Algebraic geometry is the field of mathematics which concerns the study of spaces cut out by polynomial equations. This workshop concern the interaction of two important objects in algebraic geometry - the K-groups and the Brauer group. In algebraic geometry, we study spaces via invariants - the procedure of attaching simpler, more "linear" objects to these spaces in the hope of extracting information about them. Both the K-groups and Brauer group are such examples which have had an excellent track record in being both powerful and accessible at the same time. Roughly speaking, the K-groups are built out from \emph{vector bundles} on such a space - a continuous assignment of vector spaces on each point of the space. On the other hand the Brauer groups are built from "twisted" vector bundles.

Both invariants have had a history of interaction and cross-pollination and the goal of the workshop is to bring together researchers in both areas to share their research and pave the way for even more fruitful interaction in the future. We are particularly excited about the prospect of new, field-driving questions to come out from this workshop.

The IAS/Park City Mathematics Institute is a three-week residential summer session with a graduate summer school, a research program, an undergraduate summer school, and an undergraduate faculty program. More information can be found on the conference website. PCMI encourages applications from all those with interest in the program, both from the US and internationally. This year it is organized by Benjamin Antieau (Northwestern University), Marc Levine (Universität Duisburg-Essen), Oliver Röndigs (Universität Osnabrück), Alexander Vishik (University of Nottingham), and Kirsten Wickelgren (Duke University).

The Young Topologists Meeting is an annual international conference aimed at early-career researchers in topology - both pure and applied - covering the whole breadth of the subject. It serves as a platform for graduate, PhD students, and early postdocs to present their research, exchange ideas, and build international connections.

Previous editions of the conference have been organized by the EPFL, Switzerland, the University of Copenhagen, Denmark, and jointly by the University of Stockholm and the Royal Institute of Technology, Sweden. Next up: Münster, Germany.

The research school is an introductory meeting to the thematic program “Representation Theory and Noncommutative Geometry” to be held at IHP from January to March 2025. The trimester is part of an ongoing effort to bridge two fiels of Mathematics : the representation theory of locally compact groups and the theory of operator algebras. These research domains share origins in harmonic analysis, spectral theory and quantum mechanics but grew in separate directions. Recent progress in representation theory, involving especially non-Riemannian symmetric spaces and spherical varieties, and new tools developed in operator algebras, especially those involving K-theory and the other methods of non-commutative geometry, offer exciting prospects for new work at the interface between the two fields.

Mini-course are:

• Erik P. van den Ban (Universiteit Utrecht): Harmonic analysis of non-Riemannian symmetric spaces
• Tyrone Crisp (University of Maine): Tempered representations from the point of view of Langlands, and from the point of view of operator algebras
• Omar Mohsen (Université de Paris-Saclay): Introduction to hypoelliptic operators and their index theory
• Hang Wang (East China Normal University): Groups C*-algebras and their K-theory

The classification of tempered irreducible representations for real reductive groups was completed in the 1970s by Knapp and Zuckerman, following Harish-Chandra's work on the Plancherel formula. But some aspects of the subject are now undergoing a re-examination, following the discovery of new perspectives. C*-algebras and K-theory are valuable tools in Representation Theory, as shown, for instance, by the Mackey bijection. Indeed, it was the Connes-Kasparov isomorphism in K-theory that motivated the search for a natural bijection between the tempered dual of a real reductive group and the unitary dual of its Cartan motion group, as initially suggested by Mackey in the 1970s.

The meeting will focus on recent developments in which K-theoretic ideas have offered new perspectives on the tempered dual for reductive groups or symmetric spaces, and conversely on new approaches to operator-algebraic problems using contemporary tools in representation theory.

Topics will include:

• New approaches to the Mackey bijection through pseudodifferential operator theory, which has itself undergone an extensive conceptual redesign in the past decade, thanks again to C*-algebra and K-theory connections;
• New perspectives on the the Connes-Kasparov isomorphism using Dirac cohomology and cohomological induction;
• Higher orbital intergrals, which make it possible to go beyond the "noncommutative topology of the tempered dual'', hinting at something like the "differential geometry'' of this noncommutative space;
• Study of the Casselman-Schwartz algebras and their K-theory via Paley-Wiener theorems, and connections with the Connes-Kasparov isomorphism;
• C*-algebraic analysis of the tempered dual from the point of view of G as a symmetric space for GxG, and more generally of the tempered spectrum of symmetric spaces.

Harmonic analysis on homogeneous spaces is a fundamental area of research that simultaneously generalizes classical harmonic analysis on groups and on Riemannian symmetric spaces. It naturally relates to many areas of mathematics, playing a central role in representation theory and the theory of automorphic forms.

This workshop will be an occasion to introduce recent developments in some of these areas. It will also aim to explore new connections between them and extend the fruitful interactions between C*-algebras, harmonic analysis and representation theory beyond the classical setting of groups to the general setting of homogeneous spaces.

Topics will include:

• C*-algebraic approaches to the tempered dual of non-Riemannian symmetric spaces;
• Harmonic analysis and Plancherel theory for spherical spaces;
• Connections with the Langlands program and periods of automorphic forms;
• Recent approaches to the theta correspondence via C*-algebras