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.

Jet Functors in Noncommutative Geometry

Mauro Mantegazza
Charles University
Tuesday, 8. November 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

We construct an infinite family of endofunctors $J^n_d$ on the category of left $A$-modules, where $A$ is a unital associative algebra over a commutative ring ... more

Quantum Jet Bundles

Francisco Manuel Castela Simão
Queen Mary University of London
Tuesday, 15. November 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
We formulate a notion of jet bundles over a possibly noncommutative algebra A equipped with a torsion free connection. Among the conditions needed for 3rd-order jets and above is that the connection also be flat and its ‘generalised braiding tensor’ $\sigma\colon \Omega^1 \otimes_A \Omega^1_A \to \Omega^1 \otimes_A \Omega^1_A$ obey the Yang-Baxter equation or braid relations. We then build the jet bundle on the graded vector space of symmetric tensors as a braided... more

Tracially complete C*-algebras: Tracial Transfer and Structure

Andrea Vaccaro
University of Münster
Tuesday, 22. November 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
Tracially complete C*-algebras have been playing an increasingly central role in the latest developments on the classification of stably finite C*-algebras. In this talk I will give a quick introduction to these structures, which broadly speaking resemble bundles over closed convex sets whose fibers are von Neumann algebras. I will then proceed to show how, under suitable conditions, this intuition translates into a powerful transfer phenomenon, which allows to show that numerous properties holding for von Neumann algebras are also true for tracially complete C*-algebras. I will conclude analyzing certain structural aspects of these objects, focusing in particular on how certain key properties, such as being factorial or being a trivial W*-bundle, are not preserved when going to ultraproducts.

Developments in Noncommutative Fractal Geometry

Therese Basa Landry
Fields Institute
Tuesday, 29. November 2022 - 16:00 to 17:00
This talk will be broadcast on Zoom:
https://cesnet.zoom.us/j/91975183920?pwd=SGdlRUhLS21HUnVFTXBKcE1vYXlrQT09
Meeting ID: 919 7518 3920
Passcode: 102707
As a noncommutative fractal geometer, I look for new expressions of the geometry of a fractal through the lens of noncommutative geometry.  At the quantum scale, the wave function of a particle, but not its path in space, can be studied.  Riemannian methods often rely on smooth paths to encode the geometry of a space. Noncommutative geometry generalizes analysis on manifolds by replacing this requirement with operator algebraic data.  These same “point-free” techniques can also be used to study the geometry of spaces like fractals.  Michel Lapidus, Frédéric Latrémolière, and I identified conditions under which differential structures defined on fractal curves can be realized as a metric limit of differential structures on their approximating finite graphs.  Currently, I am using some of the same tools from that project to understand noncommutative discrete structures with Manuel Reyes.  Progress in noncommutative geometry has produced a rich dictionary of... more