NCG&T Prague

usually takes place each Tuesday at 16:00 Institute of Mathematics CAS, Blue lecture room, Zitna 25, Praha 1
Chair: Tristan Bice, Karen Stung

What is noncommutative geometry and topology? The idea stems from the Gelfand theorem which states that the category of compact Hausdorff spaces and commutative C*-algebras are dual. If we drop the condition of commutativity from our C*-algebras, we arrive at the notion of a noncommutative topological space. This can be carried further into the realm of noncommutative of geometry by equipping *-algebras with geometric structures.
Our research focusses on both quantum algebraic and operator algebraic aspects of noncommutative geometry and topology. This includes research in Hopf algebras, quantum groups, and noncommutative complex geometry, while on the operator algebra side, we study C*-algebras, with particular focus on C*-algebras arising from dynamical constructions such as minimal actions, groupoids, and semigroups.
Partially supported by GAČR project 20-17488Y Applications of C*-algebra classification: dynamics, geometry, and their quantum analogues and PRIMUS grant Spectral Noncommutative Geometry of Quantum Flag Manifolds.

Constructing compacta from relations between finite graphs

Adam Bartoš

Czech Academy of Sciences

Tuesday, 7. February 2023 - 16:00 to 17:00

This talk will take place in the blue seminar room, back building, Žitná 25.

It will also be broadcast on Zoom:
https://cesnet.zoom.us/j/91975183920?pwd=SGdlRUhLS21HUnVFTXBKcE1vYXlrQT09
Meeting ID: 919 7518 3920
Passcode: 102707

In this joint work in progress with Tristan Bice and Alessandro Vignati, we consider a construction turning every poset into a compact space. The intention is that the elements of the poset correspond to special basic open sets of the space and that the order relation corresponds to actual containment of the open sets. Moreover, we build our posets from sequences of finite graphs and special bonding relations between the graphs. The desired situation is when the graphs encode certain basic open covers and the edge-relation corresponds to actual overlapping of the induced basic open sets. Finally, we are interested in situations when the sequences of graphs are Fraïssé sequences in suitable categories of graphs. In the talk I shall give an overview of the construction, demonstrate it in several examples, and compare it to a more standard approach of taking quotients of inverse limits of graphs and homomorphisms.

On the classification of quantum lens spaces

Sophie Emma Zegers

Charles University

Tuesday, 14. February 2023 - 16:00 to 17:00

In the study of noncommutative geometry many classical spaces have been given a quantum analogue. An example is quantum lens spaces, which are defined as fixed point algebras of the quantum sphere by Vaksman and Soibelman under the actions of finite cyclic groups. Hong and Szymański described both quantum spheres and quantum lens spaces, whose weights are all coprime to the order of the acting group, as graph C*-algebras. Later on Brzeziński and Szymański generalised the result to include all quantum lens spaces. Unfortunately, as pointed out by Efren Ruiz, their description is not always correct.

In this talk, I will first present how quantum lens spaces can be described as graph C*-algebras. Then, I will apply the classification result of graph C*-algebras over finite graphs by Eilers, Restorff, Ruiz and Sørensen to obtain a number-theoretic invariant for quantum lens spaces of dimension 7, where all but a single weight are coprime to the order of the acting group.... more

A spectral gap for twisted Dolbeault–Dirac operators over irreducible quantum flag manifolds

Réamonn Ó Buachalla

Charles University

Tuesday, 21. February 2023 - 16:00 to 17:00

In this talk we present a geometric approach to understanding the analytic properties of (twisted) Dolbeault–Dirac operators D_{\overline{\partial}} over the irreducible quantum flag manifolds \mathcal{O}_q(G/L_S). We deduce from a noncommutative Akizuki–Nakano identity that twisting a Dolbeault–Dirac operator by a positive line bundle results in a spectral gap for these operators. We then conclude that each positively twisted D_{\overline{\partial}} is an (unbounded) Fredholm operator, and calculate its index using the recently established noncommutative Borel–Weil theorem. (Joint work with Biswarup Das and Petr Somberg

The Zappa–Szép product of a Fell bundle by a groupoid

Anna Duwenig

Katholieke Universiteit Leuven

Tuesday, 28. February 2023 - 16:00 to 17:00

The Zappa–Szép (ZS) product originated as a generalization of the semi-direct product of groups. Keeping in mind the relationship of such semi-direct products to crossed products of C*-algebras, we define an analogue of ZS products for operator algebras: if a groupoid H acts in a sufficiently nice way on a Fell bundle B over G, we construct a new Fell bundle over the ZS product of the groupoids G and H. For this new “ZS Fell bundle”, there is a natural relationship between its representations and those representations of B and H that are covariant in an appropriate sense. Furthermore, this ZS construction lends itself to generalizations of imprimitivity theorems à la Kaliszewski–Muhly–Quigg–Williams.
This talk is based on joint work with Boyu Li (University of Windsor)

Institute of Mathematics

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