Glances@Manifolds II

Jagiellonian University in Kraków
August 8-13, 2016, Kraków, Poland

School on C*-algebras, K-theory, Index Theory, and Manifolds – Program

Lectures by
Jarosław Kędra (University of Aberdeen)

Problem sessions
Wojciech Politarczyk (Adam Mickiewicz University in Poznań)

Course title
On group actions on manifolds

The lectures will be concerned with actions of finitely presented groups on manifolds. The question whether a finitely presented group G can act effectively on a manifold M is, in general, very difficult to answer. We will discuss examples of such actions and some results providing restrictions.  We will show that finite index subgroups in SL(n,Z) for n>2 do not act effectively on the circle and that they don’t admit effective area preserving actions on surfaces of genus at least two.

  1. Definitions and first examples.
  2. Simplicity of diffeomorphism groups.
  3. Dynamics of homeomorphisms of the circle.
  4. Groups which do not act faithfully on the circle.

Bibliography

[1] Etienne Ghys. Groups acting on the circle. Enseign. Math. (2) 47 (2001), no. 3-4, 329-407. http://perso.ens-lyon.fr/ghys/articles/groupscircle.pdf
[2] Misha Gromov. Asymptotic invariants of infinite groups. Geometric group theory, Vol. 2 (Sussex, 1991), 1–295, London Math. Soc. Lecture Note Ser., 182, Cambridge Univ. Press, Cambridge, 1993.
[3] David Fisher. Groups acting on manifolds: around the Zimmer program. http://arxiv.org/abs/0809.4849


Lectures by
Vladimir Manuilov (Moscow State University)

Problem sessions
Anton Korchagin (Moscow State University)

Course title
Introduction to C*-Algebra Theory

  1. Commutative C*-algebras, spectrum, spectral radius, continuous functional calculus for normal elements.
  2. Positivity in C*-algebras, states.
  3. Ideals and quotients, GNS construction and Gelfand-Naimark theorem.
  4. Examples of C*-algebras
  5. Projections. K-theory for C*-algebras.

Bibliography

[1] K. Davidson. C*-Algebras by Example. Fields Institute Monographs, Vol. 6., AMS. 1996
[2] N. E. Wegge-Olsen. K-Theory and C*-Algebras: A Friendly Approach. Oxford Univ. Press. 1993

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Lectures by
Piotr Nowak (IM PAN and University of Warsaw)

Problem sessions
Marek Kaluba (IM PAN and Adam Mickiewicz University in Poznań)

Course title
Property (T) and Higher Index Theory

In these lectures we will describe the notion of Kazhdan’s property (T) and some of its generalizations to Banach spaces. We will also describe how such rigidity properties are used to construct K-theory classes that are of fundamental importance for the Baum-Connes conjecture.

  1. Property (T) – introduction and motivation
  2. Property (T) in Banach spaces: properties (TE) and FE
  3. Kazhdan projections in group Banach algebras and Lafforgue’s strong Banach property (T)
  4. K-theory classes induced by property (T) and counterexamples to the coarse Baum-Connes conjecture
  5. Warped cones and ghost projections

Bibliography

[1] Bekka, Bachir; de la Harpe, Pierre; Valette, Alain. Kazhdan’s property (T).  New Mathematical Monographs, 11. Cambridge University Press, Cambridge, 2008. xiv+472 pp. ISBN: 978-0-521-88720-5
[2] Piotr Nowak. Group Actions on Banach Spaces. In: Handbook of Group Actions, vol. II, 121–149. Edited by L. Ji, A. Papadopoulos, S.-T. Yau. Advanced Lectures in Mathematics (ALM) 32, International Press, Somerville; Higher Education Press, Beijing, 2015 http://arxiv.org/abs/1302.6609
[3] C. Drutu, P. Nowak. Kazhdan projections, random walks and ergodic theorems. http://arxiv.org/abs/1501.03473

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Lectures by
Jonathan Rosenberg (University of Maryland)

Problem sessions
Aleksandra Borówka (Jagiellonian University)
Michał Marcinkowski (Universität Regensburg & Wrocław Universit)

Course title
Metrics of positive scalar curvature

  1. Lecture 1:  Index theory and obstruction theory.
  2. Lecture 2:  The bordism method and existence results.
  3. Lecture 3:  The minimal surface method and special results in dimension 4.
  4. Lecture 4:  Spaces of positive scalar curvature metrics.

Bibliography

[1] Jonathan Rosenberg. Manifolds of positive scalar curvature: A progress report. http://www.math.umd.edu/~jmr/psc2006.pdf
[2] Thomas Schick. The Topology of Positive Scalar Curvaturehttp://arxiv.org/abs/1405.4220
[3] Stephan Stolz. Manifolds of Positive Scalar Curvature. Lectures given at the School on High-Dimensional Manifold Topology, Trieste, May 21 – June 8, 2001. http://www.iaea.org/inis/collection/NCLCollectionStore/_Public/38/063/38063755.pdf

 

Organizers: Adam Mickiewicz University | Chern Institute of Mathematics | Jagiellonian University | Lomonosov Moscow State University | University of Warsaw