation is equivariantly \(KK\)-equivalent to the algebra of complex numbers so that the \((K)K\)- theory groups of the Cuntz-Pimsner algebra can be effectively computed using a Gysin exact sequence involving an analogue of the Euler class of a sphere bundle. Finally, we will discuss why and how \(C^*\)-algebras in this class satisfy \(KK\)-theoretic Poincaré duality.
• July 18: Yufan Ge (Leiden University) - Zoom Talk
  \(SU(2)\)-symmetries of \(C^*\)-algebras: from bricks to buildings.
  Abstract

  In this talk, we will consider subproduct systems coming from \(SU(2)\)-representations and discuss the associated \(C^*\)-algebras. We will first review results concerning irreducible representations from Arici-Kaad, then provide some further results about more general cases. More specifically, we will discuss the structure of the \(SU(2)\)-subproduct systems associated to isotypic representations and multiplicity-free representations. Finally, we will provide results about the \(K\)-theory groups of their Toeplitz algebras. This is joint work in progress with Francesca Arici.
• Postponed: David Pask (University of Wollongong) - Zoom Talk
  Certain coloured graphs giving higher-rank graphs, and their fundamental groups. Exploring planar higher-rank trees and some other things - Part 3
  Abstract

  Building on our work from the first two talks we show that the \(k\)-graphs \(2 \le k \le 4\) we have built are planar \(k\)-trees. We then talk briefly about gluing coloured graphs. We then show that there is a planar 2-tree which admits a (fixed point) free automorphism of order three, something which does not occur in dimension one.
• July 4: Adam Rennie (University of Wollongong)
  More about Podleś
  Abstract

  See title.
• June 27: David Pask (University of Wollongong) - Zoom Talk
  Certain coloured graphs giving higher-rank graphs, and their fundamental groups. Exploring planar higher-rank trees and some other things - Part 2
  Abstract

  Following on from the introduction in the first talk, we introduce a large family of finite planar higher-rank trees inspired by the work of Johnstone and Lambek. They are constructed from ordinary planar graphs, and we show how to demonstrate their properties.
• June 20: Ada Masters (University of Wollongong)
  Spectral triples on group \(C^*\)-algebras from geometric group theory
  Abstract

  Since Connes's 1989 paper, the use of length functions to build spectral triples for group \(C^*\)-algebras has become commonplace in noncommutative geometry. This construction, although sound at the level of quantum metric spaces, always gives rise to trivial \(K\)-homology. Using ingredients from geometric group theory, this can be remedied for many CAT(0) groups, including non-discrete groups. In the process, a novel group invariant from (quantum-group-equivariant) \(KK\)-theory is uncovered. Furthermore, the understanding of group extensions in this framework is a microcosm of the more general problem of the constructive unbounded Kasparov product.
• June 13: David Pask (University of Wollongong) - Zoom Talk
  Certain coloured graphs giving higher-rank graphs, and their fundamental groups. Exploring planar higher-rank trees and some other things - Part 1
  Abstract

  In this first of three talks I shall run through some background on certain coloured directed graphs which give rise to higher rank-graphs. There will also be some examples and motivation for my investigation. The examples given have the property that there is no path of length \(3\), which makes their fundamental group quite easy to compute and, for the examples provided is always trivial. To explain where these examples come from we must first indulge in a little more background on planarity, duality and colourability of graphs before moving on next week.
• June 6: Adam Rennie (University of Wollongong)
  The Levi-Civita connection on noncommutative differential forms
  Abstract

  By combining Hilbert module and algebraic techniques, we give necessary and sufficient conditions for the existence of Hermitian and torsion-free connections on noncommutative one-forms, such as those arising from spectral triples. With additional structure we give a sufficient condition for uniqueness. Our methods are constructive, use standard definitions, and allow computation of curvature with comparable difficulty to the differential geometry case. Joint work with Bram Mesland. arXiv:2403.13735 and 2404.07957
• May 30: Adam Rennie (University of Wollongong)
  Differential calculi for algebras
  Abstract

  As a warm-up to next week's talk I will run over the basic elements of calculus for algebras.
• May 23: Dimitris Gerontogiannis (Leiden University) - Zoom Talk
  The log-Laplacian on Ahlfors regular spaces and noncommutative boundaries
  Abstract

  The Laplace-Beltrami operator is a fundamental tool in the study of compact Riemannian manifolds. In this talk, I will introduce the logarithmic analogue of this operator on Ahlfors regular spaces. These are metric-measure spaces that might lack any differential or algebraic structure. Examples are compact Riemannian manifolds, several fractals, self-similar Smale spaces and limit sets of hyperbolic isometry groups. Further, this new operator is intrinsically defined, its spectral properties are analogous to those of elliptic pseudo-differential operators on manifolds and exhibits compatibility with non-isometric actions in the sense of noncommutative geometry. This is joint work with Bram Mesland (Leiden). If time allows, I will also discuss a recent joint work with Magnus Goffeng (Lund) and Bram Mesland on the spectral geometry of Cuntz-Krieger algebras with respect to the log-Laplacian
• May 16: Aidan Sims (University of Wollongong)
  Hilbert modules and Morita Equivalence
  Abstract

  One of the most ubiquitous tools in the theory of C*-algebras is the theory of Hilbert modules, induced representations, and Morita equivalence. Since a lot of the usual crowd are away at the moment, and since some of our honours students have just finished learning about GNS representations, I will try to give an overview of what Hilbert modules are, how they are used to induce representations, what a Morita equivalence is, and what it has to do with full corners and with stable isomorphism.
• May 9: Alexander Mundey (University of Wollongong)
  Zappa-Szép products of categories and their homology
  Abstract

  The Zappa-Szép product, a generalisation of the semidirect product, traditionally arises in group theory. Rather than starting with one group acting on another by automorphisms, it features two groups acting on each other in a mutually compatible way.
  
  In this talk, I will discuss the extension of Zappa-Szép products to categories and demonstrate how this framework encompasses a variety of dynamical systems of interest to C*-algebraists, such as self-similar actions and k-graphs. I will also outline how the homology of a Zappa-Szép product of categories can be derived from the homological data of its constituent categories.
  
  This talk is based on joint work with Aidan Sims.
• May 2: Eva-Maria Hekkelman (University of New South Wales)
  The Joy of MOIs
  Abstract

  Operator integrals appear in various contexts of noncommutative geometry. Proofs of the local index formula, expansions of the spectral action and asymptotics of the heat trace are just a few examples. Thankfully, back in the day, the Soviet school of mathematics cooked up a theory of Multiple Operator Integrals (MOIs) which provides a big toolbox for handling complicated integrals of bounded operators. This is still an active field of research. Sadly, it currently isn't applicable to NCG since our integrals have unbounded operators all over the place. In joint work with Ed McDonald and Teun van Nuland we adapted the theory of MOIs to the framework of abstract pseudodifferential operators Connes–Moscovici/Higson style, which makes MOI theory usable in NCG contexts. We hope this greatly reduces the time noncommutative geometers have to spend fiddling with nasty operator integral analysis.
• April 18: Becky Armstrong (Victoria University of Wellington)
  Representing topological full groups in Steinberg algebras and C*-algebras
  Abstract

  Topological full groups are a useful groupoid invariant that have been used to solve important open problems in group theory. Steinberg algebras are a purely algebraic analogue of groupoid C*-algebras that generalise both Leavitt path algebras and Kumjian–Pask algebras. The Steinberg algebra of an ample Hausdorff groupoid is a quotient of the algebra generated by the inverse semigroup of compact open bisections of the groupoid. Since the topological full group of an ample Hausdorff groupoid sits inside this inverse semigroup, it is natural to ask what the relationship is between the algebra of the topological full group and the Steinberg algebra of the groupoid. In this talk I will present recent results answering this question. (This is joint work with Lisa Orloff Clark, Mahya Ghandehari, Eun Ji Kang, and Dilian Yang.)
• April 11: Isaac Bankier (University of Wollongong)
  The Groupoid Interpretation of Equality
  Abstract

  Primary doctrines are a construction presenting a system of first order logic over a base category. Primary doctrines need not have internal notions of equality, however this can be added cofreely, by taking "quotients" by all possible internal equivalence relations. In the proof relevant generalisation of primary doctrines, called primary fibrations, we will see that internal groupoids play the role of equivalence relations.
• April 4: Nate Brown (Pennsylvania State University) - Colloquium Talk
  Inclusive teaching: a mathematician's view
  Abstract

  After 20 years of research in theoretical maths, I switched to educational research. Specifically, inequity in STEM education at the university level. In this talk I'll present a representative sample of the body of research demonstrating inequities and their impacts. Correcting course requires a critical examination of higher education from all angles, but in the second half of my talk I'll focus on inclusive teaching as a lever for change. I'll discuss my own journey to becoming a more inclusive teacher and some cornerstones of my approach.
• March 28: Arnaud Brothier (University of New South Wales)
  Forest-skein categories and groups
  Abstract

  Vaughan Jones found unexpected connections between subfactor theory and Richard Thompson's group while attempting to construct conformal field theories (CFT). This led to numerous fruitful applications and among others provided a novel way to construct group actions using categories. I am initiating a program strengthening Jones' visionary work where the Thompson group is replaced by a family of groups that I name "forest-skein groups". These groups are interesting on their own, satisfying exceptional properties and having powerful extra-structures for studying them. They are isotropy subgroup of forest-skein categories: categories of planar diagrams mod out by certain skein relations (just like planar algebras are in Jones' subfactor framework).
  I will briefly say a word about the story of Jones's discovery and then present forest-skein categories and groups using explicit examples.
• March 21: Elizabeth Mabbutt (University of Wollongong)
  What is a Sobolev Space and Why Do We Care?
  Abstract

  In simple terms, a Sobolev space is a normed vector space where two functions are “close together” with respect to the Sobolev norm if and only if their derivatives up to order \(m\) are “close together” with respect to the \(L^p\) norm. To understand what this means, a lot of background in functional analysis is required. In this talk, I will explain this background, eventually coming up with a rigorous definition of what a Sobolev space is, and will then show why we care about these spaces.
• March 14: Nicholas Seaton (University of Wollongong)
  Turning nasty groups into nice ones
  Abstract

  In 2011, an Heuf, Kumjian and Sims provided a classification for Fell algebras. They showed that two such algebras are Morita equivalent over their spectrum \(X\) if and only if they have the same class in the equivariant sheaf cohomology group \(H^2(X)\). This group is nasty, it is very difficult to work with as it is defined using abstract nonsense. In this talk, I will show that this group is isomorphic to a nice group, that is, a direct limit of twists groups.
• March 7: Alex Paviour (University of Wollongong)
  A glimpse at Axiomatic QFT
  Abstract

  I will discuss the motivation for studying quantum field theory by describing the physical problems it tries to address. I will then describe an influential attempt at a mathematically rigorous basis for QFT, the Wightman axioms, and their physical interpretation. I'll give some physically uninteresting but mathematically challenging examples of quantum fields satisfying these axioms, discuss the benefits and shortcomings of this approach to QFT, and point to some important theorems.
• February 29: Angus Alexander (University of Wollongong)
  Stationary scattering theory
  Abstract

  In this talk I will discuss some basic quantities in scattering theory and their physical motivations, as well as how to use the stationary approach to obtain scattering information.
• February 22: Aidan Sims (University of Wollongong)
  When do \(k\)-graphs embed in their fundamental groupoids, and why does it matter?
  Abstract

  Every graph \(C^*\)-algebra is Morita equivalent to a crossed product by its fundamental group of a \(C^*\)-algebra that is itself Morita equivalent to a commutative AF algebra. It turns out that this is because graphs always embed in their fundamental groupoids. Higher-rank graphs (aka \(k\)-graphs) are a different story. I’ll explain why, starting with what a \(k\)-graph and its fundamental groupoid are. This is joint work with Nathan Brownlowe, Alex Kumjian and David Pask.
• Talks from previous years