Wollongong OANCG Seminar

Talks older than 2023 were from before the creation of this website.

Past organisers: Angus Alexander, Abraham Ng, Anna Duwenig, Kevin Brix, James Gabe, Adam Sierakowski, Nathan Brownlowe, Mike Whittaker.

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Talks from previous years

2024

• December 5: Malcolm Jones (Victoria University of Wellington) - 4:00pm start
  Path groupoids of nonfinitely aligned P-graphs
  Abstract

  The directed graph approach to C*-algebras was generalised to P-graphs by Brownlowe–Sims–Vittadello in 2013. The unit space of the usual path groupoid of a P-graph can fail to be locally compact if the P-graph is not finitely aligned. In 2020, Spielberg developed a groupoid for any left cancellative small category (for example, any P-graph) that is ample and not necessarily Hausdorff. In 2023, Neshveyev–Schwartz gave a nonfinitely aligned left cancellative small category whose reduced C*-algebra is not modelled by Spielberg's groupoid. Moreover, Spielberg's groupoid is complicated to construct. In this talk, we associate path groupoids to (not necessarily finitely aligned) P-graphs that are ample, Hausdorff and relatively easy to construct. Our path groupoid coincides with the path groupoid of finitely aligned P-graphs and is very different from Spielberg's groupoid for nonfinitely aligned P-graphs. These results appear in my thesis, which I successfully defended in October this year, supervised by Lisa Orloff Clark and Astrid an Huef.
• November 28: Aidan Sims (University of Wollongong)
  Renault’s disintegration theorem for étale groupoids
  Abstract

  The classical result about representations of commutative C*-algebras is that they all arise as direct integrals of representations of the complex numbers by scalar multiples of the identity on fibres of a Borel Hilbert bundle – that is, they “disintegrate.” To show that representations of groupoid algebras that are continuous in the inductive-limit topology are bounded by the groupoid analogue of the 1-norm, Renault needed to know that these representations also disintegrate appropriately. For general groupoids, the proof (which first appears in detail in a paper of Muhly-Williams) is very involved, the resulting disintegrated representation is pretty complicated, and it’s all-around hard to see what is going on. But it turns out that, as usual, for Hausdorff étale groupoids everything is both simpler and more enlightening. In this talk, I will try to justify that assertion.
• November 21: Ada Masters (University of Wollongong)
  Conformal symmetry of the Podleś sphere
  Abstract

  I will describe the notion of conformal group equivariance for unbounded KK-theory and its generalisation to quantum groups, extending the work of Baaj and Skandalis in the bounded picture. I will focus on the example of the Podleś sphere, whose equivariance at the level of bounded KK-theory was demonstrated by Nest and Voigt.
• November 14: Alexander Mundey (University of Wollongong)
  Cohomology of self-similar actions and Zappa-Szép products
  Abstract

  In his 2004 article, Nekrashevych introduced representations of self-similar actions and their associated universal C*-algebras. We became interested in twisting these representations by 2-cocycles but encountered the challenge of identifying exactly where these cocycles should live.
  
  In this talk, I will describe this cohomology theory for self-similar actions and the more general Zappa-Szép products. I will also introduce an analogue of the Eilenberg-Zilber theorem within this framework, which provides an explicit way of constructing cocycles.
  
  This is joint work with Aidan Sims.
• October 31: Adam Rennie (University of Wollongong)
  Towards 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.
• 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.

2023

• November 30: Adam Rennie (University of Wollongong)
  Some NCG
  Abstract

  Some NCG will be presented, content dependent on the audience for the last seminar of the year!
• November 23: Angus Alexander (University of Wollongong)
  The spectral shift function for Schrödinger operators
  Abstract

  In this talk I will give brief description of the spectral shift function for a pair of self-adjoint operators. In particular, I will describe how for Schrödinger operators the behaviour of the spectral shift function at zero is related to the existence of zero-energy resonances.
• November 16: Roozbeh Hazrat (Western Sydney University)
  Bergman algebras
  Abstract

  Exactly a half a century ago, George Bergman introduced a stunning machinery which would realise any commutative conical monoid as a non-stable K-theory of an algebras. The algebras constructed is “minimal” or “universal”. He showed many interesting algebras such as those of Leavitt can be constructed from his machinery. We will look at his paper. We then extend the results to the graded setting, where one can capture dynamics within algebras.
• November 9: Aidan Sims (University of Wollongong)
  A “what is” seminar: what is a (k-)graph C*-algebra
  Abstract

  One of the things we talk about at Wollongong all the time is “graph C*-algebra,” but we are so used to them that we don’t often explain what they are. I will try to outline what a graph C*-algebra is and what some of the main theorems about them say; if the vibe seems right, I might say something at the end about their slightly weirder cousins, k-graph C*-algebras. This talk will be aimed at someone who knows what a Hilbert space is and knows what a topological space is, but maybe not what a C*-algebra is.
• November 2: Diego Martinez (University of Münster)
  Generalized dynamics yielding nuclear crossed products
  Abstract

  Given a (generalized) dynamical system defined on a commutative C*-algebra, one can construct a suitable notion of reduced crossed product. This crossed product, however, may fail to admit a conditional expectation onto the original C*-algebra, but it does admit a weak variant. In this talk we will define these systems, and construct their associated C*-algebras. We will then give a sufficient condition for the nuclearity of these algebras that generalizes amenability for group actions on C*-algebras and twisted étale groupoids. This talk will be based on joint work with Alcides Buss.
• October 26: Memorial seminar in honour of Iain Raeburn
  
  • Time: 2:30pm - 4:30pm
  • Location: Room 20-4
  Abstract

  A series of talks by Nathan Brownlowe, Alan Carey, Adam Rennie, and Aidan Sims, celebrating the life and works of Iain Raeburn.
• October 19: Aidan Sims (University of Wollongong)
  Bibbidi bobbidi boo
  Abstract

  Connes' theorem reconstructs a manifold from a spectral triple. Renault's theorem reconstructs an étale groupoid and a twist from a Cartan pair. Put ‘em together, and what have you got? This is joint work with Anna Duwenig.
• October 12: Victor Wu (University of Sydney)
  From directed graphs of groups to Kirchberg algebras
  Abstract

  Directed graph algebras have long been studied as tractable examples of C*-algebras, but they are limited by their inability to have torsion in their \(K_1\)-group. Graphs of groups, which are famed in geometric group theory because of their intimate connection with group actions on trees, are a more recent addition to the C*-algebra scene. In this talk, I will introduce the child of these two concepts—directed graphs of groups—and describe how their algebras inherit the best properties of its parents’, with a view to outlining how we can use these algebras to model a class of C*-algebras (stable UCT Kirchberg algebras) which is classified completely by K-theory.
• October 5: Rodrigo Frausino (University of Wollongong)
  Thermodynamic Formalism for Generalized Markov Shifts with countable alphabet
  Abstract

  We shall begin by introducing the area of thermodynamic formalism for Markov shifts. When the alphabet is infinite, these spaces are usually not compact and not even locally compact. We review a compactification of the Markov space for an infinite transition matrix, introduced by Marcelo Laca and Ruy Exel in 1999, which comes from C*-algebraic theory. We end by providing some evidence, via an example, on why someone from area of thermodynamic formalism for Markov shifts might be interested in such a compactification.
• September 21: Adam Rennie (University of Wollongong)
  They may be Fredholm, but their index is not what you think
  Abstract

  From Fredholm operators on complex Hilbert spaces to Fredholm operators on complex Hilbert modules introduces some pain and technicality. From Fredholm operators on complex Hilbert spaces to real Hilbert spaces introduces minus signs and Clifford algebras. Nonetheless, all these "index environments" look and feel much the same. For real Hilbert modules it just gets weird.
• September 14: Dilshan Wijesena (University of New South Wales) - 2:30pm start
  Classifying representations of the Thompson groups and the Cuntz algebra
  Abstract

  Richard Thompson’s groups \(F\), \(T\) and \(V\) are one of the most remarkable discrete infinite groups for their several unusual properties. On the other hand, the celebrated Cuntz algebra has many fascinating properties and it is known that \(V\) embeds inside the Cuntz algebra. However, classifying the representations of the Thompson groups and the Cuntz algebra have proven to be very difficult.
  
  Luckily, thanks to the novel technology of Vaughan Jones, a rich family of so-called Pythagorean representation of the Thompson groups and the Cuntz algebra can be constructed by simply specifying a pair of finite-dimensional operators satisfying a certain equality. These representations carry a powerful diagrammatic calculus which we use to develop techniques to study their properties. This permits to reduce very difficult questions concerning irreducibility and equivalence of infinite-dimensional representations into problems in finite-dimensional linear algebra. Moreover, we introduce the Pythagorean dimension which is a new invariant for all representations of the Cuntz algebra. For each dimension \(d\), we show the irreducible classes form a moduli space of a real manifold of dimension \(2d^2+1\). Finally, we introduce the first known notion of a tensor product for representations of the Cuntz algebra.
• September 7: Dan Ursu (University of Münster) - Zoom broadcast
  Simplicity of crossed products by FC-hypercentral groups
  Abstract

  Results from a few years ago of Kennedy and Schafhauser characterize simplicity of reduced crossed products AxG, where A is a unital C*-algebra and G is a discrete group, under an assumption which they call vanishing obstruction. However, this is a strong condition that often fails, even in cases of A being finite-dimensional and G being finite.
  
  In joint work with Shirly Geffen, we find the correct two-way characterization of when the crossed product is simple, in the case of G being an FC-hypercentral group. This is a large class of amenable groups that, in the finitely-generated setting, is known to coincide with the set of groups which have polynomial growth. With some additional effort, we can characterize the intersection property for AxG in the non-minimal setting, for the slightly less general class of FC-groups. Finally, for minimal actions of arbitrary discrete groups on unital C*-algebras, we are able to generalize a result by Hamana for finite groups, and characterize when the crossed product AxG is prime.
  
  All of our characterizations are initially given in terms of the dynamics of G on I(A), the injective envelope of A. This gives the most elegant characterization from a theory perspective, but I(A) is in general a very mysterious object that is hard to explicitly describe. If A is separable, our characterizations are shown to be equivalent to an intrinsic condition on the dynamics of G on A itself.
• August 31: Owen Tanner (University of Glasgow) - 4:30pm start - Zoom broadcast
  Topological full groups by example
  Abstract

  Are there infinite, finitely generated simple groups?
  
  This seemingly innocuous question troubled some of history’s best group theorists for more than 50 years. In this talk, I will explain how ample groupoids provide a unifying framework to generate and study interesting examples. This framework is called “topological full groups” and was pioneered by Hiroki Matui in the late 00’s. It allowed us to answer fundamental questions like “Are there amenable, infinite, finitely generated simple groups?”.
  
  I will then give some concrete examples of these “topological full groups” which I found interesting to study. These examples come from concrete C*-algebras and groupoids that might be familiar to some of the audience. I will assume little prior group theory knowledge. This talk is based on some research from my thesis, and some research that I did with Eusebio Gardella.
• August 24: Angus Alexander (University of Wollongong)
  Spectral flow for Schrodinger operators
  Abstract

  In 2009 Carey, Potapov and Sukochev provided a formula for the spectral flow along a path of unbounded self-adjoint Fredholm operators. In this talk we consider the path \(H_t = -\Delta + t V\) where \(\Delta\) is the Laplacian and \(V\) is a real function which decays sufficiently fast. We demonstrate how techniques from scattering theory can be applied to evaluate the terms appearing in this spectral flow formula and obtain an expression for the number of eigenvalues of the operator \(H = -\Delta + V\). Interestingly, such a formula is sensitive to resonances, and in low dimensions we recover Levinson's theorem, a well-known result in quantum scattering theory.
• August 17: Alex Kumjian (University of Nevada, Reno)
  Homotopy of groupoid cocycles
  Abstract

  Let \(G\) be locally compact Hausdorff groupoid and let \(c: G^n \to \mathbb{T}\) be an \(n\)-cocycle. We prove that \(c\) is homotopic to the trivial cocycle iff there is an \(n\)-cocycle \(h: G^n \to \mathbb{R}\) such that \(c(x) = exp(2{\pi}ih(x))\) for all \(x \in G^n\). This is joint work with Elizabeth Gillaspy.
• August 10: Ada Masters (University of Wollongong) - Zoom broadcast
  Conformal geometry and unbounded equivariant KK-theory
  Abstract

  In which we see that the right definition is sometimes elusive, and more weight is put upon an already overburdened concept. In the development of unbounded KK-theory, an aspect which has been left unresolved is the definition of equivariance. One reason for this is that the natural definition fails to capture all the degrees of freedom available in the usual, bounded equivariant KK-theory. This is apparent already in the classical case, as we shall see. In light of the forgoing, conformal geometry will be briefly introduced. Two proposed definitions of conformal equivalence in noncommutative geometry will be compared and shown to be closely related. A small sample of unanswered questions will be provided.
• August 3: Hao Guo (Tsinghua University) - Zoom broadcast
  Quantitative scalar curvature problems and operator algebras
  Abstract

  Operator algebras have historically found significant applications to problems on smooth manifolds, particularly in relation to the problem of determining when a metric of everywhere-positive scalar curvature exists on such a manifold. More recently, researchers have been interested in the problem of quantifying more precisely the size of scalar curvature—not just its positivity—by relating it to other geometric quantities, such as lengths and volumes on the manifold. In spirit, these inquiries are analogous to more classical theorems, such as the Bonnet–Myers theorem, that deal with sectional and Ricci curvatures, but are comparatively subtler due to scalar curvature being a weaker invariant. Based on previous work with Guoliang Yu and Zhizhang Xie (Texas A&M), I will describe how the quantitative behaviour of scalar curvature can be linked to new invariants in operator algebras, specifically in quantitative K-theory groups.
• July 27: Robert Neagu (University of Oxford) - 4:30pm start - Zoom broadcast
  On topologically zero-dimensional *-homomorphisms
  Abstract

  The notion of covering dimension of a topological space dates back to the work of Lebesgue in the 1920s. C*-algebras are objects with an intrinsic topological structure and an important philosophy is that they can be seen as non-commutative topological spaces. In this talk we are going to show how Lebesgue's covering dimension can be extended to C*-algebras and, in fact, maps between them. Using classification theory, we are going to show how certain *-homomorphisms with nuclear dimension equal to zero are unitarily equivalent to morphisms which factor through a simple AF-algebra.
• July 20: Shanshan Hua (University of Oxford) - Zoom broadcast
  Nonstable K-theory for Z-stable C*-algebras
  Abstract

  In this talk, The Unital Classification Theorem for C*-algebras (by many hands) will be presented, with the motivations coming from the classification of von-Neumann algebras. Among all regularity conditions needed for the classification program, I will talk about Z-stability in more details. Then we will focus on certain special properties of Z-stable C*-algebras, i.e. K1-injectivity and K1-surjectivity.
  In Jiang's unpublished paper (1997), it is shown that any Z-stable C*-algebra A is both K1-injective and K1-surjective, which means that the K1-group can be calculated by just looking at homotopy equivalence classes of U(A), without matrix amplifications. For such A, it is also shown that the higher homotopy groups of U(A) are isomorphic to either K0(A) or K1(A), depending on the dimension of the higher homotopy group. I will present Jiang's result for Z-stable C*-algebras and explain briefly our new strategies to reprove his theorems, which uses an alternative picture of the Jiang-Su algebra as an inductive limit of generalized dimension drop algebras.
• July 13: Sean Harris (Australian National University) - Zoom broadcast
  Non-commutative metric spaces and fractals therein
  Abstract

  Non-commutative geometry considers certain non-commutative algebras and acts as if these were the algebras of continuous functions on "non-commutative" topological spaces, bringing many tools from geometry and topology to the study of such algebras (by duality, analogy, and a lot of functional analysis). I will explain how metric geometry appears in the non-commutative setting, and present my recent and ongoing work on fractals and fractal tilings in these "non-commutative metric spaces".
• July 6: Alan Stoneham (University of New South Wales)
  The Type (B) Problem for Well-bounded Operators
  Abstract

  A scalar-type spectral operator T is an operator acting on a Banach space that can be represented as an integral over σ(T) with respect to a spectral measure. When the Banach space is reflexive, it had been shown by Dunford that T being scalar-type spectral is equivalent to T having a C(σ(T)) functional calculus. In 1994, Doust and deLaubenfels showed that this equivalence holds precisely on Banach spaces that do not have a subspace isomorphic to c0. A well-bounded operator T on a Banach space is an operator that has an AC[a, b] functional calculus, and it is said to be of type (B) if it can be written as an integral with respect to a spectral family of projections. On a reflexive Banach space, all well-bounded operators are of type (B). However, classifying the Banach spaces on which all well-bounded operators are of type (B) is still an open problem, and it has been conjectured by Doust and deLaubenfels that these are precisely the reflexive Banach spaces. In this talk, we will discuss the current standing of this problem and some recent progress.
• June 29: Alexander Mundey (University of Wollongong)
  The lacunary algebra of an iterated function system
  Abstract

  Iterated function systems (IFS) are a class of dynamical systems with the remarkable feature that they admit unique—often fractal—attractors. Analysing the interplay between the dynamics and topology of IFS and their attractors using operator algebraic techniques is difficult. Subtle topological information is often discarded by the usual C*-algebraic constructions. In this talk I will introduce the Lacunary Algebra of an IFS and show how it captures more information than previous constructions. This is (as yet) unpublished work from my PhD.
• June 22: Thomas Futcher (University of Wollongong)
  A study of discrete and integral transforms with logarithmic separable kernels
  Abstract

  A class of discrete and integral transforms, which includes the Laplace and Mellin transforms is defined on an appropriate weighted \(L^1\) space. A generalised convolution product is formulated, and examples are given for specific transforms. A space in which the discrete and integral operators are injective is then presented as well as an inversion formula in special cases. The transforms and convolution products are extended to the theory of distributions. Differentiability properties of the convolution of distributions with test functions are established and our class of transforms is shown to be an isomorphism between some spaces of distributions. The set of complex Borel measures \(M(I)\) are introduced and an appropriate convolution formula is defined on \(M(I)\). The weighted \(L^1\) space in which the transforms are defined on can be embedded into the complex Borel measures and is an ideal of \(M(I)\). A transform is defined on the complex Borel measures which satisfies the homomorphism property with respect to the convolution product.
• June 15: Abraham Ng (University of Wollongong)
  \(K\)-theory for algebras associated to commuting Hilbert bimodules
  Abstract

  In the quest to characterise stably-finite extensions, we developed machinery that captures the \(K\)-theory arising out of James Fletcher’s work with algebras associated to commuting Hilbert bimodules. An application of this general machinery to rank-\(2\) Deaconu–Renault groupoids with totally disconnected locally compact Hausdorff unit space gives us both a solid grasp of their \(K\)-theory and a stable-finiteness result about their extensions. This is joint work with Astrid an Huef and Aidan Sims.
• 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 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.

2022

• Feb 24 - Abraham Ng
• Mar 3 - Lara Ismert
• Mar 10 - Valerio Proietti
• Mar 17 - Rodrigo Frausino
• Mar 24 - Arnaud Brothier
• Mar 31 - Rafael Pereira Lima
• Apr 7 - Ulrik Enstad
• Apr 28 - Efren Ruiz
• May 5 - Johannes Christensen
• May 19 - Kevin Brix
• May 26 - Hang Wang
• Jun 2 - Sean Harris
• Jun 12 - Koen van den Dungen
• Jun 23 - Alex Kumjian
• Jun 30 - Chris Bourne
• Jul 7 - Adam Rennie
• Jul 21 - Victor Wu
• Jul 28 - Kang Li
• Sep 29 - Ali Raad
• Oct 6 - Dimitris Gerontogiannis
• Oct 13 - Roozbeh Hazrat
• Oct 20 - Aidan Sims
• Oct 27 - Teun van Nuland
• Nov 3 - Ada Masters
• Nov 10 - Andrew Stocker
• Nov 17 - Angus Alexander
• Nov 18 - Lachlan MacDonald
• Nov 24 - Are Austad

2021

• Feb 4 - Chris Bourne
• Feb 11 - Boyu Li
• Feb 18 - Dan Hudson
• Feb 25 - Adam Rennie
• Mar 4 - Chris Bruce
• Mar 11 - Takuya Takeishi
• Mar 18 - Aidan Sims
• Mar 25 - Roozbeh Hazrat
• Apr 1 - Heath Emerson
• Apr 8 - Ulrik Enstad
• Apr 15 - Camila Sehnem
• Apr 22 - Kevin Brix
• Apr 29 - Shirly Geffen
• May 5 - Adam Dor On
• May 13 - Sophie Raynor
• May 27 - Nathan Brownlowe
• Jun 3 - Nicholas Seaton
• Jun 10 - Magnus Goeffeng
• Jun 17 - Anna Duwenig
• Jun 24 - Rufus Willett
• Jul 8 - Adam Rennie
• Jul 22 - Adam Rennie
• Jul 29 - Jonathan Taylor
• Aug 5 - Adam Rennie
• Aug 12 - Jamie Gabe
• Aug 19 - Astrid an Huef
• Aug 26 - Thomas Scheckter
• Sep 2 - Bram Mesland
• Sep 9 - Karen Strung
• Sep 16 - Hao Guo
• Sep 23 - Kristin Courtney
• Sep 30 - Aidan Sims
• Oct 7 - Aidan Sims
• Oct 14 - Angus Alexander
• Oct 21 - Aidan Sims
• Oct 28 - Joel Zimmerman
• Nov 4 - Ada Masters
• Nov 11 - Alex Mundey
• Nov 18 - Adam Rennie
• Nov 25 - Pieter Spaas
• Dec 2 - Marcelo Laca

▶ 2011–2015

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