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Mathematisches Forschungsinstitut Oberwolfach Noncommutative Mathematisches Forschungsinstitut Oberwolfach Report No. 29/2015 DOI: 10.4171/OWR/2015/29 Noncommutative Geometry Organised by Alain Connes, Paris Joachim Cuntz, M¨unster Stefaan Vaes, Leuven Guoliang Yu, College Station 14 June – 20 June 2015 Abstract. These reports contain an account of 2015’s meeting on noncom- mutative geometry. Noncommutative geometry has developed itself over the years to a completely new branch of mathematics shedding light on many other areas as number theory, differential geometry and operator algebras. A connection that was highlighted in particular in this meeting was the connec- tion with the theory of II1-factors and geometric group theory. Mathematics Subject Classification (2010): 58B34. Introduction by the Organisers Noncommutative geometry applies ideas from geometry to mathematical struc- tures determined by noncommuting variables. Within mathematics, it is a highly inter-disciplinary subject drawing ideas and methods from many areas of math- ematics and physics. Natural questions involving noncommuting variables arise in abundance in many parts of mathematics and theoretical quantum physics. On the basis of ideas and methods from algebraic and differential topology and Riemannian geometry, as well as from the theory of operator algebras and from homological algebra, an extensive machinery has been developed which permits the formulation and investigation of the geometric properties of noncommutative structures. This includes K-theory, cyclic homology and the theory of spectral triples. Areas of intense research in recent years are related to topics such as in- dex theory, quantum groups and Hopf algebras, the Novikov and Baum-Connes conjectures as well as to the study of specific questions in other fields such as number theory, modular forms, topological dynamical systems, renormalization 1632 Oberwolfach Report 29/2015 theory, theoretical high-energy physics and string theory. Many results elucidate important properties of specific classes of examples that arise in many applications. But the properties of many important classes of examples still remain mysterious, and are currently under intense investigation. This meeting concentrated on se- lected aspects of Noncommutative Geometry. Special emphasis this time was laid on connections to von Neumann algebras and to classification questions for group measure space II1 factors, as well as to geometric group theory and the study of embeddings of groups into Hilbert and Banach spaces. There are indications for a deep connection between recent progress in that direction and the role of factors, ergodic theory and quantum statistical mechanics in the approach to number the- ory and L-functions from noncommutative geometry. In addition quite a few other topics of current interest in Noncommutative Geometry were covered as well. Acknowledgement: The MFO and the workshop organizers would like to thank the National Science Foundation for supporting the participation of junior researchers in the workshop by the grant DMS-1049268, “US Junior Oberwolfach Fellows”. Noncommutative Geometry 1633 Workshop: Noncommutative Geometry Table of Contents Martijn Caspers (joint with Javier Parcet, Mathilde Perrin, Eric Ricard) Noncommutative De Leeuw theorems ...............................1635 Alain Connes (joint with Caterina Consani) The scaling Site .................................................1637 Guillermo Corti˜nas (joint with Joachim Cuntz, Ralf Meyer) Analytic cohomology in characteristic p> 0. ........................1638 Claire Debord (joint with Georges Skandalis) Groupoids and pseudodifferential calculus ...........................1640 Tim de Laat (joint with Uffe Haagerup, Søren Knudby) A complete characterization of connected Lie groups with the Approximation Property ..........................................1643 Mikael de la Salle (joint with Tim de Laat, Vincent Lafforgue) Approximation properties for group non-commutative Lp spaces .......1646 Simon Henry Toposes in Noncommutative geometry ..............................1647 Adrian Ioana (joint with R´emi Boutonnet and Alireza Salehi-Golsefidy) Local spectral gap in simple Lie groups .............................1650 David Kerr Quasidiagonality, unique ergodicty, and crossed products ..............1653 Matthias Lesch (joint with Henri Moscovici) The resolvent expansion for second order elliptic differential multipliers 1654 Hanfeng Li (joint with Bingbing Liang) Sofic mean length ...............................................1657 Xin Li Semigroup C*-algebras, Cartan subalgebras, and continuous orbit equivalence ................................................... .1660 Volodymyr Nekrashevych Hyperbolic groupoids and operator algebras ..........................1662 Sergey Neshveyev Categorical Poisson boundaries and applications .....................1664 Narutaka Ozawa (joint with Emmanuel Breuillard, Mehrdad Kalantar, Matthew Kennedy) The Furstenberg boundary and C∗-simplicity ........................1666 1634 Oberwolfach Report 29/2015 Ulrich Pennig (joint with Marius Dadarlat) A Dixmier-Douady Theory for strongly self-absorbing C∗-algebras .....1668 Denis Perrot Noncommutative residues and higher indices ........................1670 Sven Raum Locally compact C∗-simple groups .................................1671 Takuya Takeishi Irreducible Representations of Bost-Connes systems ..................1672 Georg Tamme (joint with Moritz Kerz, Shuji Saito) Topological K-theory for non-archimedean algebras and spaces .........1674 Gisela Tartaglia (joint with Guillermo Corti˜nas) Isomorphism conjectures in algebraic and topological K-theory ........1677 Andreas Thom Non-commutative real algebraic geometry ...........................1679 Jeffrey Tolliver A link between Krasner’s valued hyperfields and Deligne’s triples ......1680 Stefaan Vaes (joint with Sorin Popa and Dimitri Shlyakhtenko) Representation theory and (co)homology for subfactors, λ-lattices and C∗-tensor categories. ............................................1682 Rufus Willett (joint with Erik Guentner, Guoliang Yu) Dynamic Asymptotic dimension ...................................1685 Tang, Xiang (joint with Ronald G. Douglas and Guoliang Yu) An Analytic Grothendieck Riemann Roch Theorem ..................1687 Zhizhang Xie (joint with Nigel Higson) Higher Signatures of Witt spaces ..................................1689 Guoliang Yu (joint with Shmuel Weinberger) Non-rigidity of manifolds and K-theory of group C∗-algebras ..........1692.
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