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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.

A Borel-Weil theorem for the irreducible quantum flag manifolds

Alessandro Carotenuto
Charles University
Tuesday, 1. March 2022 - 16:00
This talk will take place in the blue seminar room, back building, Žitná 25, and on Zoom.
The classical Borel–Weil theorem is a foundational result in geometric representation theory which realises each irreducible representation of a complex semisimple Lie algebra g as the space of holomorphic sections of a line bundle over a flag manifold. I will give a noncommutative generalisation of the Borel–Weil theorem for the Heckenberger–Kolb calculi of the irreducible quantum flag manifolds. The proof is formulated in terms of quantum principal bundles, and the recently introduced notion of a principal pair, and uses the Heckenberger and Kolb first-order differential calculus for the quantum Possion homogeneous spaces O_q(G/L^s_S).

This talk is based on a joint work with Díaz García and Ó Buachalla

Synchronizing Dynamical Systems

Andrew Stocker
University of Colorado Boulder
Tuesday, 8. March 2022 - 16:00
This talk will take place on Zoom:

Meeting ID: 919 7518 3920
Passcode: 102707
Expansive systems are a class of topological dynamical systems which exhibit sensitivity to initial conditions.  Following work by Klaus Thomsen, we will construct a C*-algebra from several different equivalence relations on the points in a given expansive system.  This generalizes the C*-algebraic construction made by Ian Putnam for Smale Spaces, which are themselves an important special case of expansive systems.  We will then present some results obtained for a type of expansive system called synchronizing systems, and we will compute K-theoretic invariants for some specific examples.  No knowledge of dynamical systems will be assumed for this talk.

On differentiable Cuntz–Pimsner constructions for Hermitian line bimodules with connection

Branimir Ćaćić
University of New Brunswick
Tuesday, 15. March 2022 - 16:00 to 17:00
This talk will take place on Zoom:
Meeting ID: 919 7518 3920
Passcode: 102707
The Cuntz–Pimsner construction is ubiquitous and indispensable in current work on C^\ast-algebras and their noncommutative algebraic topology. Surprisingly (or not), its well-known application to topological quantum circle bundles admits the following formal distillate: the groupoid of all homomorphisms from \mathbb{Z} to a coherent 2-group G canonically deformation retracts onto G itself. Now, apply this proposition to the coherent 2-group of Hermitian line modules with bimodule connection on a unital pre-C^\ast-algebra with \ast-differential calculus. I hope to show that this one application permits the wholesale adaptation of the Cuntz–Pimsner construction with its structural implications to the noncommutative differential and Riemannian geometry of differentiable quantum circle bundles. If time permits, I’ll also sketch how these considerations imply the impossibility of coherently representing the q-monopole of Brzeziński–Majid in terms of spectral triples.

Essential groupoid C*-algebras of non-Hausdorff groupoids

Jonathan Taylor
University of Göttingen
Tuesday, 22. March 2022 - 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:
Meeting ID: 919 7518 3920
Passcode: 102707

Many interesting groupoids arise from common constructions from dynamical systems, and these constructions do not always guarentee the result groupoid is Hausdorff, but rather non-Hausdorff ‘in a controlled way’. The typical construction of the reduced groupoid C*-algebra then no longer gives the ‘smallest’ groupoid C*-algebraic construction associated to such a groupoid, leading us to define a smaller one. We will explore the definition and some properties of the essential groupoid C*-algebra, as well as some conditions for when an open subgroupoid gives the same groupoid C*-algebra