Saturday November 15th

Title: Monads and etale groupoid Fell bundles

Abstract: Several constructions in operator algebras have given rise to closely linked (``associated'') subcategories. These involve universal properties, which are sometimes not easy to verify. We introduce a new method - involving monads and comonads - to make this easier. And we apply it to a new example: Fell bundles over etale groupoids. Work in progress. Joint work with Erik Bedos and S. Kaliszewski.

Title: Diagonals in inductive limits of continuous trace algebras

Abstract: We discuss certain groupoid models of inductive limits of continuous trace

algebras which give rise diagonals and recall certain groupoid invariants. This work is inspired by the work of Li, Barlak and Raad on certain inductive limit groupoids as well as the fascinating examples of Villadsen, Rørdam, and Toms.

Title: Cohomology of ample groupoids, joint with M. Ionescu

Abstract: We recall the definitions of homology and cohomology for ´etale groupoids

G. We introduce a cochain complex for ample groupoids G using a resolution defining their homology with coefficients in Z. Using the equivalence between G-sheaves and G-modules, we show that the cohomology of this cochain complex coincides with previous versions of cohomology. We prove an exact sequence for the cohomology of skew products by a Z-valued cocycle using pullback modules. We compute the cohomology with coefficients in a G-module for some AF-groupoids and for some action groupoids. We introduce the relative cohomology for a subgroupoid and prove the existence of a long exact sequence (work in progress). We conjecture the existence of a UCT exact sequence relating the homology and cohomology of ample groupoids.


Lunch Break (About 2 hours)

Title: Intermediate Subalgebras of Cartan embeddings in rings and C*-algebras

Abstract: Let D ⊆A be a quasi-Cartan pair of algebras. Then there

exists a unique discrete groupoid twist Σ → G whose twisted Steinberg

algebra is isomorphic to A in a way that preserves D. In this paper, we

show there is a lattice isomorphism between wide open subgroupoids of

G and subalgebras C such that D ⊆C ⊆A and D ⊆C is a quasi-Cartan

pair. We also characterise which algebraic diagonal/algebraic Cartan/quasi-

Cartan pairs have the property that every subalgebra C with D ⊆C ⊆A

has D ⊆C a diagonal/Cartan/quasi-Cartan pair. In the diagonal case,

when the coefficient ring is a field, it is all of them. Beyond that, only

pairs that are close to being diagonal have this property. We then apply our

techniques to C*-algebraic inclusions and give a complete characterization

of which Cartan pairs D ⊆A have the property that every C*-subalgebra

C with D ⊆C ⊆A has D ⊆C a Cartan pair. This work is joint with C.

Orloff Clark and A. Fuller.

Title: Spectral continuity for étale groupoids with the Rapid decay property

Abstract: We show that the reduced groupoid C*-algebras of continuous fields of étale groupoids satisfying the rapid decay property yield continuous fields of C*-algebras. This establishes a new sufficient criterion that applies in the non-amenable case where the full and reduced groupoid algebras may differ. Potential applications include convergence of spectra in inverse systems of finite-index subgroups and magnetic models on hyperbolic lattices.

Title: The Burnside group/ring of a definable groupoid

 

Abstract: If G is a finite group, the Burnside ring of  G is the ring of equivalence classes of finite G-sets under disjoint union and cartesian product. For a G-manifold M, the equivariant Euler characteristic of the orbifold M/G is an orbifold invariant that takes values in the Burnside ring of G. The Burnside ring has been generalized to the case of compact Lie groups by Tom Dieck as well as finite groupoids by El Kaoutit and Spinosa.

 

We will present a generalization of the Burnside ring and equivariant Euler characteristic to the case of a topological groupoid that is definable in an o-minimal structure (e.g., the object spaces, structure maps, and orbit space are semialgebraic). In this case, we obtain an abelian group with a partially-defined multiplication, the Burnside group of the groupoid, which is in some cases a ring. We will describe the construction and structure of the Burnside group. Joint work with Carla Farsi and Emily Proctor.


Sunday November 16th

Title: Principal Actions on Topological Quivers 

Abstract: We classify the principal group actions on topological quivers and examine the consequences in the operator algebraic setting. We produce some distinguished classes of topological quivers and discuss their features in light of the classification above. 

Title: Amalgamated Free Products of twisted Étale Groupoid C∗-Algebras

Abstract: The amalgamated free product of discrete groups is a construction that generalizes the usual free product by identifying a common subgroup within the factors. Specifically, the resulting group is the universal object in the category of discrete groups that contains the underlying groups as subgroups, subject to the relations imposed by the common subgroup. In this talk, I will present a concrete construction of amalgamated free products of étale groupoids over a common unit space. The construction will be described explicitly. In the second part, we first discuss the structure of the amalgamated free products of C*-algebras, and then the groupoid C*-algebras. Finally, we will discuss how the twists over étale groupoids behave under amalgamation. If time permits, I will discuss KK-bifunctors for amalgamated free product of C*-algebras. Joint work with Alex Kumjian.


Title: Characterization of zero singular ideal in non-Hausdorff groupoid C*-algebras

Abstract: Non-Hausdorff etale groupoids arise naturally from interesting dynamical systems and as models of important classes of C*-algebras. One of the main obstacles in understanding the associated C*-algebras in terms of their groupoids is the existence of a possibly non-zero ideal consisting of functions supported on meagre sets which, for instance, obstructs characterizing simplicity in terms of the usual groupoid properties in the Hausdorff setting. In this talk, I discuss my result characterizing when this "singular" ideal is zero in terms of a groupoid property. This talk is based on my preprint https://arxiv.org/abs/2509.07262.