Profile
Profile

Prof. Dr. (em.) Werner Müller

E-mail: mueller(at)math.uni-bonn.de
Phone: +49 228 73 2840
Homepage: http://www.math.uni-bonn.de/people/mueller/
Room: 1.035
Location: Mathematics Center
Institute: Mathematical Institute
Research Areas: Former Research Area D (Leader)
Research Area A
Research Area DE

Academic Career

1977

PhD, HU Berlin

1977 - 1986

Research Scholar, Academy of Sciences of GDR, Berlin

1987 - 1989

Professor, Academy of Sciences of GDR, Berlin

1989 - 1990

Member, Institute of Advanced Study, Princeton, NJ, USA

1990 - 1993

Member, Max Planck Institute for Mathematics, Bonn

1993 - 2016

Professor (C4), University of Bonn

Since 2016

Professor Emeritus

Research Profile

My main interest is in global analysis and the theory of automorphic forms. Global analysis is concerned with the study of geometric differential operators on manifolds. The investigation of solutions of partial differential equations of geometric origin is the source of important connections between geometry, topology and analysis. I am especially interested in harmonic analysis on locally symmetric spaces and the theory of automorphic forms. The Arthur-Selberg trace formula is one of the most important tools in the theory of automorphic forms.

In joint work with T. Finis and E. Lapid I have used the trace formula to study the asymptotic distribution of automorphic forms for GL(n). This includes the Weyl law and the limit multiplicity problem. A crucial input is the refined spectral side of the trace formula, which was established in joint work with T. Finis and E. Lapid. A very challenging problem is to extend these results to other classical groups. Among other things, this requires detailed knowledge of the analytic properties of the L-functions occurring on the spectral side of the trace formula. To this end one can use Arthur's work on the endoscopic classification of automorphic representations of symplectic and orthogonal groups to relate the L-functions to L-functions for GL(n).

Another key topic of my research in recent years has been the study of analytic torsion of compact locally symmetric manifolds. Analytic torsion is a sophisticated spectral invariant of a compact Riemannian manifold and a flat bundle over this manifold. A basic problem is the approximation of L^2-torsion by the analytic torsion of finite coverings in a tower. This is a special case of the kind of problems studied to a great extent by W. Lück. Bergeron and Venkatesh used this to study the torsion in the cohomology of co-compact arithmetic groups if the level is increased. J. Pfaff and I studied the same problem if the arithmetic group is fixed and the local system varies.

Many arithmetic groups are not co-compact and the long-term goal is to extend these results to the finite volume case. The main tool is again the trace formula. Its application leads to problems related to the refined spectral side and the study of weighted orbital integrals, which appear on the geometric side of the trace formula.

Research Projects and Activities

DFG Priority Programme SPP 1154 “Global Differential Geometry”
Project leader

DFG Collaborative Research Center SFB 611 “Singular phenomena and scaling in mathematical models”
Project leader

GIF Research Project “Analytic aspects of automorphic forms and the trace formula”
Project leader, 2004 - 2008

GIF Research Project “Spectral methods in automorphic forms”
Project leader, 2008 - 2011

Research Areas A and D, DFG Cluster of Excellence “Hausdorff Center for Mathematics”
Principal Investigator

Contribution to Research Areas

Research Area A
The focus of my research in this area is on the study of the spectrum of geometric differential operators on manifolds and the relation to geometry. Of particular interest are classes of non-compact Riemannian manifolds with special structures at infinity such as manifolds with singularities and locally symmetric spaces of finite volume. In the non-compact case it is important to determine the structure of the continuous spectrum. The main tool is scattering theory.
In [1], we developed scattering theory for Laplace operators on manifolds of bounded curvature. In [2], we have studied scattering theory for differential forms on manifolds with cylindrical ends.
In particular, we have given a cohomological interpretation of the Eisenbud-Wigner time delay operator. In [3], I have extended the Selberg trace formula to non-unitary representations of the fundamental group. This has applications to dynamical zeta functions.

Research Area DE
In joint work with T. Finis and E. Lapid [4], we settled the limit multiplicity problem for the groups GL(n) and SL(n). In [5] I studied with S. Marshall the growth of the torsion subgroup in the cohomology of a compact arithmetic hyperbolic 3-manifold. With J. Pfaff [6] we extended this partially to other compact arithmetic locally symmetric manifolds. With J. Matz we have introduced the analytic torsion for congruence quotients of SL(n,R)/SO(n) and studied the approximation of the L^2-torsion.

Selected Publications

[1] Werner Müller, Gorm Salomonsen
Scattering theory for the Laplacian on manifolds with bounded curvature
J. Funct. Anal. , 253: (1): 158--206
2007
[2] Werner Müller, Alexander Strohmaier
Scattering at low energies on manifolds with cylindrical ends and stable systoles
Geom. Funct. Anal. , 20: (3): 741--778
2010
[3] Werner Müller
A Selberg trace formula for non-unitary twists
Int. Math. Res. Not. IMRN (9): 2068--2109
2011
[4] Tobias Finis, Erez Lapid, Werner Müller
Limit multiplicities for principal congruence subgroups of {GL(n)} and {SL(n)}
J. Inst. Math. Jussieu
, 14: (3): 589--638
2015
[5] Simon Marshall, Werner Müller
On the torsion in the cohomology of arithmetic hyperbolic 3-manifolds
Duke Math. J. , 162: (5): 863--888
2013
[6] Werner Müller, Jonathan Pfaff
On the growth of torsion in the cohomology of arithmetic groups
Math. Ann. , 359: (1-2): 537--555
2014
[7] Werner Müller, Jonathan Pfaff
Analytic torsion and L2-torsion of compact locally symmetric manifolds
J. Differential Geom. , 95: (1): 71--119
2013
[8] Tobias Finis, Erez Lapid, Werner Müller
On the spectral side of Arthur's trace formula---absolute convergence
Ann. of Math. (2) , 174: (1): 173--195
2011
[9] Erez Lapid, Werner Müller
Spectral asymptotics for arithmetic quotients of {SL(n,\Bbb R)/SO(n)}
Duke Math. J. , 149: (1): 117--155
2009
[10] Werner Müller
Weyl's law for the cuspidal spectrum of {SL_n}
Ann. of Math. (2) , 165: (1): 275--333
2007
[11] Werner Müller
Analytic torsion and R-torsion for unimodular representations
J. Amer. Math. Soc. , 6: (3): 721--753
1993
[12] Werner Müller
The trace class conjecture in the theory of automorphic forms
Ann. of Math. (2) , 130: (3): 473--529
1989
[14] Werner Müller Jasmin Matz
Analytic torsion of arithmetic quotients of the symmetric space \mathrmSL(n,\mathbbR)/\mathrmSO(n)
arXiv: 1607:04676, to appear in GAFA
2016
[15] Werner Müller Jasmin Matz
Approximation of L2-analytic torsion for arithmetic quotients ofthe symmetric space \mathrmSL(n,\mathbbR)/\mathrmSO(n)
arXiv: 1709:07764
2017

Publication List

MathSciNet Publication List (external link)

ArXiv Preprint List (external link)

Editorships

• Mathematische Nachrichten (1990 - 2005)
• Inventiones Mathematicae (1991 - 2007)
• Compositio Mathematicae (1993 - 1998)
• Intern. Math. Research Notices (1993 - 1998)
• Analysis & PDE (since 2008)

Awards

1983

Euler-Medal, Academy of Sciences of GDR

1991

Max Planck Research Award (together with J. Cheeger, Courant Institute)

1993

Member of the Berlin-Brandenburg Academy of Sciences and Humanities

2003

Member of the German National Academy of Sciences Leopoldina

2015

Member of the Academia Europaea

Selected Invited Lectures

1983

ICM, invited speaker, Warsaw, Poland

1988

Taneguichi Symposium, Japan

1992

ECM, invited speaker, Paris, France

1999

Conference in honor of M. Atiyah, R. Bott, F. Hirzebruch, and I. M. Singer, Harvard, MA, USA

2004

Conference in honor of J. Arthur, Toronto, ON, Canada

2008

Clay senior scholar, Lectures at MSRI, Berkeley, CA, USA

2009

Distinguished Ordway Lecturer, University of Minnesota, Minneapolis, MN, USA

2013

Conference in honor of J.-M. Bismut, Paris, France

2016

Conference in honor of J. Schwermer, Max Planck Institute for Mathematics, Bonn

Habilitations

Kai Köhler (1999), now Professor (C3), University of Düsseldorf

Selected PhD students

Werner Hoffmann (1986): “Die Spurformel für Hecke-Operatoren über Gittern vom Rang”,
now Professor, University of Bielefeld

Gorm Salomonsen (1996): “Dirac operators and analysis on open manifolds”

Boris Vaillant (2001): “Index and Spectral Theory for Manifolds with Fibred Cusps”

Jörn Müller (2008): “Zur Kohomologie und Spektraltheorie des Hodge-Laplaceoperators von Mannigfaltigkeiten mit gefaserter Spitzenmetrik”,
now Research Assistant , HU Berlin

Clara Aldana (2009): “Inverse Spectral Theory And Relative Determinants Of Elliptic Operators On Surfaces With Cusps”,
now Postdoctoral Researcher, Mathematics Research Unit, University of Luxembourg, Luxembourg

Jonathan Pfaff (2012): “Selberg and Ruelle zeta functions and the relative analytic torsion
on complete odd-dimensional hyperbolic manifolds of finite volume”

Ksenia Fedosova (2016): “Selber zeta functions and relative analytic torsion for hyperbolic
odd-dimensional orbifolds”,
now Research Assistant, University of Freiburg

Supervised Theses

  • Master theses: 10
  • Diplom theses: 12
  • PhD theses: 14
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