slideshow 3

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.

Free actions of groups on separated graphs and their C*-algebras
Alcides Buss
Federal University of Santa Catarina
Tuesday, 24. May 2022 - 16:00 to 17:00
This talk will take place on Zoom:
https://cesnet.zoom.us/j/91975183920?pwd=SGdlRUhLS21HUnVFTXBKcE1vYXlrQT09
Meeting ID: 919 7518 3920
Passcode: 102707
In this talk we are consider free actions of a group on a separated graph and explain how this structure reflects on the level of their associated C*-algebras. 
We first give a structure theorem for free actions on separated graphs, proving that all those separated graphs are skew products of the group with a certain labeling function to another separated graph, the orbit separated graph. We then explain how to recover the C*-algebra of such separated graphs as a crossed product by a coaction associated to the labeling function. Moreover, we describe the spectral decomposition of those coactions providing in this way a Fell bundle structure for their C*-algebras. All this works for full and reduced C*-algebras of separated graphs.

Differentiable positive maps, vector fields, divergences, exponentials, parallel transport, quantum mechanics and state spaces
Edwin Beggs
Swansea University
Tuesday, 31. May 2022 - 16:00 to 17:00
This talk will take place on Zoom:
https://cesnet.zoom.us/j/91975183920?pwd=SGdlRUhLS21HUnVFTXBKcE1vYXlrQT09
Meeting ID: 919 7518 3920
Passcode: 102707
The talk discusses a mixture of topics in noncommutative differential geometry which are all related by the idea of connection on bimodules. We see how exponentiation of invariant vector fields on the integers is almost, but not quite, the same as diffusion. We consider the exponential map on quantum SU_2. We see how the universal calculus on matrices is related to the usual calculus on complex projective space. We look at noncommutative fluid mechanics (well a little bit)… And we ask whether there really is a probabilistic interpretation of the Klein Gordon equation.

Joint work with Shahn Majid and Ghaliah Alhamzi

Genericity in the classifiable category
Bhishan Jacelon
Institute of Mathematics of the Czech Academy of Sciences
Tuesday, 27. September 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:
https://cesnet.zoom.us/j/91975183920?pwd=SGdlRUhLS21HUnVFTXBKcE1vYXlrQT09
Meeting ID: 919 7518 3920
Passcode: 102707
Using Fraïssé theory, Markov chains and results from topological dynamics, I will provide some answers to the question, “What do generic objects and morphisms in the category of simple, separable, nuclear, \mathcal{Z}-stable C*-algebras satisfying the UCT look like?”

Realising quantum flag manifolds as graph C*-algebras
Karen Strung
Institute of Mathematics of the Czech Academy of Sciences
Tuesday, 4. October 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:
https://cesnet.zoom.us/j/91975183920?pwd=SGdlRUhLS21HUnVFTXBKcE1vYXlrQT09
Meeting ID: 919 7518 3920
Passcode: 102707
In this talk we show how the C*-completions of the Drinfeld–Jimbo quantum flag manifolds can be realised as graph C*-algebras. We begin by recalling how to construct a C*-algebra from a directed graph, how to read the K-theory groups of the C*-algebra directly from the graph, and how to see its ideal structure. We then briefly recall the construction of a quantum flag manifold, and how to compute the primitive ideal space by using Dijkhuizen and Stokmann’s description of a complete set of irreducible *-representations. Finally, we show how to construct a graph directly from the Weyl group of the associated Lie algebra, and show that we recover some known isomorphisms between the C*-algebras of quantum flag manifolds, as well as determining surprising new ones.

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