Mini Courses
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David Kerr (University of Münster)
Title: Z-stability, amenability, and measure-preserving dynamics
Abstract:
The main theme of this lecture series will be the interplay between topology and measure that has characterized much of the joint development between dynamics and $C^*$-algebras over the last decade.
I will discuss the origins of the theory in the Rokhlin lemma and highlight both the similarities and differences in the way that the analysis has played out in ergodic theory and topological dynamics. In particular I will explain the ways in which the transfer of structure from dynamics to operator algebras undergoes a decisive technical shift when passing from von Neumann algebras to C*-algebras, even though the basic picture remains conceptually united under the common umbrella of orbit equivalence.
Reference: D. Kerr. Dimension, comparison, and almost finiteness. J. Eur. Math. Soc. 22 (2020), 3697-3745. -
Karen Strung (IMCAS in Prague)
Title: The classification program for C*-algebras with examples from dynamics
Abstract:
The Elliott classification programme asks to what extent a $C^*$-algebra can be recovered, up to isomorphism, from K-theoretic and tracial data. In these lectures, I will introduce the origins and current form of the programme, beginning with the classification of AF algebras and proceeding to the Elliott invariant and the modern classification theorem. Along the way, we will discuss the roles played by nuclearity, nuclear dimension, $\mathcal{Z}$-stability and the Universal Coefficient Theorem.
We will then consider $C^*$-algebras arising from topological dynamics. Transformation groupoids and crossed products provide a bridge from dynamical systems to operator algebras, allowing properties of the resulting $C^*$-algebras to be studied in terms of the underlying dynamics. We will focus particularly on minimal homeomorphisms and their orbit-breaking subalgebras, examining how their K-theory and traces enter the classification picture. Time permitting, we will also look at systems generated by surjective local homeomorphisms, the Deaconu—Renault groupoid, and the associated $C^*$-algebras.
Finally, we will turn to hyperbolic dynamical systems, in the form of Smale spaces, and the stable, unstable and homoclinic $C^*$-algebras associated to them. These examples illustrate how structural and classification results can be applied beyond ordinary crossed products. The emphasis throughout will be on the main ideas, examples and consequences of classification rather than on the technical details of its proofs.Background and suggested reading:
Some basics of C*-algebras would be useful, for example:- the first 2-3 chapters of Murphy’s book C*-Algebras and Operator Theory,
- Davidson’s C*-algebras by Example,
- or my An Introduction to C*-Algebras and the Classification Program (there’s a draft of it floating around somewhere on the internet in case people don’t have a copy).
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Fernando Lledó (University Carlos III of Madrid)
Title: The emergence of Operator Algebras in theory of Quantum Spin Systems
Abstract:
The theory of operator algebras was developed by J. von Neumann in the 1930s in close relationship with fundamental questions in various branches of mathematics and, also, with the emerging new theory of quantum mechanics. In this series of lectures I will present how $C^*$-algebras enter into the description of infinite systems of spins on discrete structures. I will begin with a brief introduction to the basic axioms of quantum theory (in finite dimension) and the structure of spin systems on (countable) infinite discrete sets. I will then focus on Kitaev’s quantum double model, an important prototype in this theory, and address the question of construction and classification of superselection sectors (anyonic excitations) and ground states. Time permitting I will comment on recent results in this theory in the presence of boundaries (joint work with Joan Claramunt (UPC) and Laura Sáenz (UC3M-ICMAT)).
References:- Ola Bratteli and Derek W. Robinson, Operator Algebras and Quantum Statistical Mechanics, Vols. 1 and 2 2nd ed., Springer, New York (1987 and 1997)
- Bruno Nachtergaele and Robert Sims, An Introduction to Quantum Spin Systems (John von Neumann Guest Lectures at TU-München), 2016; available at https://www.math.ucdavis.edu/bxn/LectureNotes
- Pieter Naaijkens, Quantum Spin Systems on Infinite Lattices. A Concise Introduction, Springer, 2017.
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Sanaz Pooya (University of Potsdam)
Title: Higher Kazhdan projections, $K$-theory, and delocalised $\ell^2$-Betti numbers
Abstract:Kazhdan’s property (T) admits several characterisations. One of them, arising from operator algebras, is the existence of a distinguished projection known as the Kazhdan projection. This construction has a natural higher-degree analogue, leading to the notion of higher Kazhdan projections. These projections are constructed from the reduced cohomology of groups with coefficients in unitary representations. In the presence of a suitable spectral gap, they belong to matrix algebras over the reduced group $C^*$-algebra and hence define $K$-theory classes.
In these lectures, I will introduce higher Kazhdan projections, beginning with the classical picture and explaining the role of higher group cohomology in their construction. For virtually free groups, I will present two different approaches to computing the $K$-theory classes of these projections.
I will then turn to two applications of this construction. The first lies in coarse geometry, where higher Kazhdan projections reveal a connection between the coarse Baum--Connes conjecture and Lück’s approximation theorem for $\ell^2$-Betti numbers. The second concerns delocalised $\ell^2$-Betti numbers, where explicit descriptions of the $K$-theory classes of higher Kazhdan projections lead to concrete calculations as well as new vanishing and non-vanishing results.
Background and suggested reading:The minicourse will be largely self-contained. The following references provide background on some of the topics appearing in the lectures and may be useful for preparation or further reading.
- B. Bekka, P. de la Harpe, and A. Valette, Kazhdan’s Property (T), New Mathematical Monographs, vol. 11, Cambridge University Press, 2008.
- M. Rørdam, F. Larsen, and N. J. Laustsen, An Introduction to $K$-Theory for $C^∗$-Algebras, London Mathematical Society Student Texts, vol. 49, Cambridge University Press, 2000.
- K. S. Brown, Cohomology of Groups, Graduate Texts in Mathematics, vol. 87, Springer, 1982.
- M. P. Gómez Aparicio, P. Julg, and A. Valette, The Baum–Connes Conjecture: An Extended Survey, in Advances in Noncommutative Geometry, Springer, 2019, pp. 127–244.
References:
- K. Li, P. W. Nowak, and S. Pooya, Higher Kazhdan projections, $\ell^2$-Betti numbers and Baum–Connes conjectures, Journal of Noncommutative Geometry 18 (2024), 313–336.
- S. Pooya and H. Wang, Higher Kazhdan projections and delocalised $\ell^2$-Betti numbers, Annals of K-Theory 11 (2026), no. 3, 395–418.
- S. Pooya, B. Ren, and H. Wang, Euler characteristics, higher Kazhdan projections and delocalised $\ell^2$-Betti numbers, to appear in Transactions of the American Mathematical Society; arXiv:2507.20119.