Prof. Dr. Wolfgang Lück

Director of HIM

E-mail: wolfgang.lueck(at)him.uni-bonn.de
Phone: +49 228 73 62240
Fax: +49 228 73 4406
Homepage: http://www.him.uni-bonn.de/lueck/
Room: 3.016
Location: Mathematics Center
Institute: Mathematical Institute
Hausdorff Research Institute (for Mathematics)
Research Areas: Research Area F* (Leader)
Research Area A
Research Area C
Date of birth: 19.Feb 1957
Mathscinet-Number: 116630

Publication List

Academic Career

1981

Diploma

1982 - 1985

Research Assistant, Mathematical Institute, University of Göttingen

1984

Dr. rer. nat., University of Göttingen

1985 - 1989

Assistant Professor, Mathematical Institute, University of Göttingen

1989

Habilitation, Department of Mathematics , University of Göttingen

1989 - 1990

Assistant Professor (C2-Oberassistent), Department of Mathematics, University of Göttingen

1990 - 1991

Associate Professor (with tenure), University of Kentucky, Lexington, KY, USA

1991 - 1996

Professor (C3), University of Mainz

1996 - 2010

Professor (C4/W3), University of Münster

Since 2010

Professor (W3), University of Bonn

Since 2011

Director, Hausdorff Research Institute for Mathematics (HIM), Bonn

Research Profile

One of my main project has been and will be the Farrell-Jones Conjecture for algebraic K-theory and L-theory. I made substantial contributions to its proof for hyperbolic, CAT(0), S-arithmetic groups, lattices in almost connected Lie groups, and fundamental groups of manifolds of dimension less or equal to three and to the proof of inheritance properties like passage to subgroups or directed colimits. The importance of this conjecture is illustrated by the facts that it implies other prominent conjectures such as the ones due to Bass, Borel and Novikov and has many significant applications to problems in topology, geometry, and group theory. We have also used the cycloctomic trace for topological cyclic homology to compute algebraic K-groups of integral group rings. The second key topic of my research are L^2-invariants. These are invariants defined in terms of the heat kernel or the simplicial chain complex of the universal covering of a closed Riemannian manifold using the theory of von Neumann algebras. They generalize classical invariants such as Betti numbers and Reidemeister torsion. I have analyzed questions about approximating L^2-invariants by their finite-dimensional analogues and given applications to group theory. A current and ongoing project is to link invariants from low-dimensional topology such as the Thurston norm and the Thurston polytope to generalized L^2-torsion invariants.

The Farrell-Jones Conjecture has only been formulated and investigated for discrete groups so far. We want to establish a version for totally disconnected groups and their Hecke algebras. The ultimate goal is to prove it for reductive p-adic groups. This would open a door to get new information about the representation theory of such groups. We also want to establish a version of the Baum-Connes Conjecture for Frechet algebras and prove it for case for which the Baum-Connes Conjecture is still open, for instance for CAT(0)-groups and lattices in almost connected Lie groups. All these activities are linked to the general problem to establish equivariant homotopy theory for proper actions of not necessarily finite or compact groups. This concerns both the general structure of the equivariant stable homotopy category and explicite computations based on finding good models for classifying spaces of families via geometry and the construction of equivariant Chern characters. The latest proof of the Farrell-Jones Conjecture for Waldhausen's A-theory for a large class of groups will be the basis of getting new information about the automorphism groups of closed aspherical manifolds. There are a variety of prominent open conjectures about L^2-invariants such as the ones due to Atiyah, Bergeron-Venkatesh, and Singer which we want to attack. The proposed methods are either algebraic or analytic. The project about low-dimensional manifolds and generalized notions of L^2-torsion has just been started and will lead to further interactions and results. In particular we would like to study L^2-torsion twisted with non necessarily unitary finite-dimensional representations and investigate the function it will give on the representation variety of a given group. Moreover, we will analyze further what these invariants such as the L^2-torsion or the L^2-polytope say about group automorphisms.

Selected Publications

[1] Wolfgang Lück, Holger Reich, John Rognes, Marco Varisco
Algebraic K-theory of group rings and the cyclotomic trace map
Adv. Math. , 304: : 930--1020
2017
DOI: 10.1016/j.aim.2016.09.004
[2] Holger Kammeyer, Wolfgang Lück, Henrik Rüping
The Farrell-Jones conjecture for arbitrary lattices in virtually connected Lie groups
Geom. Topol. , 20: (3): 1275--1287
2016
DOI: 10.2140/gt.2016.20.1275
[3] Arthur Bartels, Wolfgang Lück, Holger Reich, Henrik Rüping
K- and L-theory of group rings over {GL_n(\bf Z)}
Publ. Math. Inst. Hautes Études Sci. , 119: : 97--125
2014
DOI: 10.1007/s10240-013-0055-0
[4] A. Bartels, F. T. Farrell, W. Lück
The Farrell-Jones conjecture for cocompact lattices in virtually connected Lie groups
J. Amer. Math. Soc. , 27: (2): 339--388
2014
DOI: 10.1090/S0894-0347-2014-00782-7
[5] W. Lück
Approximating L2-invariants and homology growth
Geom. Funct. Anal. , 23: (2): 622--663
2013
DOI: 10.1007/s00039-013-0218-7
[6] Arthur Bartels, Wolfgang Lück
The Borel conjecture for hyperbolic and {CAT(0)}-groups
Ann. of Math. (2) , 175: (2): 631--689
2012
DOI: 10.4007/annals.2012.175.2.5
[7] Arthur Bartels, Wolfgang Lück, Holger Reich
The K-theoretic Farrell-Jones conjecture for hyperbolic groups
Invent. Math. , 172: (1): 29--70
2008
DOI: 10.1007/s00222-007-0093-7
[8] Wolfgang Lück
The relation between the Baum-Connes conjecture and the trace conjecture
Invent. Math. , 149: (1): 123--152
2002
DOI: 10.1007/s002220200215
[9] John Lott, Wolfgang Lück
L2-topological invariants of 3-manifolds
Invent. Math. , 120: (1): 15--60
1995
DOI: 10.1007/BF01241121
[10] W. Lück
Approximating L2-invariants by their finite-dimensional analogues
Geom. Funct. Anal. , 4: (4): 455--481
1994
DOI: 10.1007/BF01896404

Awards

2003

Max Planck Research Award

2008

Leibniz Prize

2010

Member of the German National Academy of Sciences Leopoldina

2012

Fellow of the American Mathematical Society

2012

Max Planck Fellow

2013

Member of the North Rhine-Westphalian Academy of Sciences, Humanities and the Arts

2015

ERC Advanced Investigator Grant for his project “K-theory, $L^2$-invariants, manifolds, groups and their interactions”

Offers

2001

University of Bonn

2003

ETH Zürich, Switzerland

2009

University of Göttingen

2010

University of Bonn

Selected Invited Lectures

2006

25th anniversary of Max Planck Institute for Mathematics, Bonn

2008

5th European Congress of Mathematics, invited lecture, Amsterdam, Netherlands

2010

International Congress of Mathematicians, Topology section, Hyderabad, India

2012

Homological growth and $L^2$-invariants, Hirzebruch Lecture, Münster

2013

Survey on $L^2$-invariants, 20th anniversary of ESI, Erwin Schrödinger Institute, Vienna, Austria

2014

Heat kernels and their applications in geometry, topology and group theory, Bethe Kolloquium of Institute for Physics, Bonn

Editorships

• Journal Mathematische Annalen, Springer (1997 - 2008)
• Proceedings “Tel Aviv Topology Conference: Rothenberg Festschrift”, Contemp. Mathematics 231 (1999)
• Proceedings of the School / Conference “High-dimensional manifold theory”, Trieste (May and June 2001)
• Topology, Elsevier (2002 - 2006)
• Commentarii Mathematici Helvetici, European Mathematical Society Publishing House (2003 - 2008)
• Mathematische Annalen (Managing Editor, 2004 - 2008)
• Geometry and Topology (2005 - 2008)
• Groups, Geometry and Dynamics, European Mathematical Society Publishing House (2006 - 2012)
• K-Theory (Interim Editor, August - December 2007)
• Münster Journal of Mathematics (2007 - 2012)
• Journal of Topology, LMS (2007 - 2012)
• Proceedings of Fourth Arolla Conference on Algebraic Topology (2012)

Research Projects and Activities

DFG Collaborative Research Center SFB 478 “Geometric structures in mathematics” in Münster
One of the two vice-speakers, 1997 - 2009

DFG Research Training Group GRK 627 “Analytic Topology and Metageometry” in Münster
Speaker, 2000 - 2009

DFG Collaborative Research Center SFB 878 “Groups, Geometry, and Actions” in Münster
Coordinator, July 1, 2010 - September 30, 2010

ERC Advanced Investigator Grant “K-theory, L^2-invariants, manifolds, groups and their interactions”

“Hausdorff Research Institute for Mathematics”
Director

DFG Cluster of Excellence “Hausdorff Center for Mathematics”
Principal Investigator

Contribution to Research Areas

Research Area A
We mainly focus on L^2-invariants. These are invariants obtained from the spectrum of the Laplace operators on the universal covering of a closed Riemannian manifold, and often have geometric or topological interpretations. Examples are L^2-Betti numbers and L^2-torsion. We will work on the Atiyah Conjecture, and on approximation problems for L^2-invariants for towers of coverings.
Research Area C
We will work on questions around the Baum-Connes Conjecture. In particular we will develop tools for and carry out computation of the topological K-theory of group C^*-algebras using this conjecture. These are not only interesting in their own right but have applications to classification of C^*-algebras, classifications of manifolds, and to the (unstable) Gromov-Lawson-Rosenberg Conjecture about closed Riemannian manifolds with positive scalar curvature.
Research Area F*
The Farrell-Jones Conjecture predicts the algebraic K- and L-theory of group rings. It implies other prominent conjectures due to Borel (topological rigidity of aspherical manifolds), Kaplansky (idempotents in group rings) and Novikov (homotopy invariance of higher signatures). We will extend the class of groups for which the conjecture is known and work on its analogue for Waldhausen's A-theory and for pseudo-isotopy.

Habilitations

Thomas Schick (2000), now Professor, University of Göttingen

Michael Joachim (2003), now Professor (Apl), University of Münster

Arthur Bartels (2005), now Professor, University of Münster

Holger Reich (2005), now Professor, FU Berlin

Tilman Bauer (2008), now Professor, KTH Royal Institute of Technology, Stockholm, Sweden

Roman Sauer (2009), now Professor, Karlsruhe Institute of Technology

Selected PhD students

Thomas Schick (1996): “Analysis on manifolds of bounded geometry, Hodge-deRham isomorphism and L²-index theorem”,
now Professor (W3), University of Göttingen

Holger Reich (1999): “Group von Neumann algebras and related algebras”,
now Professor (W3), FU Berlin

Roman Sauer (2003): “Invariance properties of L²-Betti numbers and Novikov-Shubin invariants under orbit equivalence and quasi-isometry”,
now Professor (W3), Karlsruhe Institute of Technology

Marco Varisco (2006): “Algebraic L-theory and triangular Witt groups”,
now Associate Professor, University at Albany, NY, USA

Clara Löh (2007): “L²-invariants, simplical volume and measure theory”,
now Professor (W2), University of Regensburg

Wolfgang Steimle (2010): “Obstructions to Stably Fibering Manifolds”,
now Professor (W2), University of Augsburg

Henrik Rüping (2011): “The Farrell-Jones conjecture for some general linear groups”,
now Postdoc, University of Bonn

Philipp Kühl (2014): “The hotel of algebraic surgery”

Markus Land (2016): “On the relation between K- and L-theory of complex C^*-algebras”,
now Postdoc, University of Regensburg

Supervised Theses

  • Master theses: 8, currently 2
  • Diplom theses: 30
  • PhD theses: 20, currently 7
Download Profile