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23-27 juin 2025 Paris (France)
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MinicoursesCamille Horbez : Measure equivalence rigidity among mapping class groupsMeasure equivalence is a central concept in measured group theory. It was introduced by Gromov as a measure-theoretic analogue of the geometric notion of quasi-isometry between groups, and it is also tightly related to the study of the orbit structure of probability measure-preserving group actions. We will study this notion in the context of groups that have a geometric origin, namely groups of homeomorphisms of manifolds, considered up to homotopy (a.k.a. mapping class groups). In dimension 2, a theorem of Kida asserts that the mapping class group Mod(S) of a closed, connected, oriented surface S of genus at least 2 is rigid in measure equivalence: any countable group that is measure equivalent to Mod(S) must be (virtually) isomorphic to Mod(S). Kida's proof yields a measure-theoretic extension of a theorem of Ivanov that computes the automorphism group of Mod(S). In this minicourse, I will introduce measure equivalence and mapping class groups, explain Kida's theorem and its proof, and present some extensions to 3-dimensional manifolds obtained in joint works with Vincent Guirardel and Sebastian Hensel. References A. Furman, "A survey of measured group theory", Chicago Lectures in Math. (2011), 296-374, University of Chicago Press, Chicago, IL. Xin Li : Topological groupoids and their invariantsTopological groupoids describe orbit structures of dynamical systems by capturing their local symmetries, and build a bridge between topological dynamics and C*-algebra theory. The group of global symmetries, which are pieced together from local ones, is called the topological full group. This construction gives rise to new examples of groups with very interesting properties, solving outstanding open problems in group theory. This series is about a new connection, in the context of homological invariants, between groupoids and topological full groups on the one hand and algebraic K-theory spectra and infinite loop spaces on the other hand. Several applications will be discussed. Parts of this connection already feature in work of Szymik and Wahl on the homology of Higman-Thompson groups. My plan is to first give an introduction to groupoids and their topological full groups, and then focus on homological invariants. References M. Crainic and I. Moerdijk, "A homology theory for étale groupoids", J. Reine Angew. Math. 521 (2000), 25–46. Clara Löh : The dynamical view on simplicial volumeThe simplicial volume is a homotopy invariant of closed manifolds. Simplicial volume is zero in the presence of enough amenability and it is non-zero in the presence of enough negative curvature. In this mini-course, we will introduce the basics on simplicial volume. We will then focus on a dynamical version of simplicial volume and its relation with other invariants from measured group theory (such as L2-Betti numbers and cost). References S. Braun and R. Sauer, Volume and macroscopic scalar curvature, Geom. Funct. Anal., 31(6), 1321--1376, 2021. M. Gromov, Volume and bounded cohomology, Publ. Math. IHES, 56, 5--99, 1982. C. Löh, Ergodic theoretic methods in group homology, a minicourse on L2-Betti numbers in group theory, SpringerBriefs in Mathematics, Springer, 2020. C. Löh, M. Moraschini, R. Sauer, Amenable covers and integral foliated simplicial volume, New York J. Math., 28, 1112--1136, 2022. M. Schmidt, L2-Betti Numbers of R-spaces and the Integral Foliated Simplicial Volume. PhD thesis, Westfälische Wilhelms-Universität Münster, 2005, available here. Paolo Piazza : K-Theoretic and homological invariants of Dirac operatorsIn this 4 lectures we will explore the mathematics that enters into defining and studying the index class of a Dirac operator on a Galois Γ-covering, for example the universal covering of smooth compact manifold, and, if such operator is invertible, his rho class. The index class should be thought of as a primary invariant whereas the rho class is a secondary invariant, defined precisely when the index class vanishes (for example, when the operator is invertible). The program is the following. I will first introduce the basics of K-theory and then move to coarse index theory and to the Higson-Roe analytic surgery sequence in K-theory. I will state a fundamental result, the Atiyah-Patodi-Singer delocalized index theorem in K-theory, and explain how it can be used in geometric questions, for example in order to map the Stolz' surgery sequence for positive scalar curvature metrics to the Higson-Roe surgery sequence. I will then give alternative treatments of the Higson-Roe surgery sequence, most notably the one due to Zenobi; this alternative treatment employs groupoid techniques. I will then tackle a fundamental question: how can we extract from these K-theory classes numeric invariants ? In order to answer this question I will talk first about cyclic (co)homology and then move to the definition of higher indices and higher rho invariants. This last part will be further explained by Vito Felice Zenobi in his talk. References P. Piazza and T. Schick, Rho-classes, index theory and Stolz’ positive scalar curvature sequence, J. Topol. 7 (2014), no. 4, 965–1004 P. Piazza, T. Schick, and V. Zenobi, Mapping analytic surgery to homology, higher rho numbers and metrics of positive scalar curvature, Mem. Amer. Math. Soc. 309 (2025), no. 1562, iv+ 151 N. Wegge-Olsen, K-Theory and C*-algebras, Oxford University Press, 1993 V. F. Zenobi, Adiabatic groupoid and secondary invariants in K-theory, Adv. Math. 347 (2019), 940–1001 |
