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.

Soft elements in a C*-algebra

Eduard Vilalta

Universitat Autonoma de Barcelona

Tuesday, 7. March 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

The family of elements whose Cuntz class is soft plays an important role in the study of (sufficiently noncommutative) C*-algebras.

I will begin the talk by recalling what these elements are and some of the important settings where they have appeared. Using the definition of Cu-softness as an inspiration, I will define softness for positive elements in a C*-algebra and discuss their relation with their Cu-counterpart.

Having an abundance of these positive elements (in an adequate sense) characterizes the Global Glimm Property, and this can in turn be used to show that a number of C*-algebraic invariants can be computed by only looking at the soft part of a C*-algebra.

The talk is based on joint work with H. Thiel and, if time allows, the last part will be on joint work with A. Asadi-Vasfi and H. Thiel.

Homogeneous ultrametric structures

Wiesław Kubiś

Czech Academy of Sciences

Tuesday, 11. April 2023 - 16:00 to 17:00

We shall present a new method of building homogeneous Polish ultrametric structures using a generic homomorphism. The method is purely category-theoretic and applies to many classical classes of first order structures (e.g. graphs, ordered sets) as well as to some topological-algebraic structures (e.g. Banach spaces).

Joint work with Maja & Christian Pech.

A brief introduction to canonical extensions with some applications

Tomáš Jakl

Czech Academy of Sciences

Tuesday, 2. May 2023 - 16:00 to 17:00

Canonical extensions are an algebraic approach to extensions of topo-algebraic dualities to structures with additional operations. It originates in the work of Jónsson and Tarski who described a representation of topological duals of Boolean algebras with modal operations. The theory of canonical extensions was later generalised to distributive lattices, posets, proximity lattices, coherent categories and more.

In this talk I will overview the basics of the theory of canonical extensions for distributive lattices or, equivalently, I will give an algebraic presentation of Priestley duality. Time permitting, I will also show some applications of canonical extensions to the theory of structural limits.

Institute of Mathematics

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