## Colloquia

**Talks will be held on Thursdays 8pm Sydney Time: (UTC 10am) Please write to R. Hazrat** **to get the Zoom address.**

**Talks will be held on Thursdays 8pm Sydney Time: (UTC 10am) Please write to R. Hazrat**

**to get the Zoom address.**

### 2020

4 May: **Toke Carlsen **(Faroe Islands) Graph algebras, groupoids, and symbolic dynamics**Abstract: **I will give an overview of some recent results that link diagonal-preserving isomorphism of graph algebras and isomorphism and equivalence of graph groupoids with continuous orbit equivalence, (eventual) conjugacy, and flow equivalence of symbolic dynamical systems constructed from directed graphs. PowerPoint

21 May: **Mark Lawson **(Edinburgh) Non-commutative Stone dualities**Abstract: **The classical Stone dualities for lattices such as frames, distributive lattices and generalized Boolean algebras can be generalized to a non-commutative setting to pseudogroups, distributive inverse semigroups and Boolean inverse semigroups, respectively. The goal of this talk is to sketch out the how and to motivate the why. I shall not assume any background from inverse semigroups or \'etale groupoids. PowerPoint

28 May: **Enrique Pardo Espino** (Cadiz): Self-similar graphs and their algebras**Abstract:** In this talk, we will explain the origins of the notion of self-similar graph. We give a groupoid model of the algebra associated to a self-similar graph and we provide a characterization of simplicity for these algebras. We briefly talk about further developments on this construction. The contents of this talk are part of a joint paper with R. Exel. PowerPoint

4 June: **Mike Whittaker (**Glasgow): Aperiodic tilings: from the Domino problem to aperiodic monotiles**Abstract: **Almost 60 years ago, Hao Wang posed the Domino Problem: is there an algorithm that determines whether a given set of square prototiles, with specified matching rules, can tile the plane. Robert Berger proved the undecidability of the Domino Problem by producing a set of 20,426 prototiles that tile the plane, but any such tiling is nonperiodic (lacks any translational symmetry). This remarkable discovery began the search for other (not necessarily square) aperiodic prototile sets, a finite collection of prototiles that tile the plane but only nonperiodically. In the 1970s, Roger Penrose reduced this number to two. Penrose's discovery led to the planar einstein (one-stone) problem: is there a single aperiodic prototile? In a crowning achievement of tiling theory, the existence of an aperiodic monotile was resolved almost a decade ago by Joshua Socolar and Joan Taylor. My talk will be somewhat expository, and culminate in both a new direction in aperiodic tiling theory and a new aperiodic monotile. PowerPoint

11 June: ** Kevin Brix **(Copenhagen/Wollongong): Fine structure of C*-algebras associated to topological dynamics**Abstract**: I will report on the story of associating C*-algebras to symbolic dynamical systems (e.g. shift spaces or directed graphs) and the recently articulated program of understanding dynamical relations (such as conjugacy or flow equivalence) in terms of structure-preserving *-isomorphisms of the corresponding C*-algebras. A large body of rigidity results have successfully been obtained for graphs and more general systems. There will be an emphasis on open questions and problems yet to be solved! PowerPoint

12 June: **James Hyde** (Cornell) Finitely Presented Groups that Contain Q **Abstract:** I will present joint work with James Belk and Fancesco Matucci. We construct embeddings of the additive group of rational numbers into several finitely presented groups. We rely upon Thompson's groups, which I will describe.

18 June: **Joan Bosa **(Barcelona) The realization problem for von Neumann regular rings**Abstract: **The realization problem for von Neumann (vN) regular rings asks whether all conical refinement monoids arise from monoids induced by the projective modules over a vN regular ring. We will quickly overview this problem, and show the last developments on it. This is joint work with P. Ara, E.Pardo and A.Sims. PowerPoint

June 25th: **Pere Ara** (Barcelona) Graded K-Theory, Filtered K-theory and the classification of graph algebras**Abstract: **We prove that an isomorphism of graded Grothendieck groups of two Leavitt path algebras induces an isomorphism of a certain quotient of algebraic filtered K-theory and consequently an isomorphism of filtered K-theory of their associated graph C*-algebras. As an application, we show that, since for a finite graph E with no sinks, the graded Grothendieck group of L(E) coincides with Krieger's dimension group of its adjacency matrix, our result relates the shift equivalence of graphs to the filtered K-theory and consequently gives that two arbitrary shift equivalent matrices give stably isomorphic graAbstract: We prove that an isomorphism of graded Grothendieck groups of two Leavitt path algebras induces an isomorphism of a certain quotient of algebraic filtered K-theory and consequently an isomorphism of filtered K-theory of their associated graph C*-algebras. As an application, we show that, since for a finite graph E with no sinks, the graded Grothendieck group of L(E) coincides with Krieger's dimension group of its adjacency matrix, our result relates the shift equivalence of graphs to the filtered K-theory and consequently gives that two arbitrary shift equivalent matrices give stably isomorphic graph C*-algebras. This result was only known for irreducible graphs. This is a joint work with Roozbeh Hazrat and Huanhuan Li. PowerPoint

July 2: **Aidan Sims **(Wollongong) Graded K-theory for Z_2-graded graph C*-algebras**Abstract:** While there is no universally agreed-upon definition of Z_2-graded K-theory for C*-algebras, a very natural way to define it is using Kasparov's celebrated KK-bifunctor: KK is naturally a Z_2-graded theory, and Kasparov proved that if applied to trivially-graded C*-algebras A, the groups KK_*(\mathbb{C}, A) are the K-groups of A. So it is natural to define K^{gr}_*(A) as KK_*(\mathbb{C}, A) for Z_2-graded C*-algebras A in general. I will discuss recent work with Adam Sierakowski and with honours students Quinn Patterson and Jonathan Taylor, building on previous work with Kumjian and Pask, that uses deep ideas of Pimsner to compute the graded K-theory, defined in this way, of relative graph C*-algebras carrying Z_2-gradings determined by binary labellings of the edges of the graph: the formulas that emerge strongly suggest that this notion of graded K-theory captures the right sort of information. PowerPoint

July 9: **Xiao-Wu Chen** (Hefei): Leavitt path algebra via the singularity category of a radical-square-zero algebra**Abstract: **We will recall some previous work primarily by Paul Smith, and show that the Leavitt path algebra is closely related to the singularity category of a finite dimensional radical-square-zero algebra. Recently, we

apply such a link to confirm Keller's conjecture for a radical-square-zero algebra. More precisely, we prove that

for such an algebra, the singular Hochschild cochain complex is B_infinity-isomorphic to the Hochschild cochain complex of the dg singularity category. This is based on a joint work with Huanhuan Li and Zhengfang Wang.PowerPoint

July 16: **Jean Renault** (d'Orleans) : Groupoids Extensions.**Abstract: **I shall present a groupoid version of the Mackey normal subgroup analysis in a C*-algebraic framework. More precisely, the main result is a description of the C*-algebra of a locally compact groupoid with Haar system, possibly endowed with a twist, which is an extension by a group bundle. The natural expression of this result uses Fell bundles over groupoids. When the group bundle is abelian, one obtains a twisted groupoid C*-algebra. I will give some applications. This talk is based on a joint work with M.Ionescu, A.Kumjian, A.Sims and D.Williams. PowerPoint

July 23: **Be'eri Greenfeld** (Bar Ilan) How do algebras grow?**Abstract: **The question of `how do algebras grow?', or, which functions can be realized as growth functions of algebras (associative/Lie/etc., or algebras having certain additional algebraic properties) is a major problem in the junction of several mathematical fields, including noncommutative algebra, combinatorics of (infinite) words, symbolic dynamics, self-similarity and more. We provide a novel paradigm for tackling this problem (in fact, family of problems), thereby resolving several open problems posed by experts regarding possible growth types of finitely generated associative algebras and Lie algebras. We also considerthe set of growth functions as a space, and point out odd properties it admits (arbitrarily rapid holes, and convergence to outer points - with respect to some plausible notion of limits). PowerPoint

July 30: **Arun Ram **(Melbourne): Teaching mathematics in the next life**Abstract:** For many years Ive been thinking about how to teach mathematics with honesty and inspiration. This has resulted in ideas like "proof machine", "marking apocalypse", "rotary dial phenomenon", and "just do it". And then a virus came, and the new life began, online, on Zoom. This will be a talk about the adventures of the past life and the preparations for the next. PowerPoint

August 6: **Bob Gray** (East Anglia) Undecidability of the word problem for one-relator inverse monoids

**Abstract: ** It is a classical result of Magnus proved in the 1930s that the word problem is decidable for one-relator groups. This result inspired a series of investigations of the word problem in other one-relator algebraic structures. For example, in the 1960s Shirshov proved the word problem is decidable in one-relator Lie algebras. In contrast, it remains a longstanding open problem whether the word problem is decidable for one-relator monoids. An important class of algebraic structures lying in between monoids and groups is that of inverse monoids. In this talk I will speak about a recent result which shows that there exist one-relator inverse monoids of the form Inv<A|w=1> with undecidable word problem. This answers a problem originally posed by Margolis, Meakin and Stephen in 1987. I will explain how this result relates to the word problem for one-relator monoids, the submonoid membership problem for one-relator groups, and to the question of which right-angled Artin groups arise as subgroups of one-relator groups.PowerPoint

August 13: **Cristobal Gil Canto **(Malaga) Invariant ideals in Leavitt path algebras.

**Abstract: ** As well-known examples of Leavitt path algebras arise the so-called primary colours: they respectively correspond to the ideal generated by the set of line points, the vertices that lie on cycles without exits and the one generated by the set in extreme cycles. It is known that these ideals are invariant under isomorphism. In this talk we will analyze the invariance of another key piece of a Leavitt path algebra. We will see that though the ideal generated by the vertices whose tree contains infinitely many bifurcation vertices or at least one infinite emitter is not invariant, we will find its natural replacement (which is indeed invariant). We will also give some procedures to construct invariant ideals from previous known invariant ideals. In order to do that, on the one hand, we will introduce a topology in the set of vertices of a graph. And on the other hand, via category theory, we will think of the saturated and hereditary set of a graph as a functor. This a joint work together with Dolores Martín Barquero and Cándido Martín González.

August 20: **Jose Burillo **(Barcelona): The irrational-slope Thompson's groups**Abstract:** Irrational-slope Thompson's groups were introduced by Cleary in two papers in 1995 and 2000, where he proved they are FP_\infty. These are groups of PL maps of [0,1] whose breakpoints are in some irrational subring of R and the slopes are also irrational numbers. Interest in these groups grew recently when it was asked whether they can be obtained as subgroups of Thompson's group F. In this paper we will introduce the golden ratio group F_\tau, describe how to work with it in terms of binary trees and also algebraically. We will show a presentation for F_\tau and show that elements admit a unique normal form, in similar fashion as F. We will study its metric properties and undistorted copies of F inside, and finally, if time permits, we will say a few words about the irrational versions of Thompson's groups T and V. This is joint work with Brita Nucinkis and Lawrence Reeves. PowerPoint

August 27: **Ralf Meyer** (Goettingen) Aperiodicity and related properties for crossed product inclusions**Abstract: **In recent work with Bartosz Kwaniewski, we have vastly generalised the condition that was introduced by Kishimoto in order to prove that reduced crossed products for outer group actions on simple C*-algebras are again simple. We call this condition aperiodicity, and it applies to arbitrary inclusions of C*-algebras, without requiring a crossed product structure. We relate this to topological non-triviality conditions in the special case of actions of inverse semigroups or étale groupoids (which are possibly non-Hausdorff). In that generality, we define an essential crossed product, which is a quotient of the reduced crossed product. If the action satisfies Kishimoto's condition, then the coefficient algebra detects ideals in this essential crossed product. And in the simple case, we also get criteria for the essential crossed product to be simple. We also relate aperiodicity to other properties that have been used to study the ideal structure of crossed products. This includes unique pseudo-expectations and the almost extension property, which assume that the set of pure states on the coefficient algebra that extend uniquely to the crossed product is dense. PowerPoint

September 3: **Lia Vas **(Philadelphia) The Graded Classification Conjecture for graph algebras**Abstract: **The ordinary (pointed) K_0-group is not a complete invariant of algebras typically associated to a directed graph. When these algebras are considered as graded algebras and the definition of the K_0-group is adjusted to reflect the existence of this grading, the situation becomes more interesting. The Graded Classification Conjecture states that this adjusted version of the (pointed) K_0-group is a complete invariant of a Leavitt path algebra over a field (and this statement can be adapted for other graph algebras). We shall discuss the context in which this conjecture has been formulated, the current status of the conjecture, and some ongoing research. PowerPoint

September 10: **Volodymyr Mazorchuk** (Uppsala): Adjunction in the absence of identity**Abstract:** In this talk I plan to present and discuss a rather weak bicategorical setup in which one can talk about genuine adjunctions. I will roughly describe the main motivation coming from representation theory of finitary 2-categories (or bicategories) and make some parallells with the structure theory of finite semigroups. I will try to explain how this approach simplifies some results but also makes some other results much more difficult.

This is a joint work with Hankyung Ko and Xiaoting Zhang.

September 17: **Murray Elder** (Sydney) Some new kinds of automatic groups**Abstract:** I will describe some generalisations of the notion of an automatic group, and how far they are away from automatic (in a precise sense). Relevant papers are

https://arxiv.org/abs/2008.02381 and https://arxiv.org/abs/2008.02511

September 24: **Jason Bell **(Waterloo): The growth of algebras**Abstract:** We give an overview of the theory of growth functions for associative algebras and explain their significance when trying to understand algebras from a combinatorial point of view. We then give a classification for which functions can occur as the growth function of a finitely generated associative algebra up to asymptotic equivalence. This is joint work with Efim Zelmanov.

30 September - October 2: **CRMDS-IPM** **Joint Workshop on Operator Algebras**

A joint workshop of CRMDS and Institute for Research in Fundamental Sciences, Tehran (IPM, conveyer, **Prof. Massoud Amini, **mamini@ipm.ir) on Operator Algebras will be held 30Sep 2-Oct via Zoom. Program