Research Interests of Jürgen Berndt
Background.
My own research is in geometry, but not restricted to it, and I often
deal with problems and methods that are related to other areas.
Geometry is one of the traditional areas of mathematics. In
ancient civilizations geometry was used for solving practical
problems. This changed
drastically with the ancient Greeks, where geometry became the
center of mathematics. Its axiomatic foundation and the theory
based on propositions that were logically deducted from it
influenced enormously our way of thinking and
lead over the last 2000 years to many
fundamental developments and discoveries in mathematics and the
sciences. For instance, during the Renaissance,
geometry influenced strongly
the development of astronomy, geodesy, cartography,
mechanics, optics and arts. Geometry nowadays has great impact
on our lives through many applications, for example in
medicine (diagnosis of cancer, brain imaging),
building and construction industry (computer aided design),
and manufacturing (robotics).
This shows the potential for geometry to get involved in collaborative
projects with partners from industry.
On the theoretical side, there is a
fundamental interaction between
geometry and physics, for instance between
Euclidean geometry and Newtonian mechanics, or
Riemannian geometry and relativity theory, which represent
remarkable scientific achievements. Currently there is an
intense interaction between geometry and modern physical theories
(string theories, supergravity theories, M-theory). Because of all these
applications and
its particular feature of visualization, geometry is also an extremely
attractive area of mathematics for teaching at all levels in higher education.
Current research interests:
1. Cohomogeneity one actions
(with
Hiroshi Tamaru, Hiroshima University, Japan).
In connection with special
structures on manifolds there is currently great interest in
geometries of cohomogeneity one. This concerns the investigation and
classification of Lie group actions on manifolds, or more general
objects, for which the orbit space is one-dimensional. Such actions
are important since they allow reductions from partial to ordinary
differential equations. In joint work with Hiroshi Tamaru
(Hiroshima) I study the problem of classifying such structures on
noncompact symmetric spaces (the compact case is already solved).
Because of the noncompactness of the involved groups and spaces the
traditional methods do not work, and we developed new methods of Lie
algebraic nature. We are in the process of finalising the complete
classification which can be described in terms of the Langlands and
Chevalley decompositions of parabolic subalgebras of real semisimple
Lie algebras. The results obtained so far have appeared in
J. Differential Geom. 63 (2003), 1-40;
Tohoku Math. J. (2) 56 (2004), 163-177;
Trans. Amer. Math. Soc. 359 (2007), 3425-3438;
and in the preprint arXiv:1006.1980v1.
Our research has been funded so far by the Engineering and Physical
Sciences Research Council (United Kingdom),
Mathematical Research Institute Oberwolfach (Research in Pairs) and
the Ministry of Education, Culture, Sports, Science and Technology of
Japan.
2. Differential systems and exceptional geometries
(with
Martin Guest, Tokyo Metropolitan University, Japan).
Motivated by separate work on some exceptional geometries
and the geometry of integrable systems, we have
formulated some conjectures concerning differential systems associated to
exceptional geometries. In particular, we believe that each exceptional
compact simple Lie group can be characterized as the automorphism
group of a certain differential system.
Bryant discussed briefly certain characterizations
of the exceptional compact simple Lie groups G2 and F4,
referring to Cartan for the details. In the case of G2,
this uses a famous Pfaffian system with five variables.
In both cases, the differential system has a
geometrical interpretation: it describes the superhorizontality
condition for a certain twistor fibration.
Bryant asked whether there are similar results
for the three remaining exceptional compact simple
Lie groups E6, E7 and E8.
We believe that there is a canonical choice of twistor
fibration in each of these three cases (as well as for G2 and
F4) whose differential system characterizes the corresponding Lie group.
The support for our conjecture depends on two techniques
(which were not available when Bryant wrote his papers). First, the loop
group approach to twistor fibrations gives convenient
coordinates on the twistor space which facilitate the
computations.
Second, a branched fibration that is discussed in a paper by
Atiyah and myself naturally leads to the twistor fibration of F4,
and a generalization
suggests the candidates for the distinguished twistor fibrations.
Our research has been funded by the Institut des Hautes Etudes Scientifiques
at Bures-sur-Yvette in France.
3. Shape and symmetry
(with
José Carlos Díaz-Ramos, University of Santiago de Compostela, Spain).
The aim of this project is to describe the
geometry of homogeneous submanifolds of symmetric spaces of
noncompact type in terms of algebraic and geometric data, and to
classify them according to these data.
This research has been funded so far by grants BFM 2003-02949 from the Spanish government,
PD/2006/9 from the Irish Research Council for Science, Engineering and Technology,
and MEIF-CT-2006-038754 from the European Commission. Our first result is the classification of
real hypersurfaces with at most three distinct constant principal
curvatures in complex hyperbolic spaces, which is published in
J. London Math. Soc. 74 (2006), 778-798 and
Proc. Amer. Math. Soc. 135 (2007), 3349-3357. In Geom. Dedicata 138 (2009), 129-150
we investigate the geometry of homogeneous hypersurfaces in complex hyperbolic spaces. We then shifted our
interest towards hyperpolar and polar actions on Riemannian symmetric spaces of noncompact type.
In joint work with Hiroshi Tamaru we classified in
J. Differential Geom. 86 (2010), 191-235,
all hyperpolar homogeneous foliations on such spaces. In the preprint
arXiv:1107.0688v1
we classify all homogeneous polar foliations on complex hyperbolic spaces. We are currently
trying to understand better the situation when the polar actions have singular orbits.
4. Submanifolds in Grassmannians
(with
Young Jin Suh, Kyungpook National University, Taegu, South Korea).
Submanifold geometry in space forms, or more general, in rank one symmetric spaces,
is reasonably well understood. However, submanifold geometry in symmetric spaces of higher rank
is less developed. This is not surprising, as the geometry of these spaces is
considerably more complicated for higher rank.
The aim of our project is to get a better understanding of how the geometry of the symmetric space
can be used to develop a good theory of submanifolds. At present we concentrate on
the particular case when the symmetric space is the complex 2-plane Grassmannian or its noncompact dual.
These spaces are particularly interesting as they are equipped with both a Kähler
structure and a quaternionic Kähler structure. Some results have been
published in Monatsh. Math. 127 (1999), 1-14;
Monatsh. Math. 137 (2002), 87-98; and in the preprint
arXiv:0911.3081v1.
Some recent research:
1. Submanifolds and holonomy (with
Sergio Console, University of Turin, Italy, and
Carlos Olmos, National University of Cordoba, Argentina).
Our work has been motivated
by recent progress in submanifold geometry in space forms,
using new methods based on the holonomy of the normal connection.
Particular progress has been made in the framework of
homogeneous submanifolds, isoparametric submanifolds and their
generalizations. In our monograph
Submanifolds and Holonomy (Chapman & Hall/CRC Research Notes in Mathematics 434,
Chapman & Hall/CRC, Boca Raton, FL, 2003, x+336 pp., ISBN: 1-58488-371-5)
we present an introduction to
this topic and a thorough survey of all main results in this area.
The proofs presented here are, to some extent, new,
resulting in a more
unified treatment of this topic. At the end of the book, we
discuss generalizations of some problems to more general
manifolds, in particular symmetric spaces.
2. Projective planes, spheres and Severi varieties
(with
Sir Michael Atiyah, University of Edinburgh, United Kingdom).
We studied certain branched fibrations of manifolds. A classical example,
due to Pontryagin, is the branched double covering of the 4-dimensional sphere
by the complex projective plane. Recently Atiyah and Witten
derived an analogous result for the quaternionic projective plane in the
framework of their studies of M-theory dynamics on G2-manifolds. We put
these two results into a unifying framework and obtain some
new aspects of the classical story about real normed
division algebras and projective planes.
This paper has appeared in
Surveys in Differential Geometry VIII,
Papers in Honor of Calabi, Lawson, Siu and Uhlenbeck
(International Press, Somerville, MA, 2003).
3. Symmetric submanifolds (with
Jost-Hinrich Eschenburg, University of Augsburg, Germany,
Hiroo Naitoh, Yamaguchi University, Japan,
and Kazumi Tsukada, Ochanomizu University, Tokyo, Japan).
A classical geometrical concept is that of
a reflection.
A submanifold of Euclidean space is said
to be symmetric if the reflection in each normal space maps the submanifold
into itself. A surprising result by Ferus relates the symmetric
submanifolds to certain real flag manifolds. We investigate symmetric
submanifolds in the more general framework of symmetric spaces. Our main
result is the classification of symmetric submanifolds in
symmetric spaces of noncompact type. The results have appeared in
Mathematische Annalen 332 (2005), 721-737.
Some areas of my research in previous years:
1. Geometry and topology of homogeneous spaces.
For example relating index numbers
and Betti numbers of flag manifolds
in the framework of symplectic geometry and Morse theory.
2. Geometry of nilpotent and solvable Lie groups. In particular
of generalized Heisenberg groups and its standard solvable extensions.
3. Curvature. For example Rank Rigidity Conjecture: construction
of the first counterexample to the local version of this conjecture.
4. Submanifold geometries in symmetric spaces.
5. Geometric PDEs. Minimal surfaces; PDEs related to
Einstein-like geometries that I could connect to classical results about
the integrability of the Hamilton-Jacobi equation and the Schroedinger
equation of the geodesic flow.
Here is a list of my
scientific collaborators.
Here is a list of my
publications.
Here is a list of
AMS MathScinet reviews
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Updated 15 July 2011