Wollongong OANCG Seminar

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

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

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Talks from previous years

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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