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| 1979 | Dr. rer. nat., University of Bonn | | 1979 - 1985 | Assistant Professor, University of Bonn | | 1984 - 1986 | Associate Professor, University of Maryland, College Park, MD, USA | | 1986 - 1987 | Professor (C3), University of Bonn | | 1987 - 1989 | Ordinarius, University of Zürich, Switzerland | | 1989 - 2016 | Professor (C4), University of Bonn | | Since 2007 | Scientific Member and Director, Max Planck Institute for Mathematics, Bonn |
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The present foci of my work are spectral theory on one hand and the generalized Blaschke conjecture on the other. In spectral theory, my recent work deals with small eigenvalues of the Laplacian on complete Riemannian surfaces and an associated new invariant, the analytic systole. I also study the bottom of the spectrum of the Laplacian under Riemannian coverings, extending and generalizing previous work of Brooks. All this is joint work and project with three young mathematicians, Sugata Mondal, Henrik Matthiesen, and Panagiotis Polymerakis.
In the near future, I intend to concentrate more on the generalized Blasche conjecture. Say that a closed Riemannian manifold is a Blaschke manifold if its injectivity radius and diameter coincide. The generalized Blaschke conjecture then asserts that a Blaschke manifold is a compact rank one symmetric space (cross). Blaschke manifolds do have the cohomology of a cross and fall into corresponding types. Whereas the spherical type was solved by Berger, Kazdan, Weinstein, and Yang, the conjecture is open for the other types. This is a joint project with Karsten Grove. So far our main result is the uncovering of a serious mistake in one of the classical papers on the subject, pertaining to the diffeomorphism class of Blaschke manifolds of the complex projective type.
See also http://people.mpim-bonn.mpg.de/hwbllmnn/archiv/blaschke1603.pdf
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The article [1] contains an introduction to the boundary value theory of Dirac type operators as Christian Baer and I developed it earlier, but contains also some related new work. In the paper [2], I discuss an  -invariant occuring in index theorems for Dirac operators on finite volume manifolds of pinched negative sectional curvature.
In the last years, I have been working on the spectrum of the Laplace operator on functions on Riemannian manifolds. My work concerns two problems, 1) small eigenvalues of surfaces and 2) the behaviour of the bottom of the spectrum of the Laplacian under coverings. My collaborators on the first problem were Henrik Matthiesen and Sugata Mondal. Generalizing work of Otal and Rosas, we showed that a complete Riemannian surface  of finite type with negative Euler characteristic  has at most  eigenvalues below a certain treshhold  , a new invariant introduced by us and baptized the analytic systole of , [3,4].
We also discussed relations of with other geometric invariants of [5]. My collaborators on the second problem were Henrik Matthiesen and Panagiotis Polymerakis. Extending work of Brooks and others, we showed that the bottom of the spectrum under a Riemannian covering  , where  is connected (but not necessarily complete), does not change if  is coamenable in  [6]. |
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[ 1] Christian Bär, Werner Ballmann
Guide to elliptic boundary value problems for Dirac-type operators
Arbeitstagung Bonn 2013
of Progr. Math. : 43--80
Publisher: Birkhäuser/Springer, Cham
2016[ 4] Werner Ballmann, Henrik Matthiesen, Sugata Mondal
Small eigenvalues of surfaces of finite type
Compos. Math. , 153: (8): 1747--1768
2017
DOI: 10.1112/S0010437X17007291[ 5] Werner Ballmann, Henrik Matthiesen, Sugata Mondal
On the analytic systole of Riemannian surfaces of finite type
Geom. Funct. Anal. , 27: (5): 1070--1105
2017
DOI: 10.1007/s00039-017-0422-y[ 6] Werner Ballmann, Henrik Matthiesen, Panagiotis Polymerakis
On the bottom of spectra under coverings
eprint arXiv, online first , 1701.02130:
2017[ 7] Werner Ballmann, Jochen Brüning, Gilles Carron
Index theorems on manifolds with straight ends
Compos. Math. , 148: (6): 1897--1968
2012
DOI: 10.1112/S0010437X12000401 |
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• International Journal of Mathematics (1989 - 1995)
• Geometriae Dedicata (1989 - 1999)
• Inventiones mathematicae (1996 - 2007)
• Mathematische Zeitschrift (1997 - 2003)
• Journal für die reine und angewandte Mathematik (Advisory Board, since 1990)
• Mathematical Proceedings, Cambridge Philosophical Society (since 2006)
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Christian Bär (1993), now Professor, University of Potsdam
Bernhard Leeb (1997), now Professor, LMU Munich
Vicente Cortés (1999), now Professor, University of Hamburg
Dorothee Schueth (2000), now Professor, HU Berlin
Chand Devchand (2003), now Privatdozent, University of Potsdam
Alina Vdovina (2005), now Lecturer, Newcastle University, England, UK
Gregor Weingart (2005), now Researcher, National Autonomous University of Mexico, Mexico
Alexander Lytchak (2007), now Professor, University of Cologne
Thomas Vogel (2014), now Professor, LMU München
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Paul Schmutz (1989): “Zur Anzahl kleiner Eigenwerte auf Riemannschen Flächen”
Christian Bär (1990): “Das Spektrum von Dirac-Operatoren”,
now Professor, University of Potsdam
Dorothee Schüth (1993): “Stetige isospektrale Deformationen”,
now Professor, HU Berlin
Vicente Cortés (1994): “Alekseevskiis quaternionische Kählermannigfaltigkeiten”,
now Professor, University of Hamburg
Gregor Weingart (1998): “Moduli Spaces of Minimal Isometric Immersions”,
now Researcher, National Autonomous University of Mexico, Mexico
Alexander Lytchak (2001): “Allgemeine Theorie der Submetrien und verwandte mathematische Probleme”,
now Professor, University of Cologne
Anna Wienhard (2004): “Bounded Cohomology and Geometry”,
now Professor, University of Heidelberg
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- Master theses: 10
- Diplom theses: 45
- PhD theses: 14, currently 3
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