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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 uniqueness theorem for twisted groupoid C*-algebras

Becky Armstrong
WWU Münster
Tuesday, 23. November 2021 - 16:00 to 17:00
Blue seminar room, back building, Žitna 25 and on Zoom:
Meeting ID: 919 7518 3920
Passcode: 102707
Twisted groupoid C*-algebras were introduced by Renault in 1980 and are a generalisation of twisted group C*-algebras, which are the C*-algebraic analogue of twisted group rings. Through the work of Renault and more recently of Li, it has emerged that every simple classifiable C*-algebra can be realised as a twisted groupoid C*-algebra, a result that has led to increased interest in the structure of these C*-algebras. In this talk I will describe the construction of reduced twisted C*-algebras of Hausdorff étale groupoids. I will then discuss my recent preprint in which I prove a uniqueness theorem for these algebras and use this to characterise simplicity in the case where the groupoid is effective.

Quantum Gromov-Hausdorff continuity of quantum SU(2)

Jens Kaad
Syddansk Universitet, Odense
Tuesday, 30. November 2021 - 16:00 to 17:00
Blue seminar room, back building, Žitna 25 and on Zoom -
Meeting ID: 919 7518 3920
Passcode: 102707
In this talk we investigate the spectral metric properties of quantum SU(2). These spectral metric properties are encoded by a particular twisted derivation coming from the pairing between the quantum enveloping algebra and quantum SU(2). We shall see that this twisted derivation provides quantum SU(2) with the structure of a compact quantum metric space. Moreover, these compact quantum metric spaces vary continuously in the q-deformation parameter with respect to the quantum Gromov-Hausdorff distance. This includes the special value q = 1, where we recover the classical 3-sphere equipped with the round metric. The talk is based on joint work with David Kyed.

A geometry for the space of metric spectral triples

Frédéric Latrémolière
University of Denver
Tuesday, 7. December 2021 - 16:00 to 17:00
Zoom -
Meeting ID: 919 7518 3920
Passcode: 102707
Spectral triples, introduced by A. Connes in 1985, have emerged as the preferred generalization of Riemannian geometry to the noncommutative realm, and provide a new formalism for the study of problems form quantum physics to fractal geometry. Many problems in physics and geometry suggests that a quantification on “how close” spectral triples can be from each others would prove very helpful; however, only heuristics can be found in the literature. Examples of problems where such a notion could prove useful include the convergence of matrix models in mathematical physics, or the geometry of fractal sets constructed, in a natural manner, as limits of simpler spaces.

In this talk, we will survey our construction of an actual metric, up to unitary equivalence, on a large class of spectral triples, which indeed provides a framework to discuss such problems . Our construction starts with our work in noncommutative metric geometry, spurred by the work of Rieffel, and our... more

The Cubic Dirac Operator for U_q(sl_2)

Andrey Krutov
Institute of Mathematics of the Czech Academy of Sciences
Tuesday, 25. January 2022 - 16:00 to 17:00
This talk will take place in the blue seminar room, back building, Žitna 25 and on Zoom:
Meeting ID: 919 7518 3920
Passcode: 102707

The cubic Dirac operator for a complex semisimple Lie algebra \mathfrak{g} was introduced (in the algebraic setting) by Kostant in 1999. It has numerous applications in representation theory.  In 2000, Alekseev and Meinrenken used it to study equivariant cohomology of G-manifolds.  In this talk we will present a ... more