Organisers:
Lisa Orloff Clark (Wellington University, New Zealand)
Anna Duwenig (University of New South Wales, Australia)
Roozbeh Hazrat (Western Sydney University, Australia)
Huanhuan Li (Hefei University, China)
Aidan Sims (University of New South Wales, Australia)

Program Description:
Algebras constructed from combinatorial data are central in C*-algebra theory, symbolic dynamics and noncommutative algebra. There is a long history of researchers from two or more of these fields independently arriving at similar questions, and obtaining similar results, but via completely different techniques. An important example is the question of indecomposability for C*-algebras, algebras and dynamics coming from directed graphs: indecomposability has a natural meaning in each case; researchers in each area independently discovered a characterisation in terms of exactly the same property of the graph; but the proofs of these characterisations seemed completely unrelated, and shed no light on why indecomposability of one object should be related to that of either of the others. Discovering the underlying link between these three notions of indecomposability brought the algebra and C*-algebra communities together across a 10-year program of research. The synergy that this engendered energised both fields.

This research program on combinatorial *-algebras will gather international and Australian researchers working at the cutting edge of the nexus between C*-algebras and abstract algebras, along with the emerging next generation of researchers in these areas. Its scientific focus will be to identify further meta-properties with natural interpretations across the two areas, and to understand the underlying mathematical connections between them. At the same time, it will explore emerging interrelation with geometric group theory via topological full groups. Specifically, we will focus on recent developments in non-Hausdorff groupoid algebras and Roe algebras, as well as open problems such as the algebraic Kirchberg-Phillips problem, Matui’s conjectures relating K-theory of groupoid algebras and groupoid homology, and Williams’ SSE-SE problem and its connections to Hazrat’s graded classification conjectures. This will strengthen ties and foster collaborations between the communities working on combinatorial *-algebras, including coarse geometers, C*-algebraists, pure algebraists and dynamicists.

Program Structure:
 

Lectures:
Becky Armstrong: Twisted groupoid C*-algebras and twisted Steinberg algebras
In this minicourse I will introduce topological groupoids and show how to construct $C^*$-algebras and abstract algebras from groupoids. Along the way I will provide examples and draw some links to graph algebras. I will also describe some ways that the underlying groupoid influences the structure of the associated algebras. I will then introduce twisted groupoids and show how to build $C^*$-algebras (called “twisted groupoid $C^*$-algebras”) and abstract algebras (called “twisted Steinberg algebras”) from these.

Ralf Meyer: A bicategorical approach to Leavitt path algebras and similar
I will explain how Leavitt path algebras and, more generally, algebraic Cuntz-Pimsner algebras may be constructed as limits in a bicategory that has unital rings as objects and bimodules as morphisms. Limits in bicategories are far more interesting than limit in ordinary categories because a commuting square in a bicategory comes with extra data, not extra conditions. So these limits add more data instead of imposing conditions. The bicategorical limit interpretation also works for algebras associated to self-similar groups or self-similar graphs, and even for their higher-rank analogues. All these algebras are known to be groupoid
algebras, and their underlying groupoids may also be realised in an analogous bicategory of ample groupoids and groupoid correspondences.

Lia Văs: The graded structure of graph algebras
In this series of talks, the term graph algebras refers to graph $C^*$-algebras and Leavitt path algebras. For these algebras, characterizing algebraic properties in terms of the properties of the underlying graph proved to yield fruitful results, especially if the graph algebras are considered together with their graded structure. Because of this, we focus on the grading of the graph algebras induced by considering the lengths of paths of the graph and on the advantages this graded structure presents: the graded ideals and their quotients have a concise description in terms of the graph elements, a wide variety of the graph algebras have finite composition length, and the graded Grothendieck groups carries a promise for the complete classification of the algebras.

Participant List:
Anna Duwenig (UNSW Sydney)
Alina Vdovina (CUNY, City College and Graduare Center)
Ying-Fen Lin (Queen’s University Belfast)
Ralf Meyer (Mathematisches Institut, University of Gottingen)
Ilija Tolich (Victoria University of Wellington)
Roozbeh Hazrat (Western Sydney University)
Josh Johansson (University of New South Wales (UNSW))
Aidan Sims (University of New South Wales)
Adam Dor-On (University of Haifa)
Naomi Reed (University of Wollongong)
Lynnel Naingue (Mindanao State University – Iligan Institute of Technology)
Lia Văs (Saint Joseph’s University)
Eun Ji Kang (Seoul National University)
Efren Ruiz (University of Hawai’i at Hilo)
Elizabeth Pacheco (University of Western Sydney)
Shanshan Hua (Universität Münster)
Astrid an Huef (Victoria University of Wellington)
Michael Ó Ceallaigh  (Victoria University of Wellington)
Søren Eilers (University of Copenhagen)
Giang Nam Trần (Institute of Mathematics, Vietnam Academy of Science and Technology)
Becky Armstrong (Victoria University of Wellington)
Boris Bilich (University of Haifa / University of Goettingen)
Kathryn McCormick (California State University Long Beach)
Roy Jansen (Victoria University of Wellington)
Elizabeth Gillaspy (University of Montana)
Ruijun Lin (University of Wollongong)
Samuel Evington (University of Münster)
Ryan Thompson (Te Herenga Waka Victoria University of Wellington)
Danielle Morrigan (UNSW Sydney)
Jeremy Hume (University of New South Wales)

Registration:

• Registration is now closed
• Arrival date: 1 March 2026
• Departure date: 13 March 2026

ASSOCIATED EVENTS
MATRIX Wine and Cheese Afternoon 3 March 2026
On the first Tuesday of each program, MATRIX provides a pre-dinner wine and cheese afternoon. Produce is locally-sourced to showcase delicacies from the region.